Braided Commutative Geometry and Drinfel'd Twist Deformations
BBraided Commutative Geometry and Drinfel’dTwist Deformations
Universit`a degli Studi di Napoli Federico IIDottorato di ricerca in Scienze Matematiche e InformaticheXXXII CicloTesi di Dottorato di Ricerca
Thomas WeberAdvisors: Referees:
Prof. Francesco D’Andrea Prof. Paolo AschieriProf. Gaetano Fiore Prof. Andrzej Borowiec
Dicembre 2019 a r X i v : . [ m a t h . QA ] F e b bstract This thesis revolves around the notion of twist star product, which is a certain typeof deformation quantization induced by quantizations of a symmetry of the system.On one hand we discuss obstructions of twist star products, while on the otherhand we provide a recipe to obtain new examples as projections from known ones.Furthermore, we construct a noncommutative Cartan calculus on braided commuta-tive algebras, generalizing the calculus on twist star product algebras. The startingpoint is the observation that Drinfel’d twists not only deform the algebraic struc-ture of quasi-triangular Hopf algebras and their representations but also induce starproducts on Poisson manifolds with symmetry. We further investigate the corre-spondence of Drinfel’d twists and classical r -matrices as well as twist deformationof Morita equivalence bimodules. It turns out that connected compact symplecticmanifolds are homogeneous spaces if they admit a twist star product and that wecan assume the corresponding classical r -matrix to be non-degenerate. Further-more, invariant line bundles with non-trivial Chern class and twists star productscannot coexist if they are based on the same symmetry. In particular, the sym-plectic 2-sphere and the connected orientable symplectic Riemann surfaces of genus g > D’Andrea, F. and Weber, T.:
Twist star products and Morita equivalence .C. R. Acad. Sci. Paris, 355(11):1178-1184, 2017.ii.)
Bieliavsky, P. and Esposito, C. and Waldmann, S. and Weber, T.:
Obstructions for twist star products . Lett. Math. Phys., 108(5):1341–1350,2018.iii.)
Weber, T.:
Braided Cartan Calculi and Submanifold Algebras . PreprintarXiv:1907.13609, 2019.iv.)
Fiore, G. and Weber, T.:
In preparation, 2019.3¨ur meine Großeltern Hedi und Toni4 eclaration
I declare that this thesis was composed by myself, that the work contained hereinis my own except where explicitly stated otherwise in the text, and that this workhas not been submitted for any other degree or professional qualification.Thomas Weber5 cknowledgements
First of all I have to thank my advisors Francesco and Gaetano for their patienceand constant support. Their guidance had a great influence on the outcome of thethesis. In the same way I have to mention Chiara Esposito, Luca Vitagliano andStefan Waldmann, who acted as if they were my co-supervisors and supported methroughout many years. Thank you! I am also grateful to the PhD committee forgiving me the opportunity to spend my PhD in such a beautiful and interesting city.Studying alongside Jonas Schnitzer, a fellow student of mine from W¨urzburg, madeit a lot easier for me to gain ground in Naples. In his own words, it is not clear ifthis instance was ”blessing or curse”. The same applies to my flatmates Valentino,Luca (alias Lupo di Mare) and Nicola, who introduced me to the city, the SouthernItalian cuisine and lifestyle. I want to thank all my colleagues for many interestingconversations and discussions at conferences, schools and workshops. I am happythat I was able to visit so many fascinating places. Last but not least I want tothank my family and friends for keeping in contact and caring about ”the emigrant”.It was amazing that so many of you were visiting me (some even more than once orfor a whole semester) despite the distance. You make me feel home wherever I go.6 ontents
A Monoidal Categories 136B Braided Graßmann and Gerstenhaber Algebras 141 hapter 1Introduction In this thesis we employ techniques from quantum group theory to give obstructionsfor twist deformation quantization on several classes of symplectic manifolds, whilenew examples of twist star products are obtained via submanifold algebra projection.Motivated from this quantization procedure, we further construct a noncommutativeCartan calculus on any braided commutative algebra, as well as an equivariant Levi-Civita covariant derivative. This generalizes and unifies the classical Cartan calculusof differential geometry and the calculus on twist star product algebras (see e.g. [7]).This first chapter will serve as a motivation, while we also lay down the agenda ofthe thesis and notation.
Hamiltonian Mechanics
Hamilton’s equations of motion provide a good description of macroscopic objects.We sketch how this axiomatic framework can be implemented in terms of Poissongeometry, following [14] Chap. 0 and [109] Chap. 1. The dynamics of a
Hamilto-nian system are controlled by a Hamiltonian function H ( q, p ) ∈ R via Hamilton’sequations of motion (cid:18) ˙ q ( t )˙ p ( t ) (cid:19) = (cid:18) n − n (cid:19) (cid:18) ∂H∂q∂H∂p (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) = X H ( q ( t ) , p ( t )) , (1.1)where ( q, p ) ∈ R n are phase-space coordinates. A solution x ( t ) = ( q ( t ) , p ( t )) of(1.1) is given by the flow of the Hamiltonian vector field X H : R n → R n , withrespect to some initial conditions x (0) = ( q , p ) ∈ R n . For example, we mayconsider a particle with mass m influenced by a conservative force F : R → R .The movement of the particle is represented by a smooth curve q : R → R , where q ( t ) ∈ R is the position of the particle at time t ∈ R . The velocity ˙ q and acceleration¨ q of the particle are defined by the first and second derivative of q , respectively. Itsmomentum is given by the product p = m ˙ q . Since F is conservative, there isa potential V : R → R such that F ( q ( t )) = − ( ∇ V )( q ( t )), where ∇ denotes thegradient. Then, Newton’s second law F ( q ( t )) = m ¨ q ( t ) (1.2)determines the dynamics of the particle. Employing the Hamiltonian function H ( q, p ) = p m + V ( q ), the second order partial differential equation (1.2) is equivalent8ntroduction 9to the system (1.1) of differential equations of order one. In other words, in this caseHamilton’s equations of motion constitute Newtonian mechanics. There is a notionof symplectic manifolds and more general of Poisson manifolds (see Section 3.1)that allow the formulation of a Hamiltonian formalism. The crucial ingredient isthe Poisson bracket {· , ·} , which is a Lie bracket on the algebra of smooth functions,satisfying a Leibniz rule in each entry. For every function f on a Poisson mani-fold ( M, {· , ·} ) with Hamiltonian function H and corresponding Hamiltonian flowΦ ∗ t : C ∞ ( M ) → C ∞ ( M ), the time evolution f ( t ) = Φ ∗ t ( f ) of f is determined bydd t Φ ∗ t ( f ) = { Φ ∗ t ( f ) , H } . (1.3)We regain (1.1) if M = R n and f = ( q, p ). In the next section we examine thepassage from classical mechanics to quantum mechanics, in the lines of [109] Chap. 5. Deformation Quantization
At the beginning of the 20th century quantum mechanics revolutionized the concep-tional and philosophical understanding of physics. It recovers classical mechanicsvia a limit procedure. In [40] Dirac discussed the canonical commutation relations [ q j , p k ] = i (cid:126) δ jk , (1.4)stating that position and momentum do not commute as quantum observables andtheir noncommutativity depends on the Planck constant (cid:126) . In this picture, statesare square integrable wave functions in L ( R , d q ) and observables are (unbounded)self-adjoint operators of wave functions. Interpreting position and momentum of aparticle as operators ˆ q j and ˆ p k , which act on a wave function ψ as (ˆ q j ψ )( q ) = q j ψ ( q )and (ˆ p k ψ )( q ) = − i (cid:126) ∂ψ∂q k ( q ), we obtain,[ˆ q j , ˆ p k ] ψ ( q ) = − i (cid:126) q j ∂ψ∂q k ( q ) + i (cid:126) (cid:18) δ jk ψ ( q ) + q j ∂ψ∂q k ( q ) (cid:19) = i (cid:126) δ jk ψ ( q ) , (1.5)which is a representation of (1.4). In the classical limit , where (cid:126) is considered to bearbitrarily small, the operators ˆ q j and ˆ p k become commutative and we recover thesituation known from classical mechanics. This procedure of canonical quantization can be formalized in the following way: a quantization map Q : C ∞ ( M ) → End C ( H ) (1.6)assigns to any smooth complex-valued function f on a Poisson manifold ( M, {· , ·} )a C -linear operator Q ( f ) = ˆ f : H → H on a complex Hilbert space H . The map(1.6) should be C -linear, satisfy Q (1) = id H and Q ( { f, g } ) = 1i (cid:126) [ Q ( f ) , Q ( g )] (1.7)on generators f, g ∈ C ∞ ( M ). However, it was pointed out in [40], and later by[62, 107], that (1.7) does not hold on the whole algebra of functions but instead onlyup to higher orders of (cid:126) , i.e. Q ( { f, g } ) = 1i (cid:126) [ Q ( f ) , Q ( g )] + O ( (cid:126) ) . (1.8)0 Chapter 1A particular quantization in accordance with (1.8) is given by the notion of defor-mation quantization: in [11] the authors suggested to deform the algebra structureof C ∞ ( M ) rather than the observables itself. Namely, one considers (cid:126) as a formalparameter and calls an associative product f (cid:63) g = ∞ (cid:88) n =0 C n ( f, g ) ∈ C ∞ ( M )[[ (cid:126) ]] (1.9)on the formal power series C ∞ ( M )[[ (cid:126) ]] a quantization or star product on ( M, {· , ·} )if C n are bidifferential operators on M and (cid:63) deforms the pointwise product offunctions, i.e. f (cid:63) g = f g + O ( (cid:126) ) ∈ C ∞ ( M )[[ (cid:126) ]], such that 1 (cid:63) f = f = f (cid:63) { f, g } = 1i (cid:126) [ f, g ] (cid:63) + O ( (cid:126) ) (1.10)hold for all f, g ∈ C ∞ ( M ), where [ f, g ] (cid:63) = f (cid:63)g − g(cid:63)f . In other words, this amounts toconsider a noncommutative algebra ( C ∞ ( M )[[ (cid:126) ]] , (cid:63) ) rather than endomorphisms ona Hilbert space and to set Q = id. This agenda is know as deformation quantization and has proven its profundity by many publications and a great interest in thecommunity of mathematical physics. We discuss this in more detail in Section 3.1. Drinfel’d Twists and Quantization
The term quantum group was proposed by Drinfel’d to refer to Hopf algebras inthe context of quantum integrable systems. In [42] he related solutions r ∈ Λ g ofthe classical Yang-Baxter equation (cid:74) r, r (cid:75) = 0 (see Section 3.2) on a Lie algebra g tonormalized 2-cocycles F = 1 ⊗ (cid:126) r + O ( (cid:126) ) ∈ ( U g ⊗ U g )[[ (cid:126) ]] (1.11)on formal power series of the universal enveloping algebra U g . Note that U g isa Hopf algebra with coproduct, counit and antipode defined on primitive elements x ∈ g by ∆( x ) = x ⊗ ⊗ x , (cid:15) ( x ) = 0 and S ( x ) = − x (see Section 2.1). Such aclassical r -matrix is equivalent to a G -invariant Poisson bivector π r on the Lie group G corresponding to g , while F corresponds to a G -invariant star product (cid:63) on G quantizing π r , where G -invariance means that the pullback of the left multiplication (cid:96) g : G (cid:51) h (cid:55)→ g · h ∈ G satisfies (cid:96) ∗ g π r = π r and (cid:96) ∗ g ( f (cid:63) f ) = ( (cid:96) ∗ g f ) (cid:63) ( (cid:96) ∗ g f ) for all f , f ∈ C ∞ ( G ) and g ∈ G . Accordingly, the 2 -cocycle condition (∆ ⊗ id)( F ) · ( F ⊗
1) = (id ⊗ ∆)( F ) · (1 ⊗ F ) (1.12)of F reflects associativity of (cid:63) , while the normalization property ( (cid:15) ⊗ id)( F ) = 1 = (id ⊗ (cid:15) )( F ) (1.13)is synonymous to the star product being unital with respect to the unit functionon G . Furthermore, F equals 1 ⊗ (cid:126) if and only if (cid:63) deforms thepointwise product of functions. In a next step we want to induce star products ona Poisson manifold ( M, {· , ·} ) via a Drinfel’d twist F on U g . For this we only needntroduction 11a Lie algebra action φ : g → Γ ∞ ( T M ) by derivations. Such a Lie algebra actionextends to a U g -module algebra action (cid:66) : U g ⊗ C ∞ ( M ) → C ∞ ( M ) and f (cid:63) F g = µ ( F − (cid:66) ( f ⊗ g )) , (1.14)is a star product on M , where f, g ∈ C ∞ ( M ) and µ : C ∞ ( M ) ⊗ → C ∞ ( M ) denotesthe pointwise multiplication. A star product which can be expressed as (1.14) is saidto be a twist star product . The corresponding Poisson bracket is induced by the r -matrix of F , namely, { f, g } = µ ( r (cid:66) ( f ⊗ g )) (1.15)for all f, g ∈ C ∞ ( M ). Chapter 3 is devoted to Drinfel’d twists, r -matrices and twiststar products and to obstructions of the latter. The main theorems provide a classof symplectic manifolds that do not admit a twist star product deformation. Theorem ([20, 38]) . There are no twist star products oni.) the symplectic connected orientable Riemann surfaces of genus g > ;ii.) the symplectic -sphere S ;iii.) the symplectic projective spaces CP n for a Drinfel’d twist based on U gl n +1 ( C ) ; Note that in the third class of examples only twists on universal enveloping alge-bras of matrix Lie algebras are obstructed, while the first two classes prohibit twistdeformation based on any universal enveloping algebra. In fact, we are providingmore general results which imply the above examples. For i.) and ii.) we prove thatconnected compact symplectic manifolds endowed with a twist star product are infact homogeneous spaces and that the classical r -matrix corresponding to the twist isnon-degenerate (see Section 3.2). The third obstruction utilizes twist deformationsof Morita equivalence bimodules, which are studied in Section 3.4. In a nutshell, weprove that a symplectic manifold cannot inherit both a complex line bundle whichis invariant under a Lie group action and has non-trivial Chern class and a twiststar product with Drinfel’d twist based on the universal enveloping algebra of thecorresponding Lie algebra. Braided Symmetries and Cartan Calculi
Motivated by the example of universal enveloping algebras, one defines Drinfel’dtwists on an arbitrary Hopf algebra H to be invertible elements F ∈ H ⊗ H suchthat (1.12) and (1.13) are satisfied. They lead to a deformed Hopf algebra H F withstructure ∆ F ( · ) = F ∆( · ) F − , S F ( · ) = βS ( · ) β − , (1.16)where β = µ ((id ⊗ S )( F )) ∈ H and µ : H ⊗ → H is the multiplication. Furthermore,any left H -module algebra ( A , · ), i.e. any associative unital algebra together witha Hopf algebra action which respects the algebra structure, can be deformed into aleft H F -module algebra A F = ( A , · F ), where a · F b = µ ( F − (cid:66) ( a ⊗ b )) (1.17)2 Chapter 1for all a, b ∈ A and µ denotes the undeformed multiplication on A . More general, an H -equivariant A -bimodule M (see Section 2.4) is deformed into an H F -equivariant A F -bimodule M F . The category H A M A of equivariant bimodules is particularlyinteresting in the case of triangular Hopf algebras . The latter are Hopf algebras H equipped with a universal R -matrix R ∈ H ⊗ H , which is an invertible elementcontrolling the noncocommutativity of H , namely R − = R and ∆ ( · ) = R ∆( · ) R − , (1.18)such that the hexagon relations (see Section 2.3) are satisfied. In (1.18), R denotesthe tensor flip of R and ∆ the flipped coproduct of H . The universal R -matrixencodes a braiding on the categorical level. Any cocommutative Hopf algebra istriangular with universal R -matrix 1 ⊗
1. Let ( H, R ) be triangular, A braidedcommutative and M braided symmetric in addition, i.e. b · a = ( R − (cid:66) a ) · ( R − (cid:66) b ) , m · a = ( R − (cid:66) a ) · ( R − (cid:66) m ) (1.19)for all a, b ∈ A and m ∈ M in leg notation. Then, the twist deformed Hopf algebra( H F , R F ) is triangular, A F is braided commutative and the twisted module M F braided symmetric with respect to the twisted universal R -matrix R F = F RF − . Example.
The smooth functions A = ( C ∞ ( M ) , · ) on a Poisson manifold ( M, {· , ·} ) with the pointwise product are commutative, in other words braided commutativewith respect to a triangular Hopf algebra H = ( U g , ⊗ . Then also twist starproducts (1.14) on a Poisson manifold M determine a braided commutative algebra A F = ( C ∞ ( M )[[ (cid:126) ]] , (cid:63) F ) with symmetry H F = ( U g F , F F − ) . In categorical language, the Drinfel’d twist defines a braided monoidal functorDrin F : H A M RA → H F A F M R F A F (1.20)between the representation theories of H and H F . Such a functor is a braidedmonoidal equivalence of categories, which implies that the two algebras in the ex-ample are equivalent on categorical level (where one extends the pointwise prod-uct (cid:126) -bilinearly). Braided multivector fields X •R ( A ) and braided differential formsΩ •R ( A ) are canonical examples of objects in H A M RA (c.f. Chapter 4) and we con-struct four H -equivariant maps (i.e. maps commuting with the Hopf algebra action) L R , i R , d , (cid:74) · , · (cid:75) R in analogy to the Lie derivative, insertion of multivector fields, deRham differential and Schouten-Nijenhuis bracket, for any braided commutativealgebra A . Their relations are clarified in the following theorem, providing a non-commutative Cartan calculus. Theorem ([111] Braided Cartan Calculus) . Let H be a triangular Hopf algebrawith universal R -matrix R . For every braided commutative left H -module algebra A the graded maps L R X : Ω •R ( A ) → Ω •− ( k − R ( A ) , i R X : Ω •R ( A ) → Ω •− k R ( A ) , where X ∈ X k R ( A ) and d : Ω •R ( A ) → Ω • +1 R ( A ) , satisfy [ L R X , L R Y ] R = L R (cid:74) X,Y (cid:75) R , [ L R X , i R Y ] R =i R (cid:74) X,Y (cid:75) R , [ L R X , d] R =0 , [i R X , i R Y ] R =0 , [i R X , d] R = L R X , [d , d] R =0 , (1.21) for all X, Y ∈ X •R ( A ) , where [ · , · ] R denotes the graded braided commutator. Thetwist deformation of this braided Cartan calculus (induced by the Drinfel’d functor Drin F ) is isomorphic to the braided Cartan calculus on A F with respect to R F . ntroduction 13Applying the theorem to the two algebras from our examples we recover theclassical Cartan calculus of differential geometry and the twisted Cartan calculus (c.f. [7], see also Section 4.6). This explains the braided symmetries appearing inthe latter, e.g. that vector fields X ∈ X ( M ) act rather as braided derivations L F X ( f g ) = ( L X f ) g + ( R − (cid:66) f ) L R − (cid:66) X g (1.22)on functions f, g ∈ C ∞ ( M ), than as derivations. We show the utility of the braidedCartan calculus and its similarity to the classical Cartan calculus by further dis-cussing equivariant covariant derivatives and submanifold algebras. An equivariantcovariant derivative on A is an H -equivariant map ∇ R : X R ( A ) ⊗ X R ( A ) → X R ( A ),which is left A -linear in the first argument and satisfies a braided Leibniz rule ∇ R X ( a · Y ) = ( L R X a ) · Y + ( R − (cid:66) a ) · ( ∇ R − (cid:66) X Y ) (1.23)in the second argument, where X, Y ∈ X R ( A ) and a ∈ A . After defining their cur-vature and torsion we prove that these objects behave similarly to their counterpartsfrom differential geometry. Theorem ([111]) . Any equivariant covariant derivative on A extends to an equiv-ariant covariant derivative on X •R ( A ) and Ω •R ( A ) . For every non-degenerate equiv-ariant metric g there exists a unique torsion-free equivariant covariant derivative ∇ LC on A , such that ∇ LC g = 0 . We call ∇ LC the equivariant Levi-Civita covariant derivative corresponding to g . If the twist deformation g F of g is non-degenerate, it follows that the twistdeformation of ∇ LC is the equivariant Levi-Civita covariant derivative correspondingto g F . Phrasing the notion of submanifold in algebraic terms, we prove that thebraided Cartan calculus on a submanifold algebra coincides with the projectionof the calculus on the ambient algebra. Under some hypotheses we are able toproject equivariant covariant derivatives and metrics to the submanifold algebra. Animportant observation is that these projections commute with twist deformation. Organization of the Thesis
In Chapter 2 we recall some notions concerning Hopf algebras. The strategy is todevelop the algebraic data parallel to the categorical data on representations. Aquasi-triangular Hopf algebra corresponds to the rigid braided monoidal category ofits finitely generated projective representations. Accordingly, Drinfel’d twists canbe understood as algebraic deformation tool and braided monoidal functor in thecategory of equivariant braided symmetric bimodules. We further give examplesof Drinfel’d twists and include ∗ -involutions in the picture. The case of Drinfel’dtwist deformation of Poisson manifolds is studied in Chapter 3. We define starproducts as formal deformations and investigate consequences if they are inducedby Drinfel’d twists. In particular, the relation of twists on formal power seriesof universal enveloping algebras and classical r -matrices is pointed out. Anotherimportant result is that the deformation theory of a commutative algebra is a Moritainvariant, i.e. Morita equivalent algebras share the same deformation theories. Inthe case of twist star products this sets severe conditions on equivariant line bundles4 Chapter 1Appendix AChapter 2Chapter 3Appendix BChapter 4Chapter 5 Section 5.4Figure 1.1: How to read the thesisover the manifold. Both approaches lead to obstructions of twist star products onseveral classes of symplectic manifolds. Chapter 4 covers some topics in braidedgeometry. We construct the braided Cartan calculus on any braided commutativealgebra and prove that it respects gauge equivalence classes of the Drinfel’d functor.Furthermore, equivariant covariant derivatives are defined on equivariant braidedsymmetric bimodules. In complete analogy to differential geometry we study theirproperties and prove e.g. the existence and uniqueness of an equivariant Levi-Civitacovariant derivative for a fixed non-degenerate equivariant metric. The Drinfel’dfunctor intertwines all constructions. Finally, in Chapter 5, we show that the braidedCartan calculus on a submanifold algebra is in fact the projection of the braidedCartan calculus of the ambient space. Employing certain assumptions we are ableto project equivariant metrics and covariant derivatives. As a central observationwe point out that the projection and twist deformation commute. In addition, anexplicit example of twist deformation quantization on a quadric surface is given.There are two appendices, Appendix A briefly covering the material on categorytheory which is necessary to understand the thesis and Appendix B providing someadditional material on braided exterior algebras and braided Gerstenhaber algebras.Depending on the personal background and intentions we recommend to readthe thesis in the following ways (see also Figure 1.1). In Appendix A the basiccategorical language, which is used throughout the thesis, is provided. If the readeris familiar with these concepts the appendix can be omitted. Chapter 2 containsthe algebraic concepts of Hopf algebra and Drinfel’d twist which are central to thiswork. The advanced reader might also skip this chapter. Section 2.6 is mainlyrelevant for the twist deformation of quadric surfaces in Section 5.4, where thelatter can be understood independently of Chapter 3-Section 5.3. Furthermore, thebraided Cartan calculus (Chapter 4) and its compatibility with submanifold algebras(Chapter 5) can be considered independently of Chapter 3, which treats obstructionsof twist star products and deformation quantization. In other words, if the readeris not interested in deformation quantization there is a shortcut from Chapter 2 toChapter 4, where one might consider Appendix B before, to learn about braidedGraßmann algebras and braided Gerstenhaber algebras.ntroduction 15 Notation
Throughout the thesis k denotes a commutative ring with unit 1. If the situationrequires to work over a field we write K instead. A k -module is an Abelian group M together with a distributive left k -action. Typically we write λ · m or simply λm for the action of an element λ ∈ k on an element m ∈ M of a k -module. Amap between the Abelian groups of two k -modules is said to be k -linear or k -modulehomomorphism if it intertwines the k -actions. If such a map is invertible in additionwe call it k -module isomorphism . The tensor product of k -modules is denoted by ⊗ .It is a monoidal structure on the category k M of k -modules. The flip isomorphism of two k -modules M and N is defined by τ M , N : M ⊗ N (cid:51) ( m ⊗ n ) (cid:55)→ ( n ⊗ m ) ∈ N ⊗ M . (1.24)Furthermore we write M ⊗ k = M⊗ . . . ⊗M for the tensor product of k > k -module M . Let M be a k -module. Then any element F ∈ M ⊗ can be expressedas a finite sum (cid:80) ni =1 F i ⊗ F i of factorizing elements F i ⊗ F i . Omitting this sumand the summation indices we end up with F = F ⊗ F , which is known as legnotation . Dealing with several copies of F we write F = F (cid:48) ⊗ F (cid:48) , etc. to distinguishthe summations. If M is an algebra with unit 1 we further write F = τ M , M ( F ), F = F ⊗ F = (id M ⊗ τ M , M )( F ⊗
1) and similarly for other permutations ofthe legs of F . If not stated otherwise every k -algebra A is assumed to be associativeand unital. This means there are k -linear maps µ : A ⊗ A → A and η : k → A ,called product and unit , respectively, such that µ ◦ ( µ ⊗ id) = µ ◦ (id ⊗ µ ) : A ⊗ → A (1.25)and µ ◦ ( η ⊗ id) = id = µ ◦ (id ⊗ η ) : A → A (1.26)hold. The first property is said to be the associativity of µ . In equation (1.26) weemployed the k -module isomorphisms k ⊗ A ∼ = A ∼ = A ⊗ k . The product of twoelements a and b of an algebra ( A , µ, η ) is sometimes denoted by a · b or ab if thereference to µ is not essential. Since η is determined by its value at the unit of k we often write 1 = η (1) ∈ A , calling this element the unit of A as well. The algebrais said to be commutative if ab = ba for all a, b ∈ A . An algebra homomorphism is a k -linear map Φ : A → B between algebras such that Φ( ab ) = Φ( a )Φ( b ) for all a, b ∈ A and Φ(1) = 1. If Φ is invertible in addition it is said to be an algebraisomorphism . The category of k -algebras is denoted by k A . It is monoidal withrespect to ⊗ , since k is an algebra and the tensor product A ⊗ B can be structuredas a k -algebra with unit 1 ⊗ a ⊗ b ) · ( a ⊗ b ) = ( a a ) ⊗ ( b b ) , (1.27)where a , a ∈ A and b , b ∈ B . hapter 2Quasi-Triangular Hopf Algebrasand their Representations In this preliminary chapter we recall the notion of Hopf algebra together with itscategory of representations. By adding more and more algebraic structure we succes-sively discuss coalgebras, bialgebras and finally Hopf algebras, together with somefundamental examples. Depicting those algebraic properties in terms of commuta-tive diagrams is a comfortable way of compressing the relations, furthermore reveal-ing their duality. This, together with proofs of some fundamental properties of theHopf algebra structure, is the agenda of Section 2.1. Afterwards, in Section 2.2, wefocus on modules of Hopf algebras. It turns out that, unlike for general algebras,the representation theory of Hopf algebras has many additional features. In fact,it is exactly the bialgebra structure which shapes the corresponding modules as amonoidal category, while the antipode gives rise to an additional rigidity propertyof the monoidal subcategory of finitely generated projective modules. Consequently,we introduce quasi-triangular Hopf algebras in Section 2.3 as those Hopf algebraswhose monoidal category of representations is braided and describe the correspond-ing algebraic structure on the algebraic level. It is interesting that this so-calleduniversal R -matrix satisfies the quantum Yang-Baxter equation, which connects ourconsiderations to the field of integrable systems. There is a notion of gauge equiv-alence respecting both pictures in equal measure: on the algebraic side, Drinfel’dtwists deform the quasi-triangular Hopf algebra structure such that the result is stilla quasi-triangular Hopf algebra, while on the categorical side the Drinfel’d functorleads to a braided monoidal equivalence of the representations of the deformed andundeformed Hopf algebra. This is what we discuss in Section 2.4. In Section 2.5, werefine the mentioned braided monoidal equivalence to braided symmetric equivariantbimodules of braided commutative module algebras of a triangular Hopf algebra. Atthe end of this chapter, in Section 2.6, it is discussed how ∗ -involutions fit into thepicture and how they can be deformed using unitary Drinfel’d twists. The mainreferences for the first four sections are [35, 68, 77, 81], while the last two sectionsare inspired by [6, 9, 60] and [56], respectively. The first formal definition of Hopf algebra goes to back to early works of Cartier(see [1] and references therein for a discussion on the origin of Hopf algebras). Hopf16uasi-Triangular Hopf Algebras and their Representations 17algebras are essential objects in a broad spectrum of disciplines in mathematics andlikewise in theoretical physics. In this thesis Hopf algebras incarnate symmetries ofthe algebra of observables of our interest. This point of view was first promoted inearly articles [42, 43, 44, 96] about quantum groups. Since then, numerous works onalgebraic deformation proved the fruitfulness of this approach. From an algebraicpoint of view, Hopf algebras are natural specifications of algebras: a coalgebra is thedual object to an algebra and the notion of bialgebra describes objects which inheritboth structures in a compatible way. The antipode, completing the Hopf algebrastructure, is an inverse of the corresponding convolution algebra of the bialgebra.We add more substance to this line of thought in this section. Namely, we give thedefinition of Hopf algebra building on intermediate steps and concrete examples.Several fundamental properties of the Hopf algebra structure are discussed. Sincethey are used throughout the thesis we give full proofs, also to get used to the devel-oped notation. We refer to the textbooks [35] Sec. 4.1, [68] Chap. III, [77] Chap. 1and [81] Chap. 1 for excellent introductions to Hopf algebras.Let us start this section by discussing two fundamental examples of Hopf al-gebras: the universal enveloping algebra of a Lie algebra and the group algebra.Having concrete examples in mind the abstract definitions of the following sectionsare easier to digest. Consider a Lie algebra g over a field K , with Lie bracket denotedby [ · , · ]. Its universal enveloping algebra U g is defined to be the tensor algebraT g = (cid:77) k ≥ g ⊗ k = K ⊕ g ⊕ ( g ⊗ g ) ⊕ · · · of g modulo the ideal generated by x ⊗ y − y ⊗ x − [ x, y ] for all x, y ∈ g . It inheritsthe structure of an associative unital algebra from T g , with product induced bythe tensor product and unit induced by 1 ∈ K . The universal enveloping algebrais, up to isomorphism, uniquely determined by the following universal property:for any associative unital algebra A , seen as a Lie algebra with the commutator,and any Lie algebra homomorphism φ : g → A , there is a unique homomorphismΦ : U g → A of associative unital algebras such that φ = Φ ◦ ι , where ι : g → U g denotes the canonical embedding of g in U g . Making use of this universal propertywe can define three K -linear maps ∆ : U g → U g ⊗ , (cid:15) : U g → K and S : U g → U g by declaring them to satisfy ∆( x ) = x ⊗ ⊗ x, (cid:15) ( x ) = 0 and S ( x ) = − x onelements x ∈ g and extending ∆ and (cid:15) as algebra homomorphisms and S as algebraanti-homomorphism. A calculation shows that the equations(∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆ and ( (cid:15) ⊗ id) ◦ ∆ = id = (id ⊗ (cid:15) ) ◦ ∆ (2.1)hold in addition as maps U g → U g ⊗ and U g → U g , respectively. By theuniversal property of U g it is sufficient to prove this on elements of g . If one denotesthe product and unit of U g by µ : U g ⊗ → U g and η : K → U g , respectively, oneeasily verifies that µ ◦ ( S ⊗ id) ◦ ∆ = η ◦ (cid:15) = µ ◦ (id ⊗ S ) ◦ ∆ : U g → U g (2.2)holds. Before commenting further on these maps and their properties we introduceanother algebra with additional maps ∆, (cid:15) and S obedient to the same relations.It is the group algebra K [ G ] of a finite group G , which is defined as the free K -module generated by the elements of G . Its associative product is given by the K -linearly extended group multiplication and the unit is the neutral element of G . On8 Chapter 2elements g ∈ G we define ∆( g ) = g ⊗ g, (cid:15) ( g ) = 1 and S ( g ) = g − and extend thosemaps as algebra (anti)-homomorphisms to ∆ : K [ G ] → K [ G ] ⊗ , (cid:15) : K [ G ] → K and S : K [ G ] → K [ G ]. It is easy to verify that they satisfy (2.1) and (2.2). The naturalquestion arises if there are more examples of associative unital algebras allowingfor such additional structure, maybe even revealing a greater concept. Besides, theequations (2.1) seem to mimic the axioms of an associative unital algebra in a dualfashion and the map S reminds of some kind of inverse. In fact it is the notion ofHopf algebra which unites and generalizes those two examples. To state a rigorousdefinition we need some preparation. First of all we operate slightly more generalby considering commutative rings k instead of fields K and k -modules rather than K -vector spaces. The reason is that we obviously enrich the number of examples inthis way, in particular allowing for formal power series V [[ (cid:126) ]] with coefficients in a K -vector space V in this way: V [[ (cid:126) ]] is a k = K [[ (cid:126) ]]-module. In the next definitionwe axiomatize equations (2.1). Definition 2.1.1 (Coalgebra) . Let C be a k -module for a commutative ring k . Itis said to be a k -coalgebra if there are k -linear maps ∆ : C → C ⊗ C and (cid:15) : C → k such that (2.1) hold. In this case ∆ is called coproduct and (cid:15) counit of C , whileto axioms (2.1) are called coassociativity and counitality, respectively. A k -linearmap φ : C → C (cid:48) between k -coalgebras ( C , ∆ , (cid:15) ) and ( C (cid:48) , ∆ (cid:48) , (cid:15) (cid:48) ) is said to be a coalgebrahomomorphism if ( φ ⊗ φ ) ◦ ∆ = ∆ (cid:48) ◦ φ and (cid:15) (cid:48) ◦ φ = (cid:15) . The category of k -coalgebrasis denoted by k C . We introduce
Sweedler’s notation ∆( c ) = c (1) ⊗ c (2) to denote the coproduct ofan element c of a coalgebra C . Namely, we omit a possibly finite sum of factorizingelements in the tensor product, similar to the leg notation which we introduced inthe introduction. Using the coassociativity of ∆ we define c (1) ⊗ c (2) ⊗ c (3) = c (1)(1) ⊗ c (1)(2) ⊗ c (2) = c (1) ⊗ c (2)(1) ⊗ c (2)(2) and similarly for higher coproducts of c . Combining Sweedler’s notation with legnotation we further write F , = (∆ ⊗ id)( F ), F , = (id ⊗ ∆)( F ) for any element F = F ⊗ F ∈ C ⊗ and F , = (id ⊗ ∆)( τ C , C ( F )), F , = ( τ C , C ⊗ id)(∆( F )), etc. ifwe consider permutations of the legs of F . A coalgebra C is said to be cocommutative if c (2) ⊗ c (1) = c (1) ⊗ c (2) for all c ∈ C . This is the case for our previous examplesof the universal enveloping algebra U g and the group algebra K [ G ]. Furthermore,motivated from exactly these two examples we call an element c ∈ C ξ - χ -primitive for two elements ξ, χ ∈ C , if ∆( c ) = c ⊗ ξ + χ ⊗ c . We call c ∈ C group-like if∆( c ) = c ⊗ c . The 1-1-primitive elements of U g are exactly the elements of g ,usually they are said to be primitive for short in this case, while the group-likeelements of K [ G ] are exactly the elements of G . Depicting the axioms of a coalgebra( C , ∆ , (cid:15) ) and a coalgebra homomorphism φ : C → C (cid:48) via commutative diagrams
C C ⊗ CC ⊗ C C ⊗ C ⊗ C ∆∆ id ⊗ ∆∆ ⊗ id , k ⊗ C C ⊗ C C ⊗ k C ∼ = (cid:15) ⊗ id id ⊗ (cid:15) ∼ =∆ and C C ⊗ CC (cid:48) C (cid:48) ⊗ C (cid:48) ∆ φ φ ⊗ φ ∆ (cid:48) , C C (cid:48) k φ(cid:15) (cid:15) (cid:48) , uasi-Triangular Hopf Algebras and their Representations 19respectively, the duality with algebras and algebra homomorphisms becomes visible:the corresponding diagrams in the category k A of algebras are obtained by reversingthe arrows, replacing ∆ by the product and (cid:15) by the unit in the above diagrams.Another observation affirming this duality is given by the following lemma (c.f.[81] Lem. 1.2.2 and the subsequent discussion). Lemma 2.1.2.
Let ( C , ∆ , (cid:15) ) be a k -coalgebra. Its dual k -module C ∗ = Hom k ( C , k ) is a k -algebra with product and unit given by the dual maps ∆ ∗ : C ∗ ⊗ C ∗ → C ∗ and (cid:15) ∗ : k → C ∗ , respectively. If ( A , µ, η ) is a finite-dimensional algebra over a field K ,its dual A ∗ = Hom K ( A , K ) is a coalgebra with coproduct and counit given by thedual maps µ ∗ : A ∗ → A ∗ ⊗ A ∗ and η ∗ : A ∗ → K , respectively.Proof. We only prove the second statement. For finite-dimensional vector spacesthere is an isomorphism (
A ⊗ A ) ∗ ∼ = A ∗ ⊗ A ∗ . This implies that the dual of themultiplication ( µ ∗ ( α ))( a ⊗ b ) = α ( µ ( a ⊗ b )) , where α ∈ A ∗ and a, b ∈ A , is a K -linear map µ ∗ : A ∗ → A ∗ ⊗ A ∗ . It is coassociative,since ( α (1)(1) ⊗ α (1)(2) ⊗ α (2) )( a ⊗ b ⊗ c ) = µ ( µ ( a ⊗ b ) ⊗ c )= µ ( a ⊗ µ ( b ⊗ c ))=( α (1) ⊗ α (2)(1) ⊗ α (2)(2) )( a ⊗ b ⊗ c )for all α ∈ A ∗ and a, b, c ∈ A by the associativity of µ , where we denoted α (1) ⊗ α (2) = µ ∗ ( α ). Furthermore, the dual η ∗ ( α ) = α (1) of the unit satisfies the counit axiom(( η ∗ ⊗ id) µ ∗ ( α ))( a ) = α (1) (1) ⊗ α (2) ( a )= α ( µ (1 ⊗ a ))= α ( a )=((id ⊗ η ∗ ) µ ∗ ( α ))( a )for all a ∈ A .However, the second part of the lemma indicates that the duality of algebrasand coalgebras should be understood with a grain of salt. For a infinite-dimensionalalgebra A it is known that A ∗ ⊗ A ∗ is a proper subspace of ( A ⊗ A ) ∗ and we cannot expect A ∗ to be a coalgebra in general. A way out of this problem is given by astronger notion of duality which is presented later in this section. Besides the dual k -module, there is another fundamental construction given by the tensor product C ⊗ D of coalgebras. It is a coalgebra with coproduct∆
C⊗D ( c ⊗ d ) = (id C ⊗ τ C , D ⊗ id D )(∆ C ⊗ ∆ D )( c ⊗ d ) = ( c (1) ⊗ d (1) ) ⊗ ( c (2) ⊗ d (2) )and counit (cid:15) C⊗D = (cid:15) C ⊗ (cid:15) D . Remark the duality of this construction to the tensorproduct of two algebras (see eq.(1.27)). Furthermore, any commutative ring k isa k -coalgebra with coproduct ∆ k ( λ ) = λ · (1 ⊗
1) and counit (cid:15) k = id k . In fact,coassociativity is trivial and counitality follows from the isomorphism k ⊗ k ∼ = k .We introduce an algebra which is useful in the theory of quantum groups. In fact,it is an essential tool in the subsequent proofs of this section.0 Chapter 2 Lemma 2.1.3 (Convolution Algebra) . Consider a k -algebra ( A , µ, η ) and a k -coalgebra ( C , ∆ , (cid:15) ) . Then, the k -linear maps Hom k ( C , A ) from C to A form an algebrawith associative product given by f (cid:63) g = µ ◦ ( f ⊗ g ) ◦ ∆ (2.3) for all f, g ∈ Hom k ( C , A ) and with unit given by η ◦ (cid:15). The algebra (Hom k ( C , A ) , (cid:63), η ◦ (cid:15) ) is said to be the convolution algebra and (cid:63) the convolution product . Using Sweedler’s notation and omitting the product in A theconvolution product reads ( f (cid:63) g )( c ) = f ( c (1) ) g ( c (2) ) for all f, g ∈ Hom k ( C , A ) and c ∈ C . As a reference consider [68] Prop. III.3.1.(a). Setting A = k we recover thesituation of the first part of Lemma 2.1.2. Proof.
Let f, g, h ∈ Hom k ( C , A ) and c ∈ C be arbitrary. As concatenation of k -(bi)linear maps, (cid:63) and η ◦ (cid:15) are k -(bi)linear and f (cid:63) g, η ◦ (cid:15) ∈ Hom k ( C , A ). Then(( f (cid:63) g ) (cid:63) h )( c ) =( f ( c (1)(1) ) g ( c (1)(2) )) h ( c (2) )= f ( c (1) )( g ( c (2)(1) ) h ( c (2)(2) ))=( f (cid:63) ( g (cid:63) h ))( c )proves that (cid:63) is associative and(( η ◦ (cid:15) ) (cid:63) f )( c ) = (cid:15) ( c (1) ) f ( c (2) )= f ( (cid:15) ( c (1) ) c (2) )= f ( c )=( f (cid:63) ( η ◦ (cid:15) ))( c )shows that η ◦ (cid:15) is a unit, concluding the proof of the lemma.Focusing again on our motivating examples we realize that their coalgebra struc-tures are not independent from their algebra structures. In fact, ∆ and (cid:15) are algebrahomomorphisms and µ and η are coalgebra homomorphisms, where we endow thetensor product and k with the corresponding (co)algebra structure. This indicatesthat we are not finished in formalizing our examples. Definition 2.1.4 (Bialgebra) . A k -module B is said to be a bialgebra if it is analgebra and a coalgebra such that the coproduct and the counit are algebra homomor-phisms and the product and unit are coalgebra homomorphisms. A homomorphismof bialgebras is an algebra homomorphism, which is a coalgebra homomorphism inaddition. The category of bialgebras is denoted by k B . In Section 2.2 we connect the notion of bialgebra with properties of its represen-tation theory. The conditions on an algebra ( B , µ, η ) with coalgebra structures ∆and (cid:15) to be a bialgebra is depicted in the commutativity of B ⊗ B B B ⊗ BB ⊗ B ⊗ B ⊗ B B ⊗ B ⊗ B ⊗ B µ ∆ ⊗ ∆ ∆id B ⊗ τ B⊗B ⊗ id B µ ⊗ µ , k B k ⊗ k B ⊗ B η ∼ = ∆ η ⊗ η , (2.4)uasi-Triangular Hopf Algebras and their Representations 21 B ⊗ B B k ⊗ k k µ(cid:15) ⊗ (cid:15) (cid:15) ∼ = and k B k η id k (cid:15) . (2.5)Remark that it is sufficient to demand µ and η to be coalgebra homomorphisms or∆ and (cid:15) to be algebra homomorphisms in Definition 2.1.4, which is clear from thesymmetry of the diagrams (2.4) and (2.5), see also [35] Sec. 4.1 Rem. 1. Lemma 2.1.5.
Let ( B , µ, η, ∆ , (cid:15) ) be a k -algebra and a k -coalgebra. Then µ and η are coalgebra homomorphisms if and only if ∆ and (cid:15) are algebra homomorphisms. We did not abstract the map S yet, which becomes the main character in thefollowing definition, finally completing the notion of Hopf algebra. Definition 2.1.6 (Hopf Algebra) . A k -bialgebra ( H, µ, η, ∆ , (cid:15) ) is said to be a k -Hopfalgebra if there is a k -linear bijection S : H → H such that (2.2) holds and we call S an antipode of H in that case. A bialgebra homomorphism between Hopf algebras issaid to be a Hopf algebra homomorphism if it intertwines the antipodes in addition.We denote the category of Hopf algebras by k H . In the following we often drop the reference to the commutative ring k and simplyrefer to Hopf algebras, etc. Remark that there are slightly weaker definitions of Hopfalgebra, not assuming the antipode to have an inverse (see [68, 77, 81]). We followthe convention of [35], arguing that in all examples which are relevant for us theantipode is invertible and we do not want to state this as an additional conditionthroughout the thesis. The antipode axioms (2.2) can be reformulated in pictoriallanguage by the commutativity of H ⊗ H H ⊗ HH k HH ⊗ H H ⊗ H S ⊗ id H µ ∆∆ (cid:15) η id H ⊗ S µ . Besides U g and K [ G ] there are further interesting examples of Hopf algebras takenfrom [35] Sec. 4.1 B and [81] Ex. 1.5.6, respectively. Example 2.1.7. i.) Let G be a finite group with neutral element e ∈ G andconsider the k -module F ( G ) of functions on G with values in k . It is a com-mutative k -algebra, where the product F · F of two functions F , F ∈ F ( G ) is defined by ( F · F )( g ) = F ( g ) F ( g ) for all g ∈ G and with unit function defined by G (cid:51) g (cid:55)→ ∈ k . We fur-ther define two k -linear maps ∆ : F ( G ) → F ( G × G ) and (cid:15) : F ( G ) → k as ∆( F )( g, h ) = F ( gh ) and (cid:15) ( F ) = F ( e ) for all F ∈ F ( G ) and g, h ∈ G . Notethat there is an isomorphism F ( G ) ⊗ F ( G ) ∼ = F ( G × G ) of k -modules givenby F ⊗ F (cid:55)→ (cid:92) F ⊗ F : G × G (cid:51) ( g, h ) (cid:55)→ F ( g ) F ( h ) ∈ k . Using this identification it follows that ((∆ ⊗ id)∆( F ))( g, h, (cid:96) ) = F (( gh ) (cid:96) ) = F ( g ( h(cid:96) )) = ((id ⊗ ∆)∆( F ))( g, h, (cid:96) ) and (( (cid:15) ⊗ id)∆( F ))( g ) = F ( eg ) = F ( g ) = F ( ge ) = ((id ⊗ (cid:15) )∆( F ))( g ) for all F ∈ F ( G ) and g, h, (cid:96) ∈ G . Furthermore ∆( F F )( g, h ) = ( F F )( gh ) = F ( gh ) F ( gh ) = (∆( F )∆( F ))( g, h ) , ∆(1)( g, h ) = 1 , (cid:15) ( F F ) = ( F F )( e ) = F ( e ) F ( e ) = (cid:15) ( F ) (cid:15) ( F ) and (cid:15) (1) = 1 hold for all F , F ∈ F ( G ) and g, h ∈ G , proving that ( F ( G ) , ∆ , (cid:15) ) is abialgebra. Finally, one defines an antipode on F ( G ) as the k -linear map S : F ( G ) → F ( G ) such that S ( F )( g ) = F ( g − ) for all F ∈ F ( G ) and g ∈ G .In fact ( S ( F (1) ) F (2) )( g ) = F ( g − g ) = F ( e ) = (cid:15) ( F )1 = ( F (1) S ( F (2) ))( g ) for all F ∈ F ( G ) and g ∈ G , where we used Sweedler’s notation for thecoproduct. This describes the Hopf algebra F ( G ) of k -valued functions on G .It is cocommutative if and only if G is Abelian. The same computations holdif G is an affine algebraic group over a field. If G is a compact topologicalgroup, the finite-dimensional real representations Rep( G ) of G are not onlya dense subalgebra of F ( G ) but even a Hopf algebra, since Rep( G × G ) ∼ =Rep( G ) ⊗ Rep( G ) . This is called the Hopf algebra of representative functionson G . Note that in general F ( G ) is not a Hopf algebra for a compact topologicalgroup G ;ii.) As an algebra, Sweedler’s Hopf algebra H is generated by a unit element and three elements g, x and gx such that the relations g = 1 , x = 0 and xg = − gx hold. The coproduct and counit are defined on generators by ∆( g ) = g ⊗ g, ∆( x ) = x ⊗ g ⊗ x, (cid:15) ( g ) = 1 and (cid:15) ( x ) = 0 , respectively, while the antipode reads S ( g ) = g and S ( x ) = − gx on generators. g is group-like, while x is (1 , g ) -primitive. In fact, all relations are easily veri-fied on generators. Obviously, H is neither commutative nor cocommutative.It is the smallest Hopf algebra with this property; Let G be a finite group and consider the two Hopf algebras F ( G ) and K [ G ]arising from it. There is a non-degenerate dual pairing (cid:104)· , ·(cid:105) : F ( G ) ⊗ K [ G ] → K between them and in a remarkable way it mirrors the algebra structure of F ( G )with the coalgebra structure of K [ G ] and vice versa. Namely, (cid:104) F F , g (cid:105) = F ( g ) F ( g ) = (cid:104) F ⊗ F , g (1) ⊗ g (2) (cid:105) , (cid:104) , g (cid:105) =1 = (cid:15) ( g ) , (cid:104) F, gh (cid:105) = F ( gh ) = (cid:104) F (1) ⊗ F (2) , g ⊗ h (cid:105) and (cid:104) F, e (cid:105) = F ( e ) = (cid:15) ( F )uasi-Triangular Hopf Algebras and their Representations 23for all F, F , F ∈ F ( G ) and g, h ∈ G . Furthermore the pairing mirrors the an-tipodes, i.e. (cid:104) S ( F ) , g (cid:105) = F ( g − ) = (cid:104) F, S ( g ) (cid:105) . Recalling Lemma 2.1.2 for a finite-dimensional Hopf algebra H over a field K encourages this duality: the bialgebrastructure of H is the transpose of the bialgebra structure of H ∗ via the dual pairingof (finite-dimensional) vector spaces. It is not hard to prove that the transpose ofan antipode on H leads to an antipode on H ∗ and vice versa. Let us formalize theseobservations. Definition 2.1.8 (Dual Pair of Hopf Algebras) . Consider two Hopf algebras H and H (cid:48) over a commutative ring k . They are called dual pair of Hopf algebras if there isa k -bilinear map (cid:104)· , ·(cid:105) : H (cid:48) ⊗ H → k such that (cid:104) ab, ξ (cid:105) = (cid:104) a ⊗ b, ξ (1) ⊗ ξ (2) (cid:105) , (cid:104) , ξ (cid:105) = (cid:15) ( ξ ) , (cid:104) a, ξχ (cid:105) = (cid:104) a (1) ⊗ a (2) , ξ ⊗ χ (cid:105) , (cid:104) a, (cid:105) = (cid:15) ( a ) and (cid:104) S ( a ) , ξ (cid:105) = (cid:104) a, S ( ξ ) (cid:105) for all ξ, χ ∈ H and a, b ∈ H (cid:48) . They are called strict dual pair of Hopf algebras ifthe pairing (cid:104)· , ·(cid:105) is non-degenerate in addition, i.e. if (cid:104) a, ξ (cid:105) = 0 for all ξ ∈ H implies a = 0 and (cid:104) a, ξ (cid:105) = 0 for all a ∈ H (cid:48) implies ξ = 0 . We already observed that ( K [ G ] , F ( G )), where G is a finite group, and ( H, H ∗ )for a finite-dimensional K -Hopf algebra H , are examples of strict dual pairs of Hopfalgebras. However, in the infinite-dimensional setting or more general for com-mutative rings k there are issues as already indicated by Lemma 2.1.2. A solutionhelping to avoid these problems is given by not considering the whole dual k -moduleHom k ( H, k ) of an arbitrary k -Hopf algebra H , but rather its finite-dual H ◦ = { α ∈ H ∗ | ∃ algebra ideal I ⊆ H such that dim( H/I ) < ∞ and α ( I ) = 0 } . It is a Hopf algebra with respect to the transposed Hopf algebra structure (c.f.[81] Thm. 9.1.3). In particular, (
H, H ◦ ) is a dual pair of Hopf algebras. After thisshort excursus on duality we return to discuss general properties of Hopf algebras.In particular we intend to prove additional features of antipodes, beginning withuniqueness, c.f. [35] Sec. 4.1 Rem. 4. Lemma 2.1.9.
Let ( B , µ, η, ∆ , (cid:15) ) be a k -bialgebra. A k -linear bijection S : H → H is an antipode for H if and only if S is the convolution inverse of the identity η ◦ (cid:15) in the convolution algebra Hom k ( H, H ) .Proof. By Lemma 2.1.3 the k -linear maps Hom k ( H, H ) from the coalgebra ( H, ∆ , (cid:15) )to the algebra ( H, µ, η ) form an associative algebra with respect to the convolutionproduct (cid:63) and with unit η ◦ (cid:15) . Let S : H → H be a k -linear bijection. Then S (cid:63) id = η ◦ (cid:15) if and only if S ( ξ (1) ) ξ (2) = (cid:15) ( ξ )1 for all ξ ∈ H and id (cid:63) S = η ◦ (cid:15) if andonly ξ (1) S ( ξ (2) ) = (cid:15) ( ξ )1 for all ξ ∈ H .Since the inverse element of an algebra element is unique, Lemma 2.1.9 impliesthe following statement.4 Chapter 2 Corollary 2.1.10.
Let ( H, µ, η, ∆ , (cid:15), S ) be a k -Hopf algebra. Then the underlyingbialgebra structure of H admits a unique antipode S . Moreover, the antipode of a Hopf algebra respects the underlying bialgebra struc-ture in the sense that it is an anti-bialgebra homomorphism , i.e. a bialgebra homo-morphism in a contravariant way. In detail, S : H → H is an anti-algebra homo-morphism if S ( ξχ ) = S ( χ ) S ( ξ ) and S (1) = 1 (2.6)hold for all ξ, χ ∈ H and it is an anti-coalgebra homomorphism if S ( ξ ) (1) ⊗ S ( ξ ) (2) = S ( ξ (2) ) ⊗ S ( ξ (1) ) and (cid:15) ( S ( ξ )) = (cid:15) ( ξ ) (2.7)hold for all ξ ∈ H . Furthermore, the examples we mentioned suggest that theinverse of the antipode is given by the antipode itself. Even if this does not holdin general, it is the case for a huge class of Hopf algebras, namely for commutativeor cocommutative ones, including our examples besides Example 2.1.7 ii.). Bothstatements are discussed in the following proposition (c.f. [68] Thm. III.3.4. and[77] Prop. 1.3.1). Proposition 2.1.11.
Let H be a k -Hopf algebra. Then its antipode S is an anti-bialgebra homomorphism. If H is either commutative or cocommutative S = id follows.Proof. Consider the coalgebra ( H ⊗ H, ∆ H ⊗ H , (cid:15) H ⊗ H ) and the convolution algebraHom k ( H ⊗ H, H ) with convolution product (cid:63) and unit η ◦ (cid:15) H ⊗ H . It is an algebraaccording to Lemma 2.1.3. We are going to prove the first equation of (2.6) by defin-ing the left- and right-hand side to be k -linear maps f, g : H ⊗ H → H , respectively,and showing that f (cid:63) h = η ◦ (cid:15) H ⊗ H = h (cid:63) g for a k -linear map h : H ⊗ H → H . Infact this is sufficient, since the left and right inverse of an algebra element coincideif both exist. For the first equation of (2.7) we use a similar strategy, consideringthe convolution algebra of k -linear maps from the coalgebra ( H, ∆ , (cid:15) ) to the algebra( H ⊗ H, µ H ⊗ H , η H ⊗ H ). We divide the proof in three parts.i.) S is an anti-algebra homomorphism: define f ( ξ ⊗ χ ) = S ( ξχ ) and g ( ξ ⊗ χ ) = S ( χ ) S ( ξ ) for all ξ, χ ∈ H . Then, using that ∆ and (cid:15) are algebrahomomorphisms and the antipode properties, we obtain( f (cid:63) µ )( ξ ⊗ χ ) = f ( ξ (1) ⊗ χ (1) ) µ ( ξ (2) ⊗ χ (2) )= S ( ξ (1) χ (1) ) ξ (2) χ (2) = S (( ξχ ) (1) )( ξχ ) (2) = η ( (cid:15) ( ξχ ))= η ( (cid:15) ( ξ ) (cid:15) ( χ ))= η ( (cid:15) H ⊗ H ( ξ ⊗ χ ))and ( µ (cid:63) g )( ξ ⊗ χ ) = µ ( ξ (1) ⊗ χ (1) ) g ( ξ (2) ⊗ χ (2) )= ξ (1) χ (1) S ( χ (2) ) S ( ξ (2) )= ξ (1) η ( (cid:15) ( χ )) S ( ξ (2) )uasi-Triangular Hopf Algebras and their Representations 25= ξ (1) S ( ξ (2) ) η ( (cid:15) ( χ ))= η ( (cid:15) ( ξ )) η ( (cid:15) ( χ ))= η ( (cid:15) ( ξ ) (cid:15) ( χ ))= η ( (cid:15) H ⊗ H ( ξ ⊗ χ ))for all ξ, χ ∈ H , implying that f = g . Furthermore, S (1) = S (1 (1) )1 (2) = η ( (cid:15) (1)) = 1implies (2.6).ii.) S is an anti-coalgebra homomorphism: define f ( ξ ) = S ( ξ ) (1) ⊗ S ( ξ ) (2) and g ( ξ ) = S ( ξ (2) ) ⊗ S ( ξ (1) ) for all ξ ∈ H . Then( f (cid:63) ∆)( ξ ) = µ H ⊗ H ( f ( ξ (1) ) ⊗ ∆( ξ (2) ))= S ( ξ (1) ) (1) ξ (2)(1) ⊗ S ( ξ (1) ) (2) ξ (2)(2) =( S ( ξ (1) ) ξ (2) ) (1) ⊗ ( S ( ξ (1) ) ξ (2) ) (2) =( η ( (cid:15) ( ξ ))) (1) ⊗ ( η ( (cid:15) ( ξ ))) (2) = (cid:15) ( ξ ) η (1) ⊗ η (1)= η H ⊗ H ( (cid:15) ( ξ ))and (∆ ⊗ g )( ξ ) = ξ (1)(1) S ( ξ (2)(2) ) ⊗ ξ (1)(2) S ( ξ (2)(1) )= ξ (1) S ( ξ (2)(2)(2) ) ⊗ ξ (2)(1) S ( ξ (2)(2)(1) )= ξ (1) S ( ξ (2)(2) ) ⊗ ξ (2)(1)(1) S ( ξ (2)(1)(2) )= ξ (1) S ( ξ (2)(2) ) ⊗ η ( (cid:15) ( ξ (2)(1) ))= ξ (1) S ( ξ (2) ) ⊗ η (1)= η ( (cid:15) ( ξ )) ⊗ η (1)= η H ⊗ H ( (cid:15) ( ξ ))imply f = g . Furthermore, (cid:15) ( S ( ξ )) = (cid:15) ( S ( (cid:15) ( ξ (1) ) ξ (2) )) = (cid:15) ( ξ (1) S ( ξ (2) )) = (cid:15) ( (cid:15) ( ξ )1) = (cid:15) ( ξ )for all ξ ∈ H , implying (2.7).iii.) For a commutative or cocommutative H the antipode is an involu-tion: for all ξ ∈ H we prove( S (cid:63) S )( ξ ) = S ( ξ (1) ) S ( ξ (2) ) = S ( S ( ξ (2) ) ξ (1) ) = S ( η ( (cid:15) ( ξ ))) = η ( (cid:15) ( ξ )) , using i.), where the third equality holds if H is commutative or cocommutative.Since S (cid:63) id = η ◦ (cid:15) by the antipode property we obtain S = id, because theright inverse of an algebra element is unique if it exists.This concludes the proof of the proposition.As a terminal observation we want to prove that commutativity of a bialgebrahomomorphism with the antipodes is in fact a redundant condition in the definitionof Hopf algebra homomorphism (compare to [35] Sec. 4.1 Rem. 2).6 Chapter 2 Lemma 2.1.12.
Let φ : H → H (cid:48) be a bialgebra homomorphism between k -Hopfalgebras ( H, µ, η, ∆ , (cid:15), S ) and ( H (cid:48) , µ (cid:48) , η (cid:48) , ∆ (cid:48) , (cid:15) (cid:48) , S (cid:48) ) . Then φ S = φ ◦ S and φ S (cid:48) = S (cid:48) ◦ φ are k -linear maps which are convolution inverse to φ . In particular φ S = φ S (cid:48) and φ is a Hopf algebra homomorphism if and only if it is a bialgebra homomorphism.Proof. The k -linearity of φ S and φ S (cid:48) is clear since they are defined as a concatenationof k -linear maps. For all ξ ∈ H one obtains( φ S (cid:63) φ )( ξ ) = φ ( S ( ξ (1) )) φ ( ξ (2) ) = φ ( S ( ξ (1) ) ξ (2) ) = φ ( η ( (cid:15) ( ξ ))) = η (cid:48) ( (cid:15) ( ξ ))using that φ is an algebra homomorphism and( φ (cid:63) φ S (cid:48) )( ξ ) = φ ( ξ (1) ) S (cid:48) ( φ ( ξ (2) )) = φ ( ξ ) (1) S (cid:48) ( φ ( ξ ) (2) ) = η (cid:48) ( (cid:15) (cid:48) ( φ ( ξ ))) = η (cid:48) ( (cid:15) ( ξ ))using that φ is a coalgebra homomorphism, implying φ S = φ S (cid:48) by the uniqueness ofthe inverse element. This shows that commuting with the antipode is a redundantcondition for a Hopf algebra homomorphism.In the next section we focus on algebra representations and characterize bialge-bras and Hopf algebras via additional properties of the category of representationsof the underlying algebras. These considerations further lead to the definition ofquasi-triangular structures in Section 2.3. We introduce Hopf algebra modules and prove that they form a monoidal categorywith monoidal structure given by the tensor product of k -modules. In fact it turnsout that bialgebras are those algebras whose category of representations is monoidalwith respect to the usual associativity and unit constraints, leading to an importantcharacterization of bialgebras. The antipode of a Hopf algebra gives rise to an addi-tional duality property on the categorical level, however only for finitely generatedprojective modules. All definitions and statements can also be found in [35] Sec. 5.1,[68] Sec. XI.3 and [77] Sec. 9.1.Consider a k -algebra ( A , µ, η ) for a commutative ring k . A k -module M is saidto be a left A -module if there exists a k -linear map λ : A ⊗ M → M such that thediagrams
A ⊗ A ⊗ M A ⊗ MA ⊗ M M id A ⊗ λµ ⊗ id M λλ and k ⊗ M A ⊗ MM η ⊗ id M ∼ = λ commute. The tuple ( M , λ ) is called a left representation of A on M and λ is saidto be a left A -module action or left A -module structure . The left A -modules forma category A M with morphisms given by left A -module homomorphism , where a k -linear map φ : M → M (cid:48) between two left representations ( M , λ M ) and ( M (cid:48) , λ M (cid:48) )of A is said to be a left A -module homomorphism if the diagram A ⊗ M MA ⊗ M (cid:48) M (cid:48) λ M id A ⊗ φ φλ M(cid:48) uasi-Triangular Hopf Algebras and their Representations 27commutes. This means that φ respects the left A -module structures of M and M (cid:48) or that φ is left A -linear in other words. If A = H is a Hopf algebra we often referto left H -module homomorphisms as H -equivariant maps . The algebra A and thecorresponding category A M can be seen as two sides of the same coin via the socalled Tannaka-Krein Duality . By assigning to any left A -module its underlying k -module and to any left A -module homomorphism itself, now seen as a k -linearmap, we obtain a functor F : A M → k M . There is a 1 : 1-correspondence between A and the natural transformations Nat( F, F ) of F . This allows us to reconstruct A from its representation theory A M and the functor F . On the other hand, we areable to structure the natural transformations of F as an algebra in this way. Thefunctor F is called the forgetful functor for obvious reasons. Proposition 2.2.1 (Tannaka Reconstruction of Algebras) . Let A be an algebra andconsider the forgetful functor F . For every a ∈ A there is a natural transformation Θ a of F given on objects M of A M by Θ a M : F ( M ) (cid:51) m (cid:55)→ a · m ∈ F ( M ) and which is the identity on morphisms. This leads to an algebra isomorphism A ∼ = Nat(
F, F ) .Proof. The proof is taken from [77] Ex. 9.1.1. Recall that F ( M ) = M since F isthe forgetful functor. Θ a is a natural transformation since for any left A -modulehomomorphism φ : M → M (cid:48) we obtain F ( φ )(Θ a M ( m )) = φ ( a · m ) = a · φ ( m ) = Θ a M (cid:48) ( F ( φ )( m ))for all m ∈ M . On the other hand, we can select an element a in A starting froma natural transformation Θ of F , by defining a = Θ A (1), where A is a left A -module via the left multiplication and 1 denotes the unit in A . We prove that theseconstructions are inverse to each other, for which the 1 : 1-correspondence follows.Any a ∈ A can be recovered via Θ a A (1) = a · a . On the other hand let Θ be anatural transformation of F and consider the corresponding element a = Θ A (1) ∈ A .ThenΘ a M ( m ) = a · m = F ( φ m )(Θ A (1)) = Θ M ( F ( φ m )(1)) = Θ M (1 · m ) = Θ M ( m )for all m ∈ M and any left A -module M , where φ m : A (cid:51) b (cid:55)→ b · m ∈ M is a left A -module homomorphism. The associative unital algebra structure on Nat( F, F )is describe as follows: two natural transformations Θ a and Θ b of F are specifiedby two elements a, b ∈ A , while their product Θ a · Θ b is defined as the naturaltransformation Θ a · b . Explicitly, for any left A -module M one has(Θ a · Θ b ) M ( m ) = Θ a · b M ( m ) = ( a · b ) · m = a · ( b · m ) = Θ a M (Θ b M ( m ))for all m ∈ M and Θ M = id M . Since the product of A is associative, so is theproduct of Nat( F, F ).This duality of algebras and their representation theory is a fundamental conceptand will be applied throughout the whole thesis. We are going to examine furtheralgebraic properties of algebras parallel to categorical properties of their modules.Shifting between these two pictures turns out to be an extremely useful tool.8 Chapter 2Analogously to left modules, one defines a right A -module to be a k -module M together with a k -linear map ρ : M ⊗ A −→ M , making
M ⊗ A ⊗ A M ⊗ AM ⊗ A M ρ ⊗ id A id M ⊗ µ ρρ and M ⊗ k M ⊗ AM id M ⊗ η ∼ = ρ commute. A right A -module homomorphism is a k -linear map φ : M → M (cid:48) , where( M , ρ M ) and ( M (cid:48) , ρ M (cid:48) ) are right A -modules, such that M ⊗ A MM (cid:48) ⊗ A M (cid:48) ρ M φ ⊗ id A φρ M(cid:48) commutes. The category of right A -modules is denoted by M A . If an object in A M is also a right A (cid:48) -module for another k -algebra ( A (cid:48) , µ (cid:48) , η (cid:48) ) it is natural to askif both module actions are compatible. This leads to the notion of bimodules. An A - A (cid:48) -bimodule is an object ( M , λ, ρ ) in A M ∩ M A (cid:48) such that A ⊗ M ⊗ A (cid:48)
M ⊗ A (cid:48)
A ⊗ M M λ ⊗ id A(cid:48) id A ⊗ ρ ρλ commutes. If A (cid:48) = A as algebras we are calling M an A -bimodule. The category of A - A (cid:48) -bimodules is denoted by A M A (cid:48) . Its morphisms are left A -module homomor-phisms which are also right A (cid:48) -module homomorphisms. Since every right A -modulecan be viewed as a left A op -module for the opposite algebra A op = ( A , µ ◦ τ A⊗A , η ),we can focus on left A -modules without loss of generality. We come back to right andbimodules in the subsequent sections. In the following we denote a left A -moduleaction by · for short, if there is no danger of confusion with other operations.The category A M of left A -modules is a subcategory of the category k M ofall k -modules for any k -algebra A . In the following lines we show how to use theadditional data of a bialgebra to structure A M even as a monoidal subcategory of k M . Assume that ( A , µ, η, ∆ , (cid:15) ) is a k -bialgebra. Then, the tensor product M ⊗ M (cid:48) of two left A -modules becomes an object in A M via a · ( m ⊗ m (cid:48) ) = ( a (1) · m ) ⊗ ( a (2) · m (cid:48) )for all a ∈ A , m ∈ M and m (cid:48) ∈ M (cid:48) , by making use of the fact that ∆ is analgebra homomorphism. Furthermore, since (cid:15) is an algebra homomorphism, thecommutative ring k becomes a left A -module via a · λ = (cid:15) ( a ) λ for all a ∈ A and λ ∈ k . This action respects the usual associativity constraint of k M , since ∆ is coassociative. Namely, a · (( m ⊗ m (cid:48) ) ⊗ m (cid:48)(cid:48) ) =(( a (1)(1) · m ) ⊗ ( a (1)(2) · m (cid:48) )) ⊗ ( a (2) · m (cid:48)(cid:48) )=( a (1) · m ) ⊗ (( a (2)(1) · m (cid:48) ) ⊗ ( a (2)(2) · m (cid:48)(cid:48) ))= a · ( m ⊗ ( m (cid:48) ⊗ m (cid:48)(cid:48) ))uasi-Triangular Hopf Algebras and their Representations 29for another left A -module M (cid:48)(cid:48) and m (cid:48)(cid:48) ∈ M (cid:48)(cid:48) . It further respects the usual unitconstraints of k M , i.e. a · ( λ ⊗ m ) =( (cid:15) ( a (1) ) λ ) ⊗ ( a (2) · m )= λ ⊗ ( a · m )= λ ( a · m )= a · ( m ⊗ λ )for all a ∈ A , m ∈ M and λ ∈ k , since (cid:15) satisfies the counit axiom. Strictlyspeaking the usual associativity and unit constraints α , (cid:96) and r are left A -modulehomomorphisms on objects and we should write α M , M (cid:48) , M (cid:48)(cid:48) ( a · (( m ⊗ m (cid:48) ) ⊗ m (cid:48)(cid:48) )) = a · α M , M (cid:48) , M (cid:48)(cid:48) (( m ⊗ m (cid:48) ) ⊗ m (cid:48)(cid:48) )and (cid:96) M ( a · ( λ ⊗ m )) = a · (cid:96) M ( λ ⊗ m ) , r M ( a · ( m ⊗ λ )) = a · r M ( m ⊗ λ ) . Instead we treat those isomorphisms as equalities in the above computations. In thenext proposition (c.f. [68] Prop. XI.3.1 and [77] Ex. 9.1.3) we show that coassocia-tivity and the counit axiom are not only sufficient but also necessary for A M to bea monoidal subcategory of k M with usual associativity and unit constraints. Proposition 2.2.2.
Let A be an algebra over a commutative ring k and considertwo algebra homomorphisms ∆ : A → A ⊗ A and (cid:15) : A → k . (2.8) Then ( A , ∆ , (cid:15) ) is a k -bialgebra if and only if A M is a monoidal subcategory of k M with respect to the usual tensor product and unit of k -modules and the usualassociativity and unit constraints.Proof. We already proved that A M is a monoidal subcategory if A is a bialgebra.So assume that A is an algebra endowed with two algebra homomorphisms (2.8)and assume further that A M is monoidal with respect to α, (cid:96), r . Since α M , M (cid:48) , M (cid:48)(cid:48) is a left A -module homomorphism for all objects M , M (cid:48) , M (cid:48)(cid:48) in A M by definition,we obtain α A , A , A (((∆ ⊗ id) ◦ ∆)( a )) = α A , A , A ( a · ((1 ⊗ ⊗ a · α A , A , A ((1 ⊗ ⊗ ⊗ ∆) ◦ ∆)( a )for all a ∈ A , by considering the left A -module A itself with multiplication from theleft as module action. Furthermore, (cid:96) A ((( (cid:15) ⊗ id) ◦ ∆)( a )) = (cid:96) A ( a · (1 ⊗ a · (cid:96) A (1 ⊗
1) = a · a and r A (((id ⊗ (cid:15) ) ◦ ∆)( a )) = r A ( a · (1 ⊗ a · e A (1 ⊗
1) = a · a follow for all a ∈ A . Since we treat α, (cid:96), r as identities on objects this means that∆ is coassociative and (cid:15) is a counit. This concludes the proof.0 Chapter 2This characterizes k -bialgebras completely. Focusing on representation theoryone might for this reason reformulate the definition of a bialgebra by declaring:an algebra A with algebra homomorphisms ∆ : A → A ⊗ A and (cid:15) : A → k is abialgebra if A M is a monoidal subcategory of k M with usual associativity and unitconstraints.Incorporating the antipode of a Hopf algebra in the picture we receive a dualityproperty on the side of monoidal categories. However, for this we have to restrict ourclass of modules from arbitrary k -modules k M for a commutative ring k to finitelygenerated projective k -modules. A k -module M is said to be finitely generatedprojective if there is a finite set { m i } i ∈ I of elements in M , called generators , and a setof elements { α i } i ∈ I called dual generators in the dual k -module M ∗ = Hom k ( M , k ),subscripted by the same finite index set I , such that α i ( m j ) = δ ij and m = (cid:88) i ∈ I α i ( m ) m i for all m ∈ M . It follows that M ∗ is finitely generated projective with generators { α i } i ∈ I and dual generators { ˆ m i } i ∈ I in ( M ∗ ) ∗ , defined by ˆ m i ( α ) = α ( m i ) for all α ∈ M ∗ . Furthermore, the map ˆ: M → ( M ∗ ) ∗ is a k -module isomorphism. Notethat this identification fails for general k -modules, even if k is a field. We denote thecategory of finitely generated projective k -modules with k -module homomorphismsas morphisms by k M f . It is a monoidal subcategory of k M . Lemma 2.2.3.
The monoidal category k M f of finitely generated projective modulesof a commutative ring k is a rigid category.Proof. We follow [35] Ex. 5.1.3. Consider a finitely generated projective k -module M with (dual) generators m i and α i and defineev M ( α ⊗ m ) = ˆ m i ( α ) α i ( m ) ,π M (1) = m i ⊗ α i , ev (cid:48)M ( m ⊗ α ) = ˆ m i ( α ) α i ( m ) and π (cid:48)M (1) = α i ⊗ m i where m = α i ( m ) m i ∈ M and α = ˆ m i ( α ) α i ∈ M ∗ and we used Einstein sumconvention. Then ((id M ⊗ ev M ) ◦ ( π M ⊗ id M ))( m ) = m and ((ev M ⊗ id M ∗ ) ◦ (id M ∗ ⊗ π M ))( α ) = α follow.Note that without a finite set of dual generators we are in general not able todefine the maps π required for rigidity. In particular, k M is not rigid in general.This motivates the passage from general k -modules to finitely generated projec-tive k -modules. As a special case we recover the rigid monoidal category of finite-dimensional K -vector spaces K Vec f if k = K is a field. In Proposition 2.2.2 we provedthat bialgebras correspond to monoidal categories of representation. The analoguefor Hopf algebras in the setting of finitely generated projective modules is given inthe following statement (c.f. [35] Ex. 5.1.4).uasi-Triangular Hopf Algebras and their Representations 31 Proposition 2.2.4.
Let H be a k -Hopf algebra and consider the monoidal category H M of left H -modules, characterized by the bialgebra structure of H . The monoidalsubcategory H M f of finitely generated projective left H -modules is rigid, where theleft and right dual M ∗ and ∗ M of an object M in H M f are defined as the finitelygenerated projective k -module M ∗ with left H -module actions given by (cid:104) ξ · α, m (cid:105) = (cid:104) α, S ( ξ ) · m (cid:105) and (cid:104) ξ · α, m (cid:105) = (cid:104) α, S − ( ξ ) · m (cid:105) for all ξ ∈ H , m ∈ M and α ∈ M ∗ , respectively. The forgetful functor F : H M f → k M f is monoidal.Proof. This follows from Lemma 2.2.3, using the same evaluation and projectionmaps. It only remains to prove that they are morphisms in the right category,i.e. that they are left H -module homomorphisms. Consider a finitely generatedprojective left H -module M and let ξ ∈ H , m ∈ M and α ∈ M ∗ . Thenev M ( ξ · ( α ⊗ m )) =ev M (( ξ (1) · α ) ⊗ ( ξ (2) · m ))=( ξ (1) · α )( ξ (2) · m )= α (( S ( ξ (1) ) ξ (2) ) · m )= (cid:15) ( ξ ) · α ( m )= ξ · ev M ( α ⊗ m )for a dual set of generators { α i } i ∈ I and { m i } i ∈ I . Similarly one proves that π M , ev M ∗ and π M ∗ are H -linear.The converse also holds true, giving a Tannaka-Krein duality for Hopf algebrassimilar to Proposition 2.2.1. Theorem 2.2.5 (Tannaka-Krein Reconstruction of Hopf Algebras) . For a commuta-tive ring k consider an essentially small, k -linear, rigid, Abelian, monoidal category C together with a k -linear, exact, faithful, monoidal functor F : C → k M f . Thenthere is a k -Hopf algebra H and an equivalence G : C → H M f such that F = G (cid:48) ◦ G ,where G (cid:48) : H M f → k M f is the forgetful functor. For a proof we refer to [35] Thm. 5.1.11. This reveals the true analogue of theHopf algebra structure on the monoidal category of its representations and gives usa deeper understanding of the notion of Hopf algebra.
Following the spirit of the last section, i.e. characterizing algebraic structures byproperties of the corresponding representation theory, we introduce quasi-triangularbialgebras as those bialgebras whose corresponding monoidal category is braided.We prove that this leads to universal R -matrices, which satisfy the hexagon relationsand control the noncocommutativity of the coproduct. The corresponding referencesare [35] Sec. 5.2, [68] Sec. XIII 1.3 and [77] Sec. 9.2.2 Chapter 2 Definition 2.3.1 (Quasi-Triangular Bialgebra) . A bialgebra A is said to be quasi-triangular if its monoidal category A M of left A -modules is braided. We call A triangular if A M is braided symmetric. We expect to find additional algebraic structure underlying a quasi-triangularbialgebra. In fact, the following proposition, taken from [68] Prop. XIII.1.4, recoversthe original Definition from [43].
Proposition 2.3.2.
A bialgebra A is quasi-triangular if and only if there is aninvertible element R ∈ A ⊗ A such that ∆ ( a ) = R ∆( a ) R − (2.9) holds for all a ∈ A , and such that the equations (∆ ⊗ id)( R ) = R R and (id ⊗ ∆)( R ) = R R (2.10) are satisfied. The bialgebra A is triangular if and only if the element R satisfies R − = R (2.11) in addition. If the conditions (2.9) and (2.10) are satisfied, the element R is said to be a universal R -matrix or quasi-triangular structure for A . If (2.11) holds in addition, R is called triangular structure . A is said to be quasi-cocommutative with respectto R if (2.9) holds. The equations (2.10) are the so-called hexagon relations . Proof.
Let β be a braiding on the monoidal category A M and define R = τ A , A ( β A , A (1 ⊗ . Clearly
R ∈ A ⊗ A is invertible. Before showing that it satisfies equations (2.9) and(2.10) we prove that β M , M (cid:48) ( m ⊗ m (cid:48) ) = τ M , M (cid:48) ( R · ( m ⊗ m (cid:48) )) for all m ∈ M , m (cid:48) ∈ M and left A -modules M and M (cid:48) . Since β is a natural isomorphism of the functors ⊗ and ⊗ ◦ τ we obtain β M , M (cid:48) (( φ m ⊗ φ m (cid:48) )( a ⊗ b )) = ( φ m (cid:48) ⊗ φ m )( β A⊗A ( a ⊗ b )) , where φ m ( a ) = a · m for all a ∈ A and m ∈ M is a left A -module homomorphism.Then β M , M (cid:48) ( m ⊗ m (cid:48) ) = β M , M (cid:48) (( φ m ⊗ φ m (cid:48) )(1 ⊗ φ m (cid:48) ⊗ φ m )( β A⊗A (1 ⊗ φ m (cid:48) ⊗ φ m )( τ A , A ( R ))= τ M , M (cid:48) ( R · ( m ⊗ m (cid:48) ))follows, since τ is a natural transformation. Using this we prove∆( a ) τ A , A ( R ) = a · ( β A , A (1 ⊗ β A , A ( a · (1 ⊗ τ A , A ( R · ∆( a )) , uasi-Triangular Hopf Algebras and their Representations 33since β is a left A -module homomorphism on objects. Applying the flip isomorphismon both sides of the above equation leads to equation (2.9). By making use of thehexagon relations of β we obtain R , = α A , A , A ( β A , A⊗A ( α A , A , A ((1 ⊗ ⊗ A ⊗ β A , A )( α A , A , A (( β A , A ⊗ id A )((1 ⊗ ⊗ R R and R , = α − A , A , A ( β A⊗A , A ( α − A , A , A (1 ⊗ (1 ⊗ β A , A ⊗ id A )( α − A , A , A ((id A ⊗ β A , A )(1 ⊗ (1 ⊗ R R , concluding that R satisfies the equations (2.10): for the first we perform a tensorshift (123) → (231) to obtain R , = R R and a shift (123) → (312) for thesecond equation R , = R R . Let on the other hand A be a bialgebra andassume the existence of an invertible element R ∈ A ⊗ A satisfying equations (2.9)and (2.10). For two arbitrary left A -modules M and M (cid:48) we define an isomorphism β M , M (cid:48) : M ⊗ M (cid:48) → M (cid:48) ⊗ M of left A -modules by β M , M (cid:48) ( m ⊗ m (cid:48) ) = τ M , M (cid:48) ( R · ( m ⊗ m (cid:48) ))for all m ∈ M and m (cid:48) ∈ M (cid:48) . In fact, for all a ∈ A one obtains β M , M (cid:48) ( a · ( m ⊗ m (cid:48) )) = τ M , M (cid:48) (( R ∆( a )) · ( m ⊗ m (cid:48) ))= τ M , M (cid:48) ((∆ ( a ) R ) · ( m ⊗ m (cid:48) ))=∆( a ) · τ M , M (cid:48) ( R · ( m ⊗ m (cid:48) ))= a · β M , M (cid:48) ( m ⊗ m (cid:48) )by property (2.9) and the inverse of β M , M (cid:48) is given by M (cid:48) ⊗ M (cid:51) ( m ⊗ m (cid:48) ) (cid:55)→ R − · ( τ M (cid:48) , M ( m ⊗ m (cid:48) )) ∈ M ⊗ M (cid:48) . This implies that β : ⊗ → ⊗ ◦ τ is a natural transformation. It remains to provethat hexagon relations for β and it is not surprising that they are following by thehexagon relations of R . In detail we obtain α M (cid:48) , M (cid:48)(cid:48) , M ( β M , M (cid:48) ⊗M (cid:48)(cid:48) ( α M , M (cid:48) , M (cid:48)(cid:48) (( m ⊗ m (cid:48) ) ⊗ m (cid:48)(cid:48) )))= α M (cid:48) , M (cid:48)(cid:48) , M ( τ M , M (cid:48) ⊗M (cid:48)(cid:48) (( R · m ) ⊗ (( R · m (cid:48) ) ⊗ ( R · m (cid:48)(cid:48) ))))=( R · m (cid:48) ) ⊗ (( R · m (cid:48)(cid:48) ) ⊗ ( R · m ))=( R · m (cid:48) ) ⊗ (( R (cid:48) · m (cid:48)(cid:48) ) ⊗ (( R (cid:48) R ) · m ))=(id M (cid:48) ⊗ β M , M (cid:48)(cid:48) )(( R · m (cid:48) ) ⊗ (( R · m ) ⊗ m (cid:48)(cid:48) ))=(id M (cid:48) ⊗ β M , M (cid:48)(cid:48) )( α M (cid:48) , M , M (cid:48)(cid:48) (( β M , M (cid:48) ⊗ id M (cid:48)(cid:48) )(( m ⊗ m (cid:48) ) ⊗ m (cid:48)(cid:48) )))4 Chapter 2and α − M (cid:48)(cid:48) , M , M (cid:48) ( β M⊗M (cid:48) , M (cid:48)(cid:48) ( α − M , M (cid:48) , M (cid:48)(cid:48) ( m ⊗ ( m (cid:48) ⊗ m (cid:48)(cid:48) ))))= α − M (cid:48)(cid:48) , M , M (cid:48) (( R · m (cid:48)(cid:48) ) ⊗ (( R · m ) ⊗ ( R · m (cid:48) )))=(( R · m (cid:48)(cid:48) ) ⊗ ( R · m )) ⊗ ( R · m (cid:48) )=((( R (cid:48) R ) · m (cid:48)(cid:48) ) ⊗ ( R (cid:48) · m )) ⊗ ( R · m (cid:48) )=( β M , M (cid:48)(cid:48) ⊗ id M (cid:48) )(( m ⊗ ( R · m (cid:48)(cid:48) )) ⊗ ( R · m (cid:48) ))=( β M , M (cid:48)(cid:48) ⊗ id M (cid:48) ) α − M , M (cid:48)(cid:48) , M (cid:48) (id M ⊗ β M (cid:48) , M (cid:48)(cid:48) )( m ⊗ ( m (cid:48) ⊗ m (cid:48)(cid:48) ))for another left A -module M (cid:48)(cid:48) , where we used equations (2.10) in leg notation, i.e. R ⊗ R ⊗ R = ( R (cid:48) R ) ⊗ R ⊗ R (cid:48) and R ⊗ R ⊗ R = R (cid:48) ⊗ R ⊗ ( R (cid:48) R ) . This concludes the characterization of quasi-triangular bialgebras.Every universal R -matrix satisfies an additional equation, connecting it to thetheory of integrable systems (see [104] for more information). Corollary 2.3.3 (QYBE) . A universal R -matrix R on a quasi-triangular bialgebrasatisfies the quantum Yang-Baxter equation R R R = R R R . Proof.
Using (2.9) and (2.10) we obtain R R R = R R , = R , R = R R R . Definition 2.3.4 (Quasi-Triangular Hopf Algebra) . A Hopf algebra is said to be(quasi-)triangular if its underlying bialgebra is.
We conclude this section with a specification of Proposition 2.2.4 in the settingof quasi-triangular Hopf algebras (compare also to [35] Ex. 5.1.4).
Corollary 2.3.5.
Let H be a quasi-triangular Hopf algebra. Then H M f is a rigidbraided monoidal category.Proof. From Proposition 2.2.4 it follows that H M f is a rigid monoidal category andby definition H M is braided. Since H M f is a monoidal subcategory of H M thelatter also applies to H M f . In this section we discuss gauge transformations of quasi-triangular bialgebras. Thecorresponding algebraic tool is given by a normalized 2-cocycle, called Drinfel’dtwist. Following [77] Sec. 2.3, we elaborate how to twist deform the coproductsuch that the deformed structure still corresponds to a quasi-triangular bialgebra.uasi-Triangular Hopf Algebras and their Representations 35Similarly, one twist deforms bialgebra modules, which leads to the definition of theDrinfel’d functor. We prove that this functor is braided monoidal and gives rise toa braided monoidal equivalence of the representation theory of the deformed andundeformed bialgebra. A twist deformation of antipodes is incorporated in the nextsection. We also refer to [68] Sec. XV.3. In the following, B denotes a bialgebra overa commutative ring k with coproduct ∆ and counit (cid:15) . Definition 2.4.1 (Drinfel’d Twist) . An invertible element
F ∈ B ⊗ B is said tobe a Drinfel’d twist, or twist for short, if it is normalized, i.e. ( (cid:15) ⊗ id)( F ) = 1 =(id ⊗ (cid:15) )( F ) and satisfies the -cocycle condition ( F ⊗ ⊗ id)( F ) = (1 ⊗ F )(id ⊗ ∆)( F ) . The original definition goes back to Drinfel’d [42], where the author introducestwists as quantizations of solutions of the classical Yang-Baxter equation. Thosetwists where considered as elements on (formal power series of) universal envelop-ing algebras of Lie algebras (see also Section 3.2). The generalization to arbitrarybialgebras was undertaken by Giaquinto and Zhang in [60]. Drinfel’d twists onbialgebroids are considered in e.g. [25, 113]. We start by giving some examples ofDrinfel’d twists to convince the reader of the richness of this concept.
Example 2.4.2. i.) On every bialgebra B there is a twist given by the unit ele-ment ⊗ . We refer to it as the trivial twist in the following;ii.) [c.f. [60] Thm. 2.1] Let k be a commutative ring such that Q ⊆ k and considera commutative bialgebra B . Denote the primitive elements of B by P . Then exp( (cid:126) r ) = ∞ (cid:88) n =0 (cid:126) n n ! r n ∈ ( B ⊗ B )[[ (cid:126) ]] is a twist on B [[ (cid:126) ]] for any r ∈ P ⊗ P . Here B [[ (cid:126) ]] is a topologically free moduleand we consider the completed tensor product (c.f. [50] Sec. 1.1);Proof. Let r = r ⊗ r ∈ P ⊗ P . In particular ( (cid:15) ⊗ id)( r ) = 0 = (id ⊗ (cid:15) )( r ) andsince (cid:15) is an algebra homomorphism this implies that exp( (cid:126) r ) is normalized.Then(∆ ⊗ id) exp( (cid:126) r ) = ∞ (cid:88) n =0 (cid:126) n n ! ∆( r n ) ⊗ r n = ∞ (cid:88) n =0 (cid:126) n n ! ∆( r ) n ⊗ r n = exp((∆ ⊗ id)( (cid:126) r ))since ∆ is an algebra homomorphism and similarly (id ⊗ ∆) exp( (cid:126) r ) = exp((id ⊗ ∆)( (cid:126) r )) follows. Since B is commutative, the Baker-Campbell-Hausdorff seriesof B [[ (cid:126) ]] is trivial and(∆ ⊗ id)(exp( (cid:126) r ))(exp( (cid:126) r ) ⊗
1) = exp((∆ ⊗ id)( (cid:126) r )) exp( (cid:126) r ⊗ ⊗ id)( (cid:126) r ) + (cid:126) r ⊗ ⊗ ∆)(exp( (cid:126) r ))(1 ⊗ exp( (cid:126) r )) = exp((id ⊗ ∆)( (cid:126) r ) + 1 ⊗ (cid:126) r ) follow.Now r , r ∈ P , which implies that (∆ ⊗ id)( (cid:126) r )+ (cid:126) r ⊗ ⊗ ∆)( (cid:126) r )+1 ⊗ (cid:126) r .This proves that exp( (cid:126) r ) is a twist on B [[ (cid:126) ]].6 Chapter 2 iii.) A variation of ii.) is described in [95]: let k be a commutative ring such that Q ⊆ k and consider a Lie algebra g over k together with a set of elements x , . . . , x n , y , . . . , y n ∈ g , where n ∈ N , such that [ x i , x j ] = [ x i , y j ] = [ y i , y j ] =0 for all ≤ i ≤ n . Then exp( (cid:126) r ) ∈ ( U g ⊗ U g )[[ (cid:126) ]] is a twist on U g [[ (cid:126) ]] ,where r = (cid:80) ni =1 x i ⊗ y i . In this case we refer to exp( (cid:126) r ) as an Abelian twist;iv.) [c.f. [60] Thm. 2.10] Let k be a commutative ring such that Q ⊆ k and considerthe Lie algebra g over k which is generated by two elements H, E ∈ g such that [ H, E ] = 2 E . Then ∞ (cid:88) n =0 n (cid:88) m =0 (cid:126) n n ! ( − m (cid:18) nm (cid:19) E n − m H (cid:104) m (cid:105) ⊗ E m H (cid:104) n − m (cid:105) ∈ ( U g ⊗ U g )[[ (cid:126) ]] is a twist on U g [[ (cid:126) ]] , where H (cid:104) m (cid:105) is defined inductively by H (cid:104) (cid:105) = 1 and H (cid:104) m +1 (cid:105) = H ( H + 1) · · · ( H + m ) ;v.) [c.f. [51, 86]] In the same setting as in iv.) there is a twist on U g [[ (cid:126) ]] givenby exp (cid:18) H ⊗ log(1 + (cid:126) E ) (cid:19) ∈ ( U g ⊗ U g )[[ (cid:126) ]] , which is known as Jordanian twist. We further refer to [7] Sec. 2.3.3 andreferences therein for a generalization of Jordanian twist and to [24, 80] formore recent discussions on Jordanian twists; Besides the trivial twist, all Drinfel’d twists we discussed above are formal powerseries with entries in a bialgebra or Hopf algebra. Further remark that all of themare the identity in zero order of the formal parameter. In fact, many examples ofDrinfel’d twists occur in the context of deformation quantization, which naturallyutilizes formal power series. Furthermore in the primordial article [42] Drinfel’dintroduced twists F = (cid:80) ∞ n =0 (cid:126) n F n ∈ U g ⊗ [[ (cid:126) ]] on formal power series of universalenveloping algebras of a real or complex Lie algebra g , such that F = 1 ⊗
1. Thefirst order term F satisfies the so-called classical Yang-Baxter equation and Drinfel’dproved in the mentioned article that conversely any solution of the classical Yang-Baxter equation on g can be realized as the first order term of a twist on U g [[ (cid:126) ]]starting with the identity. We revive this thought in Section 3.2, discussing thecorrespondence of (formal) Drinfel’d twists and their first order in detail. Howeverfor the rest of this section we come back to discuss general properties of Drinfel’dtwists and how they deform the underlying quasi-triangular bialgebra structure. Lemma 2.4.3 (Inverse Twist) . Let F be a twist on B . Its inverse F − is normalized,i.e. ( (cid:15) ⊗ id)( F − ) = 1 = (id ⊗ (cid:15) )( F − ) and satisfies the so called inverse -cocyclecondition (∆ ⊗ id)( F − )( F − ⊗
1) = (id ⊗ ∆)( F − )(1 ⊗ F − ) . (2.12)As already indicated, twist are deformation tools, leading to compatibility on acategorical level. We start by explaining how the twist deforms the underlying bial-gebra B before passing to module algebras and more general to equivariant bialgebramodule algebra modules in the next section. We define the twisted coproduct ∆ F ( ξ ) = F ∆( ξ ) F − (2.13)for all ξ ∈ B .uasi-Triangular Hopf Algebras and their Representations 37 Proposition 2.4.4.
Let F be a twist on B . Then B F = ( B , µ, η, ∆ F , (cid:15) ) is a bialgebra.If B is quasi-triangular with universal R -matrix R , so is B F with quasi-triangularstructure R F = F RF − . (2.14) If R is triangular, so is R F .Proof. We split the proof into four parts.i.) (∆ F , (cid:15) ) is a coalgebra structure on B : let ξ ∈ B . Then( (cid:15) ⊗ id)∆ F ( ξ ) = (cid:15) ( F ) (cid:15) ( ξ (1) ) (cid:15) ( F (cid:48) − ) F ξ (2) F (cid:48) − = (cid:15) ( F ) F (cid:15) ( ξ (1) ) ξ (2) (cid:15) ( F (cid:48) − ) F (cid:48) − = ξ and similarly one proves (id ⊗ (cid:15) )∆ F ( ξ ) = ξ .ii.) ∆ F is an algebra homomorphism: let ξ, χ ∈ B . Then∆ F ( ξχ ) = F ∆( ξχ ) F − = F ∆( ξ ) F − F ∆( χ ) F − = ∆ F ( ξ )∆ F ( χ )and ∆ F (1) = F (1 ⊗ F − = 1 ⊗
1, i.e. ∆ F respects the algebra structure of B and B ⊗ B .iii.) R F is a quasi-triangular structure: let R be a universal R -matrix on B and ξ ∈ B . Then∆ F ( ξ ) = F ∆ ( ξ ) F − = F R ∆( ξ ) R − F − = R F ∆ F ( ξ ) R − F proves that ∆ F is quasi-cocommutative with respect to R F . Moreover(∆ F ⊗ id)( R F ) = F (∆ ⊗ id)( F )(∆ ⊗ id)( R )(∆ ⊗ id)( F − ) F − = F F , R R F − , F − = F F , R R F − , F − = F R F , F − , R F − = R F F F , F − , F − R F = R F R F and (id ⊗ ∆ F )( R F ) = F (id ⊗ ∆)( F )(id ⊗ ∆)( R )(id ⊗ ∆)( F − ) F − = F F , R R F − , F − = R F F F , F − , F − R F = R F R F are the hexagon relations of R F with respect to B F .iv.) If R is triangular so is R F : assume that R is triangular. Then R F = F R F − = F R − F − = R − F , i.e. R F is triangular, too.8 Chapter 2This is all we need to conclude the proposition.The bialgebra constructed in Proposition 2.4.4 is called twisted bialgebra and wedenote it by B F . In the following we use Sweedler’s notation ξ (cid:99) (1) ⊗ ξ (cid:99) (2) = ∆ F ( ξ ) todenote the twisted coproduct of ξ ∈ B . One might ask if the process of twisting isreversible and what happens if one deforms repetitively. Both issues are discussedin the following lemma. Lemma 2.4.5. If F is a Drinfel’d twist on B and F (cid:48) a Drinfel’d twist on B F then F (cid:48) F is a Drinfel’d twist on B such that B F (cid:48) F = ( B F ) F (cid:48) . Furthermore, F − is a Drinfel’d twist on B F and ( B F ) F − = B is an equality of bialgebras.Proof. We prove that F (cid:48) F is a Drinfel’d twist on B . Since (cid:15) is an algebra homo-morphism we have ( (cid:15) ⊗ id)( F (cid:48) F ) = 1 = (id ⊗ (cid:15) )( F (cid:48) F ), which means that F (cid:48) F isnormalized. Furthermore,( F (cid:48) F ⊗ ⊗ id)( F (cid:48) F ) =( F (cid:48) F ⊗ ⊗ id)( F (cid:48) )( F − ⊗ F ⊗ ⊗ id)( F )=(1 ⊗ F (cid:48) F )(id ⊗ ∆)( F (cid:48) )(1 ⊗ F − )( F ⊗ ⊗ id)( F )=(1 ⊗ F (cid:48) F )(id ⊗ ∆)( F (cid:48) )(id ⊗ ∆)( F )=(1 ⊗ F (cid:48) F )(id ⊗ ∆)( F (cid:48) F )proves that F (cid:48) F satisfies the 2-cocycle condition with respect to B . Moreover, F − is a Drinfel’d twist on B F , since F − is normalized and( F − ⊗ F ⊗ id)( F − ) =(∆ ⊗ id)( F − )( F − ⊗ ⊗ ∆)( F − )(1 ⊗ F − )=(1 ⊗ F − )(id ⊗ ∆ F )( F − )by the inverse 2-cocycle property. Since B ⊗ = B we conclude the proof of thelemma.Fix a Drinfel’d twist F on B . In Proposition 2.4.4 we proved that B F is abialgebra, so according to Proposition 2.2.2 B M and B F M are both monoidal sub-categories of k M . Ignoring the monoidal structures for a while we can define anassignment Drin F : B M → B F M (2.15)to be the identity on objects and morphisms. It assigns to any object M in B M itself, however seen as a left B F -module and to any left B -module homomorphism φ : M → M (cid:48) itself, however viewed as a left B F -module homomorphism. Thisassignment is well-defined since B and B F coincide as algebras. To distinguish thosetwo pictures we write Drin F ( M ) = M F and Drin F ( φ ) = φ F . uasi-Triangular Hopf Algebras and their Representations 39Since Drin F is the identity on objects and morphisms it follows that Drin F is afunctor. We prove that it is even a (braided) monoidal functor, leading to a (braided)monoidal equivalence. Recall that for two left B -modules M and M (cid:48) , the tensorproduct M ⊗ M (cid:48) is again a left B -module, or equivalently a left B F -module. As a k -module it coincides with the left B F -module M F ⊗ F M (cid:48)F , where we denote themonoidal structure on B F M by ⊗ F . However the left B F -actions differ, since for all ξ ∈ B , m ∈ M and m (cid:48) ∈ M (cid:48) one obtains ξ · ( m ⊗ m (cid:48) ) = ( ξ (1) · m ) ⊗ ( ξ (2) · m (cid:48) )while ξ · ( m ⊗ F m (cid:48) ) = ( ξ (cid:99) (1) · m ) ⊗ F ( ξ (cid:99) (2) · m (cid:48) ) . To compare these two pictures we define a left B F -module isomorphism ϕ M , M (cid:48) : M F ⊗ F M (cid:48)F (cid:51) ( m ⊗ F m (cid:48) ) (cid:55)→ ( F − · m ) ⊗ ( F − · m (cid:48) ) ∈ ( M ⊗ M (cid:48) ) F . It functions as the natural transformation.
Theorem 2.4.6 (c.f. [68] Lem. XV.3.7) . Let F be a Drinfel’d twist on B . Thefunctor Drin F : ( B M , ⊗ ) → ( B F M , ⊗ F ) is monoidal and the monoidal categories ( B M , ⊗ ) and ( B F M , ⊗ F ) are monoidallyequivalent. If B is quasi-triangular, Drin F is a braided monoidal functor and thetwo categories are braided monoidally equivalent.Proof. Let M , M (cid:48) and M (cid:48)(cid:48) be objects in B M . By the inverse 2-cocycle property of F − we deduce that the diagram M F ⊗ F M (cid:48)F ⊗ F M (cid:48)(cid:48)F M F ⊗ F ( M (cid:48) ⊗ M (cid:48)(cid:48) ) F ( M ⊗ M (cid:48) ) F ⊗ F M (cid:48)(cid:48)F ( M ⊗ M (cid:48) ⊗ M (cid:48)(cid:48) ) F id ⊗ F ϕ M(cid:48) , M(cid:48)(cid:48) ϕ M , M(cid:48) ⊗ F id ϕ M , M(cid:48)⊗M(cid:48)(cid:48) ϕ M⊗M(cid:48) , M(cid:48)(cid:48) commutes, while the normalization property of F − is sufficient to make k F ⊗ F M F ( k ⊗ M ) F M F ϕ k , M ∼ = ∼ = , M F ⊗ F k F ( M ⊗ k ) F M F ϕ M , k ∼ = ∼ = commute. This proves that Drin F is a monoidal functor. Assume now that B isquasi-triangular with universal R -matrix R ∈ B ⊗ B . Equivalently, ( B M , ⊗ ) isbraided monoidal with braiding β M , M (cid:48) : M ⊗ M (cid:48) (cid:51) ( m ⊗ m (cid:48) ) (cid:55)→ R − · ( m (cid:48) ⊗ m ) ∈ M (cid:48) ⊗ M . This was proven in Proposition 2.3.2. Furthermore ( B F , ⊗ F ) is braided monoidalwith braiding β FM F , M (cid:48)F : M F ⊗ F M (cid:48)F (cid:51) ( m ⊗ F m (cid:48) ) (cid:55)→ R − F · ( m (cid:48) ⊗ F m ) ∈ M (cid:48)F ⊗ F M F R F = F RF − is the quasi-triangular struc-ture on B F . Then M F ⊗ F M (cid:48)F ( M ⊗ M (cid:48) ) F M (cid:48)F ⊗ F M F ( M (cid:48) ⊗ M ) F ϕ M , M(cid:48) β FMF , M(cid:48)F
Drin F ( β M , M(cid:48) ) ϕ M(cid:48) , M commutes by definition, proving that Drin F is a braided monoidal functor. RecallingLemma 2.4.5, the inverse of Drin F is given byDrin F − : ( B F M , ⊗ F ) → ( B M , ⊗ ) . This concludes the proof of the theorem.
In this section we discuss bialgebra module algebras, i.e. bialgebra modules with anadditional algebra structure that is respected by the bialgebra action. We prove thatthe Drinfel’d functor restricts to an isomorphism of the equivariant algebra mod-ules [60]. Passing to equivariant algebra bimodules this even becomes a monoidalequivalence if one considers the tensor product over the algebra. In the end weare interested in braided commutative algebras and equivariant algebra bimoduleswhich are braided symmetric. As concrete examples one may think of twist starproduct algebras together with twisted multivector fields and differential forms (seeChapter 4). We obtain a braided monoidal equivalence via the Drinfel’d functor inthis case (c.f. [6, 9]). At the end of this section we also include twist deformationsof antipodes and rigid categories. Since there are two different module actions in-volved in the following, one from the bialgebra and one from them module algebra,we denote the first by (cid:66) and the latter by · for convenience. In a first step we provethat a twist on a bialgebra B deforms the category of left B -module algebras. Definition 2.5.1. A k -algebra A is said to be a left B -module algebra of there is a k -linear map (cid:66) : B ⊗ A → A , structuring A as a left B -module such that ξ (cid:66) ( a · b ) = ( ξ (1) (cid:66) a ) · ( ξ (2) (cid:66) b ) and ξ (cid:66) (cid:15) ( ξ )1 hold for all ξ ∈ B and a, b ∈ A . A left B -module algebra homomorphism is an algebrahomomorphism between left B -module algebras which is a left B -module homomor-phism in addition. This constitutes the category B A of left B -module algebras. In other words, the left B -module action is respecting the algebra structure of A .Fix a Drinfel’d twist F on B and a left B -module algebra A for now. Since B and B F are representatives of the same gauge equivalence class it is natural to ask if also B F respects the algebra structure of A . However, since ∆ differs from ∆ F in general,this can not be expected. On the other hand there is a way to gauge transform A using the twist F , such that B F respects the gauge transformed algebra.uasi-Triangular Hopf Algebras and their Representations 41 Proposition 2.5.2 (c.f. [6] Thm. 3.4) . Let F be a twist on B and consider an object A ∈ B A with product · and unit . Then A F = ( A , · F , is an (associative unital)algebra, where a (cid:63) F b = ( F − (cid:66) a ) · ( F − (cid:66) b ) (2.16) for all a, b ∈ A . Moreover, A F is an object in B F A with respect to the same Hopfalgebra action, i.e. it is a left B F -module algebra.Proof. We split the proof into two parts. Denote µ F = · F and the product and unitof A by µ A and η A , respectively.i.) A F is an associative unital algebra: let a, b, c ∈ A . Then µ F ( µ F ( a ⊗ b ) ⊗ c ) = µ A ( µ A ((( F − F (cid:48) − ) (cid:66) a ) ⊗ (( F − F (cid:48) − ) (cid:66) b )) ⊗ ( F − (cid:66) c ))= µ A (( F − (cid:66) a ) ⊗ µ A ((( F − F (cid:48) − ) (cid:66) b ) ⊗ (( F − F (cid:48) − ) (cid:66) c )))= µ F ( a ⊗ µ F ( b ⊗ c ))shows that µ F is associative, where we made use of the associativity of µ A andthe 2-cocycle property of F − . Furthermore, µ F is unital with respect to η A ,since for all λ ∈ k ( µ F ◦ (id ⊗ η A ))( a ⊗ λ ) = λµ A (( F − (cid:66) a ) ⊗ ( F − (cid:66) λµ A ((( F − (cid:15) ( F − )) (cid:66) a ) ⊗ λµ A ((1 (cid:66) a ) ⊗ λa and similarly ( µ F ◦ ( η A ⊗ id))( λ ⊗ a ) = λa follows. Here we used that F − isnormalized and that µ A is unital with respect to η A .ii.) (cid:66) : B F ⊗ A F → A F respects the algebra structure of A F : let ξ ∈ B and a, b ∈ A . Then ξ (cid:66) µ F ( a ⊗ b ) = µ A ((( F (cid:48)(cid:48) − F (cid:48) ξ (1) F − ) (cid:66) a ) ⊗ (( F (cid:48)(cid:48) − F (cid:48) ξ (2) F − ) (cid:66) b ))= µ F (( ξ (cid:99) (1) (cid:66) a ) ⊗ ( ξ (cid:99) (2) (cid:66) b ))and ξ (cid:66) (cid:15) ( ξ )1 are satisfied.Since A F equals A as a k -module and B F equals B as an algebra this is all we haveto prove.More generally, we are able to deform the category of B -equivariant left A -modules. Definition 2.5.3.
Let A be a left B -module algebra and consider the left B -modules M which are left A -modules in addition, such that ξ (cid:66) ( a · m ) = ( ξ (1) (cid:66) a ) · ( ξ (2) (cid:66) m ) (2.17) holds for all ξ ∈ B , a ∈ A and m ∈ M . They are said to be B -equivariant left A -modules, forming a category BA M with morphisms being left B -linear and left A -linear maps between B -equivariant left A -modules. Proposition 2.5.4 (c.f. [6] Thm. 3.5) . Let F be a twist on B and consider a left B -module algebra A . For every object M in BA M the twisted left A F -module action a • F m = ( F − (cid:66) a ) · ( F − (cid:66) m ) , (2.18) where a ∈ A and m ∈ M , structures M as an B F -equivariant left A F -module. The B F -equivariant left A F -module M with module action • F is denoted by M F .If the left A -module action is trivial the assignment of Proposition 2.5.4 reduces tothe Drinfel’d functor Drin F : B M → B F M . On the other hand we can extend theDrinfel’d functor to B -equivariant left A -modules. Proposition 2.5.5 (c.f. [9] Prop. 3.9) . Let
A ∈ B A and F be a twist on B . Thereis a functor Drin F : BA M → B F A F M (2.19) from the category of B -equivariant left A -modules to the category of B F -equivariantleft A F -modules. It is the identity on morphisms and assigns to any B -equivariantleft A -module M the B F -equivariant left A F -module M F defined in Proposition 2.5.4.Furthermore, Drin F is an isomorphism of categories.Proof. By Proposition 2.5.4 the assignment Drin F is well-defined on objects. Toprove that it is well-defined on morphisms consider a left A -module homomorphism φ : M → M (cid:48) between B -equivariant left A -module homomorphisms, which is also aleft B -module homomorphism. Let ξ ∈ B , a ∈ A and m ∈ M . Then φ ( a • F m ) = φ (( F − (cid:66) a ) · ( F − (cid:66) m ))=( F − (cid:66) a ) · φ ( F − (cid:66) m )=( F − (cid:66) a ) · ( F − (cid:66) φ ( m ))= a • F φ ( m ) , while φ ( ξ (cid:66) m ) = ξ (cid:66) φ ( m ) holds by definition. Consequently, Drin F is also well-defined on morphisms. The functorial properties are clear since the concatenationof morphisms is well-defined. There is an inverse functorDrin F − : B F A F M → BA M , being the identity on morphisms and assigning to any B F -equivariant left A F -module( M , • ) the same left B -module but with left A -module structure a · m = ( F (cid:66) a ) • ( F (cid:66) m )for all a ∈ A and m ∈ M . In fact, this defines a left A -module action, since for all a, b ∈ A and m ∈ M one obtains a · ( b · m ) = a · (( F (cid:66) b ) • ( F (cid:66) m ))=( F (cid:48) (cid:66) a ) • ( F (cid:48) (cid:66) (( F (cid:66) b ) • ( F (cid:66) m )))=( F (cid:48) (cid:66) a ) • ((( F (cid:48) (cid:99) (1) F ) (cid:66) b ) • (( F (cid:48) (cid:99) (2) F ) (cid:66) m ))=( F (cid:48) (cid:66) a ) • ((( F F (cid:48) ) (cid:66) b ) • (( F F (cid:48) ) (cid:66) m ))uasi-Triangular Hopf Algebras and their Representations 43=(( F F (cid:48) ) (cid:66) a ) • ((( F F (cid:48) ) (cid:66) b ) • ( F (cid:48) (cid:66) m ))=((( F F (cid:48) ) (cid:66) a ) (cid:63) F (( F F (cid:48) ) (cid:66) b )) • ( F (cid:48) (cid:66) m )=(( F (cid:48) (cid:66) a ) · ( F (cid:48) (cid:66) b )) • ( F (cid:48) (cid:66) m )=( F (cid:48) (cid:66) ( a · b )) • ( F (cid:48) (cid:66) m )=( a · b ) · m and 1 · m = ( F (cid:66) • ( F (cid:66) m ) = m . Moreover, M is B -equivariant, since ξ (cid:66) ( a · m ) = ξ (cid:66) (( F (cid:66) a ) • ( F (cid:66) m ))=(( ξ (cid:99) (1) F ) (cid:66) a ) • (( ξ (cid:99) (2) F ) (cid:66) m )=(( F ξ (1) ) (cid:66) a ) • (( F ξ (2) ) (cid:66) m )=( ξ (1) (cid:66) a ) · ( ξ (2) (cid:66) m )for all ξ ∈ B , a ∈ A and m ∈ M . We denote M with the module action · by M F − . Any morphisms φ : M → M (cid:48) in B F A F M can be viewed as a morphisms φ : M F − → M (cid:48) F − in BA M . While the left B -linearity is clear we check that φ ( a · m ) = φ (( F (cid:66) a ) • ( F (cid:66) m ))=( F (cid:66) a ) • φ ( F (cid:66) m )=( F (cid:66) a ) • ( F (cid:66) φ ( m ))= a · ( φ ( m ))holds in addition for all a ∈ A and m ∈ M F − . The functors Drin F and Drin F − are inverse to each other: let M be on object in BA M . Then ( M F ) F − = M asobjects in BA M , since ( F (cid:66) a ) • F ( F (cid:66) m ) = a · m for all a ∈ A and m ∈ M . On the other hand, let M be an object in B F A F M . Then( M F − ) F = M as objects in B F A F M . To see this let a ∈ A and m ∈ M be arbitraryand consider ( F − (cid:66) a ) · ( F − (cid:66) m ) = a • m. On the level of morphisms there is nothing to prove. This concludes the proof ofthe proposition.The natural question arises if the functor from Proposition 2.5.5 is still (braided)monoidal. This has to be negated, since BA M is not monoidal in general. We needtwo further specifications, the first being to consider B -equivariant A -bimodules.Namely, we consider the subcategory BA M A of BA M , consisting of those objects M which inherit an additional right A -module action, which commutes with the left A -module action such that ξ (cid:66) ( m · a ) = ( ξ (1) (cid:66) m ) · ( ξ (2) (cid:66) a )for all ξ ∈ B , m ∈ M and a ∈ A . Morphisms in BA M A are left B -linear and left andright A -linear maps. In a second step we replace the tensor product ⊗ of k -modules4 Chapter 2with the tensor product ⊗ A over A . Namely, for two objects M and M (cid:48) in BA M A ,the product M ⊗ A M (cid:48) is defined by the quotient M ⊗ M (cid:48) / N M , M (cid:48) , where N M , M (cid:48) is the ideal in M ⊗ M (cid:48) , defined by the image of ρ M ⊗ id M (cid:48) − id M ⊗ λ M (cid:48) , where ρ M and λ M (cid:48) denote the right and left A -module action on M and M (cid:48) ,respectively. In particular this implies( m · a ) ⊗ A m (cid:48) = m ⊗ A ( a · m (cid:48) )for all a ∈ A , m ∈ M and m (cid:48) ∈ M (cid:48) . Furthermore, M ⊗ A M (cid:48) is a B -equivariant A -bimodule with induced left B -action and left and right A -actions given for all a ∈ A , m ∈ M and m (cid:48) ∈ M (cid:48) by a · ( m ⊗ A m (cid:48) ) = ( a · m ) ⊗ A m (cid:48) and ( m ⊗ A m (cid:48) ) · a = m ⊗ A ( m (cid:48) · a ) , respectively. On morphisms φ : M → N and ψ : M (cid:48) → N (cid:48) of BA M A one defines( φ ⊗ A ψ )( m ⊗ A m (cid:48) ) = φ ( m ) ⊗ A ψ ( m (cid:48) ) for all m ∈ M and m (cid:48) ∈ M (cid:48) . This impliesthe following statement. Lemma 2.5.6 (c.f. [9] Prop. 3.11) . The tuple ( BA M A , ⊗ A ) is a monoidal categoryand the Drinfel’d functor Drin F : ( BA M A , ⊗ A ) → ( B F A F M A F , ⊗ A F ) (2.20) is monoidal, leading to a monoidal equivalence. However, the monoidal functor (2.20) still fails to be braided monoidal in general.In fact ( BA M A , ⊗ A ) is not even braided in general if B is quasi-triangular. The B -equivariant A -bimodules are still too arbitrary. We have to demand even moresymmetry before. We do so by considering a braided commutative (also called quasi-commutative ) left B -module algebra A for a triangular bialgebra ( B , R ) instead ofa general left B -module algebra. This means that b · a = ( R − (cid:66) a ) · ( R − (cid:66) b ) holdsfor all elements a, b of a left B -module algebra ( A , · ). On the level of A -bimoduleswe want to keep this symmetry. Definition 2.5.7.
Let ( B , R ) be a triangular bialgebra and ( A , · ) be a braided com-mutative left B -module algebra. A B -equivariant A -bimodule M is said to be braidedsymmetric if a · m = ( R − (cid:66) m ) · ( R − (cid:66) a ) for all a ∈ A and m ∈ M . Their morphisms are left B -linear maps which are leftand right A -linear in addition. We denote the category of B -equivariant braidedsymmetric A -bimodules by BA M RA . In other words, the left and right A -module actions are related via the universal R -matrix R , mirroring the braided commutativity of A . We proceed by provingthe main theorem of this section (c.f. [9] Thm. 3.13). It states that the Drinfel’dfunctor is braided monoidal on equivariant braided symmetric bimodules.uasi-Triangular Hopf Algebras and their Representations 45 Theorem 2.5.8.
Let ( B , R ) be a triangular bialgebra and F a Drinfel’d twist on B .Then, the triple ( BA M RA , ⊗ A , β R ) is a braided monoidal category and the Drinfel’dfunctor Drin F : ( BA M RA , ⊗ A , β R ) → ( B F A F M R F A F , ⊗ A F , β F ) (2.21) is braided monoidal, leading to a braided monoidal equivalence.Proof. It is sufficient to prove that ⊗ A and Drin F are closed in the category ofequivariant braided symmetric bimodules. So let M and M (cid:48) be objects in BA M A .Then a · ( m ⊗ A m (cid:48) ) =( a · m ) ⊗ A m (cid:48) =(( R − (cid:66) m ) · ( R − (cid:66) a )) ⊗ A m (cid:48) =( R − (cid:66) m ) ⊗ A (( R − (cid:66) a ) · m (cid:48) )=( R − (cid:66) m ) ⊗ A (( R (cid:48) − (cid:66) m (cid:48) ) · (( R (cid:48) − R − ) (cid:66) a ))=( R − (cid:66) ( m ⊗ A m (cid:48) )) · ( R − (cid:66) a )for all a ∈ A , m ∈ M and m (cid:48) ∈ M (cid:48) . Furthermore k is a B -equivariant braidedsymmetric A -bimodule, since a · λ = λ · a = ( R − (cid:66) λ ) · ( R − (cid:66) a )for all a ∈ A and λ ∈ k . We further observe that a · F m =( F − (cid:66) a ) · ( F − (cid:66) m )=(( R − F − ) (cid:66) m ) · (( R − F − ) (cid:66) a )=( R − F (cid:66) m ) · F ( R − F (cid:66) a )for all a ∈ A and m ∈ M , proving that M F is an object in B F A F M R F A F .The rest of this section is devoted to include twist deformations of antipodes inthe picture. If there is an antipode S on B we define β = F S ( F ) ∈ B ⊗ B . Since B is a Hopf algebra in this case it is more convenient to write H instead of B . Lemma 2.5.9.
The element β is invertible with inverse given by β − = S ( F − ) F − .Proof. This is just a matter of computation using the axioms of Hopf algebra andDrinfel’d twist. Explicitly we obtain ββ − = F S ( F ) S ( F (cid:48) − ) F (cid:48) − = F (cid:48)(cid:48) − (cid:15) ( F (cid:48)(cid:48) − ) F S ( F ) S ( F (cid:48) − ) F (cid:48) − = F (cid:48)(cid:48) − F S ( F ) S ( F (cid:48) − ) S ( F (cid:48)(cid:48) − ) F (cid:48)(cid:48) − F (cid:48) − = F (cid:48)(cid:48) − F S ( F (cid:48)(cid:48) − F (cid:48) − F ) F (cid:48)(cid:48) − F (cid:48) − = F (cid:48)(cid:48) − F (cid:48) − F S ( F (cid:48)(cid:48) − F (cid:48) − F ) F (cid:48)(cid:48) − = F (cid:48)(cid:48) − S ( F (cid:48)(cid:48) − ) F (cid:48)(cid:48) − = (cid:15) ( F (cid:48)(cid:48) − ) F (cid:48)(cid:48) − =1and similarly β − β = 1.6 Chapter 2Let us define a k -linear map S F : H → H by S F ( ξ ) = βS ( ξ ) β − (2.22)for all ξ ∈ H . It is said to be the twisted antipode . Proposition 2.5.10.
Let ( H, µ, η, ∆ , (cid:15), S ) be a Hopf algebra and F a Drinfel’d twiston H . Then H F = ( H, µ, η, ∆ F , (cid:15), S F ) is a Hopf algebra. If R is a quasi-triangularstructure on H then R F is a quasi-triangular structure on H F . If R is triangular,so is R F .Proof. S F is an antipode on H with respect to ∆ F and (cid:15) since for all ξ ∈ H weobtain( µ ◦ ( S F ⊗ id) ◦ ∆ F )( ξ ) = S F ( F ξ (1) F (cid:48) − ) F ξ (2) F (cid:48) − = F (cid:48)(cid:48) S ( F (cid:48)(cid:48) ) S ( F ξ (1) F (cid:48) − ) S ( F (cid:48)(cid:48)(cid:48) − ) F (cid:48)(cid:48)(cid:48) − F ξ (2) F (cid:48) − = F (cid:48)(cid:48) S ( F (cid:48)(cid:48)(cid:48) − F ξ (1) F (cid:48) − F (cid:48)(cid:48) ) F (cid:48)(cid:48)(cid:48) − F ξ (2) F (cid:48) − = F (cid:48)(cid:48) S ( F (cid:48) − F (cid:48)(cid:48) ) S ( ξ (1) ) ξ (2) F (cid:48) − = (cid:15) ( ξ ) F (cid:48)(cid:48) S ( F (cid:48)(cid:48) ) S ( F (cid:48) − ) F (cid:48) − = (cid:15) ( ξ ) ββ − = (cid:15) ( ξ )1 , where we used Lemma 2.5.9. In complete analogy one proves µ ◦ (id ⊗ S F ) ◦ ∆ F = η ◦ (cid:15) .The other statements only involve the underlying bialgebra structure and have beenproven in Proposition 2.4.4.As a consequence of Theorem 2.5.8 we obtain the following result. Corollary 2.5.11.
Let H be quasi-triangular and let furthermore F be a Drinfel’dtwist on H . Then, the triple ( H A M RA , ⊗ A , β R ) is a braided monoidal category and theDrinfel’d functor Drin F : ( H A M RA , ⊗ A , β R ) → ( H F A F M R F A F , ⊗ A F , β F ) (2.23) is braided monoidal, leading to a braided monoidal equivalence. Restricting to finitelygenerated projective modules, similar results hold on the rigid subcategory. During the chapter we build up a setup to understand the braided monoidalcategory ( H A M RA , ⊗ A , β R ) and that (2.23) acts as a gauge equivalence on braidedsymmetric modules. In Chapter 4 we resume by building a noncommutative Cartancalculus for any braided commutative algebra in this category in a way that gaugeequivalence classes are respected. Before, in Chapter 3, we recall some concepts of deformation quantization and how they relate to Drinfel’d twist deformation. Inparticular we point out several cases which do not allow for a twist deformationquantization.uasi-Triangular Hopf Algebras and their Representations 47 In this short section we recall the notion of Hopf ∗ -algebras and their representations.In other words we discuss how to incorporate a ∗ -involution in the previous picture.The most remarkable innovation is that also the ∗ -involution admits a Drinfel’dtwist deformation if the twist is unitary . While we omit this additional structure inthe general framework we are utilizing it in the explicit example of Section 5.4. Asreference we name [56] Chap. 2 and [77] Sec. 1.7.Let k be a commutative unital ring endowed with a ∗ -involution , i.e. there isan (anti)automorphism k (cid:51) λ (cid:55)→ λ ∈ k , which is an involution in addition. Onelements λ, µ ∈ k this reads λ + µ = λ + µ,λµ = µλ = λµ,λ = λ, , where we used the commutativity of k in the second equation. Furthermore, a ∗ -algebra over k is an associative unital algebra ( A , · , A ) together with a ∗ -involution,i.e. a map ∗ : A → A such that( λa + µb ) ∗ = λa ∗ + µb ∗ , ( a · b ) ∗ = b ∗ · a ∗ , ( a ∗ ) ∗ = a, (1 A ) ∗ =1 A for all λ, µ ∈ k and a, b ∈ A . Note that in general ( a · b ) ∗ (cid:54) = a ∗ · b ∗ since A might be noncommutative. Coalgebras are defined as dual objects to algebras,while bialgebras unite both concepts in a compatible way and Hopf algebras inheritan additional antipode which is an anti-bialgebra homomorphism. In this light it isclear how to define Hopf ∗ -algebras . Definition 2.6.1. A k -Hopf algebra ( H, · , H , ∆ , (cid:15), S ) is said to be a Hopf ∗ -algebraif ( H, · , H ) is a ∗ -algebra with ∗ -involution ∗ : H → H such that ∆( ξ ∗ ) = ( ξ (1) ) ∗ ⊗ ( ξ (2) ) ∗ , (cid:15) ( ξ ∗ ) = (cid:15) ( ξ ) , and S (( S ( ξ ∗ )) ∗ ) = ξ (2.24) hold for all ξ ∈ H . If S = id the last equation of (2.24) becomes S ( ξ ∗ ) = S ( ξ ) ∗ , saying that S respects the ∗ -involution. This happens for example if H is cocommutative or com-mutative. We discuss an example of Hopf ∗ -algebra which is commonly used indeformation quantization. Example 2.6.2.
Consider a Lie ∗ -algebra ( g , [ · , · ] , ∗ ) , which is defined as a k -Liealgebra ( g , [ · , · ]) together with a map ∗ : g → g satisfying ( λx + µy ) ∗ = λx ∗ + µy ∗ , [ x, y ] ∗ = [ y ∗ , x ∗ ] and ( x ∗ ) ∗ = x for all λ, µ ∈ k and x, y ∈ g . The universal enveloping algebra U g of g is notonly a Hopf algebra but can even be structured as a Hopf ∗ -algebra in this case: the extension of ∗ : g → g as an algebra anti-homomorphism to ∗ : U g → U g iswell-defined since ∗ = 1 = 1 and (cid:18) x ⊗ y − y ⊗ x − [ x, y ] (cid:19) ∗ = y ∗ ⊗ x ∗ − x ∗ ⊗ y ∗ − [ y ∗ , x ∗ ] for all x, y ∈ g , where the latter implies that ∗ -respects the relation which is usedto construct U g as a quotient form the tensor algebra. It is sufficient to prove thecompatibility of the coalgebra structure and the antipode with the ∗ -involution onprimitive elements. In fact ∆( x ∗ ) = x ∗ ⊗ ⊗ x ∗ = x ∗ ⊗ ∗ + 1 ∗ ⊗ x ∗ = ( x (1) ) ∗ ⊗ ( x (2) ) ∗ ,(cid:15) ( x ∗ ) = 0 = (cid:15) ( x ) and S (( S ( x ∗ )) ∗ ) = S (( − x ∗ ) ∗ ) = ( x ∗ ) ∗ = x hold for all x ∈ g since x ∗ ∈ g by definition. Note that every R -Lie algebra can be structured as a Lie ∗ -algebra with ∗ -involution given by x ∗ = − x for all x ∈ g . Fix a Hopf ∗ -algebra H . A left H -module ∗ -algebra is a ∗ -algebra ( A , · , A , ∗ )which is a left H -module algebra, such that( ξ (cid:66) a ) ∗ = S ( ξ ) ∗ (cid:66) a ∗ (2.25)holds for all ξ ∈ H and a ∈ A . Let A be a left H -module ∗ -algebra. An H -equivariant A - ∗ -bimodule is an H -equivariant A -bimodule M endowed with a map ∗ : M → M , such that( λm + µn ) ∗ = λm ∗ + µn ∗ , ( a · m · b ) ∗ = b ∗ · m ∗ · a ∗ and ( m ∗ ) ∗ = m (2.26)for all λ, µ ∈ k , m, n ∈ M and a, b ∈ A . Consider a Drinfel’d twist F on H . Itdeforms the Hopf algebra structure of H , the algebra structure of A and the A -bimodule structure of M such that A F is a left H F -module algebra and M is an H F -equivariant A F -bimodule. The natural question arises if there is a way to twistdeform the ∗ -involution ∗ (cid:55)→ ∗ F such that ( A F , ∗ F ) is a left H F -module ∗ -algebraand M F an H F -equivariant A F - ∗ -bimodule. It turns out that this is the case if weimpose another condition on the twist F . Definition 2.6.3 (Unitary Twist) . A twist F = F ⊗ F on a Hopf ∗ -algebra H issaid to be unitary if F ∗ ⊗ F ∗ = F − . We slightly modify Example 2.4.2 iii.) and v.) to obtain two classes of examplesof unitary twists.
Example 2.6.4.
Let g be a Lie ∗ -algebra over C .i.) For n ∈ N consider a set x , . . . , x n , y , . . . , y n ∈ g of elements in g whichare all Hermitian, i.e. x ∗ i = x i , y ∗ i = y i , or anti-Hermitian, i.e. x ∗ i = − x i , y ∗ i = − y i . Further assume that [ x i , y j ] = [ x i , x j ] = [ y i , y j ] = 0 . Then F = exp(i (cid:126) r ) ∈ ( U g ⊗ U g )[[ (cid:126) ]] (2.27) is a unitary twist, where r = (cid:80) ni =1 x i ⊗ y i and i denotes the imaginary unit in C . It remains to prove that F is unitary, which follows since F ∗ ⊗ F ∗ = exp (cid:18) i (cid:126) n (cid:88) i =1 x ∗ i ⊗ y ∗ i (cid:19) = exp( − i (cid:126) r ) = F − . uasi-Triangular Hopf Algebras and their Representations 49 ii.) Let H, E ∈ g be anti-Hermitian elements such that [ H, E ] = 2 E . Then F = exp (cid:18) H ⊗ log(1 + i (cid:126) E ) (cid:19) ∈ ( U g ⊗ U g )[[ (cid:126) ]] (2.28) is a unitary twist. Using the formal power series expansion in the (cid:126) -adictopology gives (log(1 + i (cid:126) E )) ∗ = log(1 + i (cid:126) E ) = log(1 + i (cid:126) E ) . Then clearly F ∗ ⊗ F ∗ = exp (cid:0) − H ⊗ log(1 + i (cid:126) E ) (cid:1) = F − , which implies that F is unitary. After discussing two examples we return to the general theory. The claim wasthat unitary twists allow for deformations of the ∗ -involution such that A F and M F are deformed as ∗ -algebras and ∗ -bimodules. Proposition 2.6.5 ([56]) . Let H be a cocommutative Hopf ∗ -algebra and consider aleft H -module ∗ -algebra A . Then A F is a left H F -module ∗ -algebra with ∗ -involution ∗ F : A F → A F defined by a ∗ F = S ( β ) (cid:66) a ∗ (2.29) for all a ∈ A , where β = F S ( F ) . If furthermore, M is an H -equivariant A - ∗ -bimodule, M F is an H F -equivariant A F - ∗ -bimodule with ∗ -involution m ∗ F = S ( β ) (cid:66) m ∗ (2.30) for all m ∈ M . In particular( a · F b ) ∗ F = b ∗ F · F a ∗ F and ( a · F m · F b ) ∗ F = b ∗ F · m ∗ F · a ∗ F (2.31)hold for all a, b ∈ A and m ∈ M . Note that the ∗ -involution on H F is undeformed. Proof.
Let us first prove that
F · ∆( β ) · ( S ( F ) ⊗ S ( F )) = β ⊗ β. (2.32)In fact, the 2-cocycle property and the normalization property of F and the anti-bialgebra homomorphism property of S as well as the antipode property imply F ∆( β )( S ( F ) ⊗ S ( F )) = F (cid:48)(cid:48) F (cid:48) S ( F (cid:48) ) (1) S ( F ) ⊗ F (cid:48)(cid:48) F (cid:48) S ( F (cid:48) ) (2) S ( F )= F (cid:48)(cid:48) F (cid:48) S ( F F (cid:48) ) ⊗ F (cid:48)(cid:48) F (cid:48) S ( F F (cid:48) )= F (cid:48)(cid:48) ( F F (cid:48) ) (1) S ( F (cid:48) ) ⊗ F (cid:48)(cid:48) ( F F (cid:48) ) (2) S ( F F (cid:48) )= F (cid:48)(cid:48) F F (cid:48) S ( F (cid:48) ) ⊗ F (cid:48)(cid:48) F F (cid:48) S ( F (cid:48) ) S ( F )= F (cid:48)(cid:48) F F (cid:48) S ( F (cid:48) ) ⊗ F (cid:48)(cid:48) F S ( F )= F F (cid:48) S ( F (cid:48) ) ⊗ F (cid:48)(cid:48) F S ( F (cid:48)(cid:48) F )= F F (cid:48) S ( F (cid:48) ) ⊗ F (cid:48)(cid:48) F S ( F ) S ( F (cid:48)(cid:48) )= F (cid:48) S ( F (cid:48) ) ⊗ F (cid:48)(cid:48) S ( F (cid:48)(cid:48) )= β ⊗ β. Then (2.32) leads to ∆( β ) = F − · ( β ⊗ β ) · (( S ⊗ S )( F − )) (2.33)0 Chapter 2and ( a · F b ) ∗ F = S ( β ) (cid:66) ( a · F b ) ∗ = S ( β ) (cid:66) (( F − (cid:66) a ) · ( F − (cid:66) b )) ∗ = S ( β ) (cid:66) (( F − (cid:66) b ) ∗ · ( F − (cid:66) a ) ∗ )= S ( β ) (cid:66) (( S ( F − ) ∗ (cid:66) b ∗ ) · ( S ( F − ) ∗ (cid:66) a ∗ ))= S ( β ) (cid:66) (( S ( F ) (cid:66) b ∗ ) · ( S ( F ) (cid:66) a ∗ ))=(( S ( β ) (1) S ( F ) S ( β − )) (cid:66) b ∗ F ) · (( S ( β ) (2) S ( F ) S ( β − )) (cid:66) a ∗ F )=( S ( β − F β (2) ) (cid:66) b ∗ F ) · ( S ( β − F β (1) ) (cid:66) b ∗ F )=( F − (cid:66) b ∗ F ) · ( F − (cid:66) a ∗ F )= b ∗ F · F a ∗ F holds for all a, b ∈ A , using that F is unitary, eq.(2.33) and S = id, which holdssince H is cocommutative. Furthermore, for a ∈ A we obtain( a ∗ F ) ∗ F =( S ( β ) (cid:66) a ∗ ) ∗ F = S ( β ) (cid:66) ( S ( β ) (cid:66) a ∗ ) ∗ =( S ( β ) S ( β ∗ )) (cid:66) ( a ∗ ) ∗ =( S ( β ) β ∗ ) (cid:66) a = a where we employed that S ( β ) β ∗ = F S ( F ) S ( F (cid:48) ∗ ) F (cid:48) ∗ = F S ( F ) S ( F (cid:48) − ) F (cid:48) − = F (cid:48)(cid:48) − F S ( F ) S ( F (cid:48) − ) S ( F (cid:48)(cid:48) − ) F (cid:48)(cid:48) − F (cid:48) − = F (cid:48)(cid:48) − F (cid:48) − F S ( F ) S ( F (cid:48)(cid:48) − F (cid:48) − ) F (cid:48)(cid:48) − = F (cid:48)(cid:48) − S ( F (cid:48)(cid:48) − ) F (cid:48)(cid:48) − =1 . The k -linearity of ∗ F is clear since (cid:66) is k -linear and we also obtain1 ∗ F = S ( β ) (cid:66) ∗ = (cid:15) ( β )1 = 1 . This proves that ( A F , ∗ F ) is a ∗ -algebra. The left H F -action is compatible with the ∗ -involution since S F ( ξ ) ∗ (cid:66) a ∗ F =( βS ( ξ ) β − ) ∗ (cid:66) ( S ( β ) (cid:66) a ∗ )=(( β − ) ∗ S ( ξ ∗ ) β ∗ S ( β )) (cid:66) a ∗ =(( β − ) ∗ S ( ξ ∗ )) (cid:66) a ∗ = S ( β ) (cid:66) ( S ( ξ ∗ ) (cid:66) a ∗ )= S ( β ) (cid:66) ( ξ (cid:66) a ) ∗ =( ξ (cid:66) a ) ∗ F . This proves that ( A F , ∗ F ) is a left H F -module ∗ -algebra. Replacing the product ofthe algebra with the left and right A -module action on M the same calculation leadsto the conclusion that ( M F , ∗ F ) is an H F -equivariant A - ∗ -bimodule. This concludesthe proof of the proposition.uasi-Triangular Hopf Algebras and their Representations 51Note that there is another notion of twist deformed ∗ -involution, given by real Drinfel’d twists, which is discussed in [77] Prop. 2.3.7. The corresponding conditionon a twist F is S ( F ∗ ) ⊗ S ( F ∗ ) = F . Note that following this convention onehas to deform the ∗ -involution on the Hopf algebra as well. We end this section bydiscussing a classical example from differential geometry that also appears in thelater sections. Example 2.6.6.
Let M be a smooth manifold and consider the algebra C ∞ ( M ) of smooth complex-valued functions on M . It is commutative with respect to thepointwise product and there is a ∗ -involution ∗ : C ∞ ( M ) → C ∞ ( M ) defined by f ∗ ( p ) = f ( p ) for all f ∈ C ∞ ( M ) and p ∈ M . This structures C ∞ ( M ) as a ∗ -algebra over C . On smooth vector fields X ∈ X ( M ) we define L X ∗ f = − ( L X ( f ∗ )) ∗ (2.34) for all f ∈ C ∞ ( M ) , where L : X ( M ) → Der( C ∞ ( M )) denotes the Lie deriva-tive. One easily proves that this structures X ( M ) as a Lie ∗ -algebra. In particu-lar [ X, Y ] ∗ = [ Y ∗ , X ∗ ] for all X, Y ∈ X ( M ) . Moreover, with the usual C ∞ ( M ) -bimodule actions L f · X · g h = f ( L X h ) g, where f, g, h ∈ C ∞ ( M ) and X ∈ X ( M ) , X ( M ) becomes an A - ∗ -bimodule. This can be extended to multivector X • ( M ) fieldsby defining ( X ∧ Y ) ∗ = Y ∗ ∧ X ∗ for X, Y ∈ X • ( M ) . Remark that the Gerstenhaberbracket satisfies (cid:74) X, Y (cid:75) ∗ = (cid:74) Y ∗ , X ∗ (cid:75) for all X, Y ∈ X • ( M ) . Using the dual pairing (cid:104)· , ·(cid:105) : Ω ( M ) × X ( M ) → C ∞ ( M ) the ∗ -involution on X ( M ) induces a ∗ -involution on Ω ( M ) . Namely, for ω ∈ Ω ( M ) we define ω ∗ ∈ Ω ( M ) by (cid:104) ω ∗ , X (cid:105) = (cid:104) ω, X ∗ (cid:105) for all X ∈ X ( M ) . Inductively this extends to Ω • ( M ) , structuring it as a C ∞ ( M ) - ∗ -bimodule with respect to the usual left and right C ∞ ( M ) -module actions, definedby ( f · ω · g )( X ) = f · ω ( X ) · g for all f, g ∈ C ∞ ( M ) , ω ∈ Ω ( M ) and X ∈ X ( M ) .Assume now the existence of a left H -module ∗ -algebra action (cid:66) on C ∞ ( M ) fora cocommutative Hopf ∗ -algebra H and let F be a unitary Drinfel’d twist on H . Anatural candidate is given by the universal enveloping algebra of the Lie ∗ -algebra X ( M ) . The adjoint action L ξ (cid:66) X f = ξ (1) (cid:66) ( L X ( S ( ξ (2) ) (cid:66) f )) structures X ( M ) as an H -equivariant A - ∗ -bimodule. Again, the dual pairing inducesthe same structure on Ω ( M ) via (cid:104) ξ (cid:66) ω, X (cid:105) = ξ (1) (cid:66) (cid:104) ω, S ( ξ (2) ) (cid:66) X (cid:105) . Extending the H -action via the coproduct to higher wedge products we end up with H -equivariant A - ∗ -bimodules X • ( M ) and Ω • ( M ) . By Proposition 2.6.5 the unitarytwist F deforms them into H F -equivariant A F - ∗ -bimodules. In Chapter 4 we come back to this example and further examine how the struc-ture of the Cartan calculus on M can be twist deformed. Motivated by this, weconstruct a noncommutative Cartan calculus on a huge class of noncommutativealgebras. hapter 3Obstructions of Twist StarProducts The aim of this chapter is to give a quick recap on symplectic geometry and its quan-tization in the form of Drinfel’d twist deformation quantization, before discussingseveral obstructions to this construction. We start by reviewing basic definitionsand properties of Poisson manifolds in Section 3.1. As examples, constant Poissonstructures on R n , the KKS Poisson structure on the dual of a Lie algebra and theconnected orientable symplectic Riemann surfaces are depicted. Equivalently to thePoisson bracket we may consider a bivector, squaring to zero under the Schouten-Nijenhuis bracket. If this so called Poisson bivector is non-degenerate, we obtaina symplectic manifold. Star products are introduced as formal deformations ofthe algebra of smooth functions, deforming the underlying Poisson bracket in ad-dition. Then, examples of star products on some of the former Poisson manifoldsare given. We end the first section with results about existence and classification ofstar products in the symplectic case. In Section 3.2, classical r -matrices are definedas skew-symmetric solutions of the classical Yang-Baxter equation. Equivalently,one can think of G -equivariant Poisson bivectors on a Lie group G . Then it isnot surprising that they appear as skew-symmetrization of the first order of formal R -matrices and Drinfel’d twists on universal enveloping algebras. We discuss quan-tization of r -matrices before turning to the notion of twist star products. The latterare star products on Poisson manifolds which are induced by Drinfel’d twists. It fol-lows that the underlying Poisson bracket is induced by the corresponding r -matrix.In the symplectic case it is not hard to see that this implies that the manifold is ahomogeneous space under certain conditions, leading to a first obstruction: the sym-plectic Pretzel surfaces of genus g > As indicated in the introduction,
Poisson geometry is a decent mathematical frame-work to describe classical mechanics, while deformation quantization pictures thequantized system. In this section we briefly recall the notion of Poisson manifold.As corresponding literature we mention [14] Chap. 2-3 and [109] Chap. 3-4. After-wards, following [47] Sec. 2.3, [64] Chap. II 2 and [109] Sec 6.1, star products areintroduced as formal deformations of the algebra of functions on a Poisson manifoldand examples are given. We discuss the notion of equivalence of star products andtheir classification on symplectic manifolds. A more detailed overview of the fieldof deformation quantization is given in [110].A
Poisson manifold is a smooth manifold M [75], together with a Poissonbracket on its algebra C ∞ ( M ) of smooth functions. The latter is a Lie bracket {· , ·} : C ∞ ( M ) × C ∞ ( M ) → C ∞ ( M ) satisfying a Leibniz rule { f, gh } = { f, g } h + g { f, h } (3.1)in addition, where f, g, h ∈ C ∞ ( M ). The Leibniz rule implies that X f := −{ f, ·} : C ∞ ( M ) → C ∞ ( M )is a smooth vector field on M for all f ∈ C ∞ ( M ), called Hamiltonian vector field .On a local chart (
U, x ) around a point p ∈ M the Poisson bracket reads { f, g }| U = (cid:88) ij π ij ∂f∂x i ∂g∂x j for all f, g ∈ C ∞ ( M ), with π ij = { x i , x j } = − π ji ∈ C ∞ ( U ). This determines abivector π U = 12 (cid:88) ij π ij ∂∂x i ∧ ∂∂x j ∈ X ( U )on U , which corresponds to a global bivector π ∈ X ( M ) such that { f, g } = π (d f, d g )for all f, g ∈ C ∞ ( M ). One can prove that the Jacobi identity of {· , ·} is equivalentto the vanishing (cid:74) π, π (cid:75) = 0 of the Schouten-Nijenhuis bracket of π with itself. Thelatter is defined as the extension (cid:74) X ∧ . . . ∧ X k , Y ∧ . . . ∧ Y (cid:96) (cid:75) = k (cid:88) i =1 (cid:96) (cid:88) j =1 ( − i + j [ X i , Y j ] ∧ X ∧ . . . ∧ (cid:99) X i ∧ . . . ∧ X k ∧ Y ∧ . . . ∧ (cid:98) Y j ∧ . . . ∧ Y (cid:96) of the Lie bracket [ · , · ] of vector fields, where X , . . . , X k , Y , . . . , Y (cid:96) ∈ X ( M ) and (cid:99) X i , (cid:98) Y j means that X i and Y j are omitted in the above wedge product. It is definedto vanish if one of its entries is a function. In other words, the data π ∈ X ( M )and (cid:74) π, π (cid:75) = 0 is equivalent to the properties of {· , ·} . For this reason we sometimesrefer to ( M, π ) as a Poisson manifold and call π the corresponding Poisson bivector .4 Chapter 3
Example 3.1.1. i.) Consider R n with global coordinates ( x , . . . , x n ) . Any skew-symmetric matrix ( π ij ) ij ∈ M n ( R ) leads to a Poisson bivector π = (cid:88) i 0, where we set Ω − ( M ) = 0. In the presence of a Liegroup action Φ : G × M → M the de Rham differential forms a cochain complex withrespect to G -invariant differential forms Ω • ,G ( M ), i.e. elements ω ∈ Ω • ( M ) such thatΦ ∗ g ω = ω for all g ∈ G . It is well-defined since d commutes with pullbacks. Thecorresponding G -invariant de Rham cohomology is denoted by H • ,G dR ( M ). Note thatthere is also a Poisson cohomology on ( M, π ) defined by the differential d π = (cid:74) π, · (cid:75) (c.f. [76]).In a next step we describe how to quantize Poisson manifolds. To be moreprecise, we want to perform formal deformations of the algebra of smooth functions C ∞ ( M ). Since the definition is given in algebraic terms we first formulate it forgeneral associative algebras (following [58]), before coming back to smooth functions.Let ( A , · , 1) be an associative unital algebra. A formal deformation of ( A , · , 1) is anassociative product (cid:63) on A = A [[ (cid:126) ]] such that a (cid:63) a = 1 (cid:63) a and a (cid:63) b = a · b + O ( (cid:126) ) (3.5)for all a, b ∈ A . The classical limit assigns to a formal deformation (cid:63) its lowest order · and we sometimes write lim (cid:126) → a (cid:63) b = a · b . Heuristically, this amounts to set thevalue of (cid:126) to zero. If A is commutative and (cid:63) a formal deformation of A , the firstorder { a, b } = 1 (cid:126) ( a (cid:63) b − b (cid:63) a ) + O ( (cid:126) ) (3.6)6 Chapter 3of the (cid:63) -commutator [ a, b ] (cid:63) = a (cid:63) b − b (cid:63) a defines a Poisson bracket on A , i.e. a Liebracket satisfying a Leibniz rule { a, bc } = { a, b } c + b { a, c } for all a, b, c ∈ A . Equation (3.6) is said to be the correspondence principle , whichcan be expressed as lim (cid:126) → (cid:126) [ a, b ] (cid:63) = { a, b } , (3.7)using the classical limit. Two formal deformations (cid:63) and (cid:63) (cid:48) are said to be equivalent if there are k -linear maps T n : A → A for all n > T = id + (cid:80) n> (cid:126) n T n satisfies T ( a (cid:63) b ) = T ( a ) (cid:63) (cid:48) T ( b ) (3.8)for all a, b ∈ A . Since T starts with the identity it is invertible in the (cid:126) -adictopology and the (cid:126) -linear extension T : ( A , (cid:63) ) → ( A , (cid:63) (cid:48) ) is an algebra isomorphism.The equivalence class of formal deformations which are equivalent to (cid:63) is denoted by[ (cid:63) ]. From any automorphism φ of A and any formal deformation (cid:63) of A we constructanother formal deformation (cid:63) φ by a (cid:63) φ b = φ − ( φ ( a ) (cid:63) φ ( b ))for all a, b ∈ A . Lemma 3.1.3 ([27] Prop. 2.14) . Two formal deformations (cid:63) and (cid:63) (cid:48) of A are iso-morphic if and only if there exists an automorphism φ of A such that [ (cid:63) φ ] = [ (cid:63) (cid:48) ] . In the case A = C ∞ ( M ) for a smooth manifold M , formal deformations of thepointwise product should be considered in the smooth category. Definition 3.1.4 ([11]) . A formal deformation f (cid:63) g = f · g + (cid:88) r> (cid:126) r C r ( f, g ) of ( C ∞ ( M ) , · ) is said to be a star product on M if C r are bidifferential operators.The corresponding Poisson bracket {· , ·} structures M as a Poisson manifold. Ac-cordingly, a formal deformation of a Poisson manifold ( M, {· , ·} ) is a star product (cid:63) , such that lim (cid:126) → (cid:126) [ f, g ] (cid:63) = { f, g } (3.9) holds for all f, g ∈ C ∞ ( M ) . Two formal deformations (cid:63) , (cid:63) (cid:48) of a Poisson manifold ( M, {· , ·} ) are equivalentas star products if there is a formal equivalence via differential operators. They areisomorphic if and only if there is a Poisson diffeomorphism φ : M → M such that[ (cid:63) φ ] = [ (cid:63) (cid:48) ], where φ ∗ ( f (cid:63) φ g ) = φ ∗ ( f ) (cid:63) φ ∗ ( g )for all f, g ∈ C ∞ ( M ). The set of equivalence classes of star products on ( M, {· , ·} )is denoted by Def( M, {· . ·} ) and by Def( M, ω ) in the symplectic case. We give twoexamples of star products to become familiar with the notion. Namely, we quantizethe Poisson structures discussed in Example 3.1.1.bstructions of Twist Star Products 57 Example 3.1.5. i.) Consider R n with coordinates ( x , . . . , x n ) and a constantPoisson bivector π = (cid:80) i Proposition 3.1.6 ([16, 39]) . Let ( M, ω ) be a symplectic manifold. Then there isa bijection c : Def( M, ω ) → [ ω ] (cid:126) + H ( M )[[ (cid:126) ]] , (3.12) which assigns to any star product (cid:63) on ( M, ω ) its characteristic class c ( (cid:63) ) . In particular, the above theorem reveals that there is a star product on everysymplectic manifold ( M, ω ). If H ( M ) = { } , all star products on ( M, ω ) areequivalent. There is a geometric construction [52] of Fedosov, leading to the result ofProposition 3.1.6. Moreover, it provides a recursive formula to successively assemblethe star product.If there is a Lie group action Φ : G × M → M on a Poisson manifold ( M, π ), astar product (cid:63) on ( M, π ) is said to be G -invariant ifΦ ∗ g ( f (cid:63) h ) = (Φ ∗ g ( f )) (cid:63) (Φ ∗ g ( h )) (3.13)8 Chapter 3for all g ∈ G and f, h ∈ C ∞ ( M ), where Φ ∗ g : C ∞ ( M ) → C ∞ ( M ) denotes the pullbackof the diffeomorphism Φ g : M → M . Two G -invariant star products (cid:63) and (cid:63) (cid:48) aresaid to be G -invariantly equivalent if there exists an equivalence T = id + (cid:80) r> (cid:126) r T r of star products consisting of G -invariant operators, i.e. Φ ∗ g ( T r ( f )) = T r (Φ ∗ g ( f )) forall g ∈ G , f ∈ C ∞ ( M ) and r > 0. The corresponding set of G -invariant equivalenceclasses of G -invariant star products on ( M, π ) is denoted by Def G ( M, π ). Proposition 3.1.7 ([15]) . Let ( M, ω ) be a symplectic manifold and consider a Liegroup action Φ : G × M → M on M such that Φ ∗ g ω = ω for all g ∈ G . Then thereis a bijection c G : Def G ( M, ω ) → [ ω ] G (cid:126) + H ,G dR ( M )[[ (cid:126) ]] , (3.14) which assigns to any G -invariant star product (cid:63) on ( M, ω ) its G -invariant charac-teristic class c G ( (cid:63) ) . There is another existence and classification theorem [94] of symplectic manifoldincorporating the notion of momentum map. In this section we examine Drinfel’d twists in the setting of deformation quantization.It turns out that invariant star products on a Lie group G can be identified withDrinfel’d twists F on U g [[ (cid:126) ]], where g denotes the Lie algebra corresponding to G .The skew-symmetrization of the first order term of a G -invariant star product is a G -invariant Poisson bivector. On the side of the universal enveloping algebra, thismeans that the skew-symmetrization of the first order term of a twist is a classical r -matrix [103]. After internalizing the relation of twists and classical r -matrices,we consider star products which are induced by Drinfel’d twists and study firstobstructions of this situation. For an introduction to classical r -matrices we refer to[50] Chap. 3 and [71] Sec. 2, while Drinfel’d twists on universal enveloping algebrasand twist star products are discussed in e.g. [5, 6, 20, 21, 22, 23, 38, 56, 67, 80].Let g be a Lie algebra over a commutative unital ring k such that Q ⊆ k anddenote the corresponding exterior algebra by Λ • g . By extending the Lie bracket [ · , · ]of g , using the following expression (cid:74) x ∧ . . . ∧ x k , y ∧ . . . ∧ y (cid:96) (cid:75) = k (cid:88) i =1 (cid:96) (cid:88) j =1 ( − i + j [ x i , y j ] ∧ x ∧ . . . ∧ (cid:98) x i ∧ . . . ∧ x k ∧ y ∧ . . . ∧ (cid:98) y j ∧ . . . ∧ y (cid:96) for all x , . . . , x k , y , . . . , y (cid:96) ∈ g , as well as (cid:74) x, λ (cid:75) = (cid:74) λ, x (cid:75) = (cid:74) λ, µ (cid:75) = 0for all x ∈ g and λ, ν ∈ k , we obtain a Gerstenhaber bracket (cid:74) · , · (cid:75) on Λ • g . Anelement r ∈ g ∧ g is said to be a classical r -matrix if (cid:74) r, r (cid:75) = 0 . (3.15)bstructions of Twist Star Products 59Clearly every scalar multiple of a classical r -matrix is a classical r -matrix as well.In particular, 0 is a trivial solution. The equation (3.15) is called classical Yang-Baxter equation on Λ g . It is the skew-symmetrization of the classical Yang-Baxterequation CYB( r ) = 0 on g ⊗ , whereCYB : g ⊗ (cid:51) r (cid:55)→ [ r , r ] + [ r , r ] + [ r , r ]=[ r , r (cid:48) ] ⊗ r ⊗ r (cid:48) + r ⊗ [ r , r (cid:48) ] ⊗ r (cid:48) + r ⊗ r (cid:48) ⊗ [ r , r (cid:48) ] ∈ g ⊗ is said to be the classical Yang-Baxter map . Example 3.2.1. Consider the k -Lie algebra g generated by two elements H, E ∈ g ,such that [ H, E ] = 2 E . Then r = H ∧ E is a classical r -matrix on g . This is obvioussince Λ g = { } . In the next theorem we prove that classical r -matrices naturally appear as the(skew-symmetrization of the) classical limit of universal R -matrices on U g [[ (cid:126) ]] (c.f.[50] Prop. 9.5). Theorem 3.2.2. Let R = 1 ⊗ (cid:126) ˜ r + O ( (cid:126) ) be a universal R -matrix on U g [[ (cid:126) ]] .Then r = ˜ r − ˜ r ∈ Λ g (3.16) is a classical r -matrix on g .Proof. We first prove ˜ r = r ⊗ r ∈ g ⊗ g by showing that r i are primitive elements of U g . The hexagon relations (∆ ⊗ id)( R ) = R R of R and (id ⊗ ∆)( R ) = R R read ∞ (cid:88) n =0 (cid:126) n ∆( R n ) ⊗ R n = ∞ (cid:88) n =0 (cid:126) n n (cid:88) m =0 ( R m ⊗ ⊗ R m )(1 ⊗ R (cid:48) m − n ⊗ R (cid:48) m − n )and ∞ (cid:88) n =0 (cid:126) n R n ⊗ ∆( R n ) = ∞ (cid:88) n =0 (cid:126) n n (cid:88) m =0 ( R m ⊗ ⊗ R m )( R (cid:48) m − n ⊗ R (cid:48) m − n ⊗ , where we denoted R = (cid:80) ∞ n =0 (cid:126) n R n ⊗ R n . In order one of (cid:126) this gives r ⊗ r ⊗ r = ( r ⊗ ⊗ r ) ⊗ r and r ⊗ r ⊗ r = r ⊗ ( r ⊗ ⊗ r ) , since R ⊗ R = 1 ⊗ R ⊗ R = r ⊗ r . We conclude that ˜ r = r ⊗ r ∈ g ⊗ g . Inparticular, its skew-symmetrization r is an element of g ∧ g . Recall that R satisfiesthe quantum Yang-Baxter equation R R R = R R R . In order two of (cid:126) this equation reads( v ⊗ v ⊗ 1) + ( v ⊗ ⊗ v ) + (1 ⊗ v ⊗ v )+ ( r ⊗ r ⊗ r (cid:48) ⊗ ⊗ r (cid:48) + 1 ⊗ r (cid:48) ⊗ r (cid:48) ) + ( r ⊗ ⊗ r )(1 ⊗ r (cid:48) ⊗ r (cid:48) )=( v ⊗ v ⊗ 1) + ( v ⊗ ⊗ v ) + (1 ⊗ v ⊗ v )+ (1 ⊗ r ⊗ r )( r (cid:48) ⊗ ⊗ r (cid:48) + r (cid:48) ⊗ r (cid:48) ⊗ 1) + ( r ⊗ ⊗ r )( r (cid:48) ⊗ r (cid:48) ⊗ , R = 1 ⊗ (cid:126) r ⊗ r + (cid:126) v ⊗ v + O ( (cid:126) ). The first threeterms on both sides of the equation cancel each other and the remaining terms giveCYB( r ⊗ r ) = 0 after applying the relation xy − yx = [ x, y ] for x, y ∈ g , which holdsin U g . In particular, the skew-symmetrization r of ˜ r = r ⊗ r satisfies (cid:74) r, r (cid:75) = 0,i.e. r is a classical r -matrix.Note that we even proved a stronger statement: the first order of any universal R -matrix is a solution of the classical Yang-Baxter equation on g ⊗ . The correspon-dence of the classical Yang-Baxter equation and quantum Yang-Baxter equation isfurther discussed in [41]. Since any Drinfel’d twist F on a cocommutative Hopfalgebra leads to a universal R -matrix F F − on the twisted Hopf algebra, a resultsimilar to Theorem 3.2.2 holds for the first order of a Drinfel’d twist on U g [[ (cid:126) ]]. Corollary 3.2.3 ([42] Thm. 5(a)) . Let F = 1 ⊗ (cid:126) ˜ r + O ( (cid:126) ) be a Drinfel’d twiston U g [[ (cid:126) ]] . Then r = ˜ r − ˜ r ∈ Λ g (3.17) is a classical r -matrix on g .Proof. Since R = F F − is a universal R -matrix, Theorem 3.2.2 implies that theskew-symmetrization of its first order is a classical r -matrix. Let F = (cid:80) ∞ n =0 (cid:126) n F n ⊗ F n and F − = (cid:80) ∞ n =0 (cid:126) n F n ⊗ F n . Then, the first order in (cid:126) of R = ∞ (cid:88) n =0 (cid:126) n n (cid:88) m =0 ( F m ⊗ F m )( F n − m ⊗ F n − m )is F ⊗ F + F ⊗ F = ˜ r − ˜ r ∈ g ⊗ g , where we used that the first order of theinverse of F is given by − ˜ r .In the next remark we resume the thought of the introduction of this sectionand identify twists with invariant star products. Furthermore, we discuss severalimportant constructions and classifications of Drinfel’d twist on universal envelopingalgebras. Remark 3.2.4. i.) Consider a G -invariant star product (cid:63) on a Lie group G ,where the Lie group action is given by left multiplication (cid:96) g : G (cid:51) h (cid:55)→ gh ∈ G on G . It corresponds to a Drinfel’d twist F = 1 ⊗ O ( (cid:126) ) ∈ ( U g ⊗ U g )[[ (cid:126) ]] ,where g is the Lie algebra corresponding to G [15, 16]. To see this, recall thatfor every k > there is an isomorphism Γ ∞ ( T G ⊗ k ) G (cid:51) X (cid:55)→ X ( e ) ∈ T e G ⊗ k between G -invariant sections Γ ∞ ( T G ⊗ k ) G and the tangent space at the unitelement e ∈ G . The inverse is given by T e G ⊗ k (cid:51) ξ (cid:55)→ X ξ ∈ Γ ∞ ( T G ⊗ k ) G , where X ξ ( g ) = ( T e (cid:96) g ) ⊗ k ξ for all g ∈ G and T e (cid:96) g : T e G → T g G denotes the tangentmap of (cid:96) g at e . Since this isomorphism is in fact a Lie algebra isomorphism,i.e. [ X ξ , X η ] = X [ ξ,η ] for all ξ, η ∈ g , it extends to an isomorphism of G -invariant bidifferential operators and elements in U g ⊗ , leading to the statedcorrespondence (cid:63) (cid:55)→ F . Since f (cid:63) g = f · g + O ( (cid:126) ) it follows that F =1 ⊗ O ( (cid:126) ) , while (cid:63) f = f = f (cid:63) leads to ( (cid:15) ⊗ id)( F ) = 1 = (id ⊗ (cid:15) )( F ) and the associativity of (cid:63) corresponds to the -cocycle condition of F . It is bstructions of Twist Star Products 61 clear that the data of a G -invariant star product on G and a Drinfel’d twist F = 1 ⊗ O ( (cid:126) ) on U g are equivalent. In the same spirit, classical r -matrices on g correspond to G -invariant Poisson bivectors on G . Using theright multiplication one proves that right invariant Poisson bivectors on G also correspond to classical r -matrices and that the difference of an inducedright invariant and induced left invariant Poisson bivector structures G as aPoisson-Lie group (see e.g. [50] Prop. 3.1);ii.) In [42] Thm. 6 it is proven that for any (non-degenerate) classical r -matrix r ∈ g ∧ g there exists a twist F = 1 ⊗ (cid:126) r + O ( (cid:126) ) on U g [[ (cid:126) ]] . This is doneby extending the Lie algebra g via the inverse of r , constructing the Gutt starproduct on the dual Lie algebra of the extension and restricting this productto an affine subspace, which is locally diffeomorphic to G . In the end, theidentification i. ) is employed;iii.) In [48] there is an alternative proof of [42] Thm. 6, using a Fedosov-like con-struction. Furthermore, in [48] Thm. 5.6 the authors give a classificationof twists on U g [[ (cid:126) ]] which quantize a fixed non-degenerate classical r -matrix r ∈ g ∧ g . Namely, the twists on U g [[ (cid:126) ]] quantizing r are in bijection withthe second Chevalley-Eilenberg cohomology H ( g )[[ (cid:126) ]] . Note that this classi-fication is undertaken up to equivalence, where two twists F = 1 ⊗ O ( (cid:126) ) and F (cid:48) = 1 ⊗ O ( (cid:126) ) are said to be equivalent if there exists an element S = 1 + O ( (cid:126) ) ∈ U g [[ (cid:126) ]] , such that (cid:15) ( S ) = 1 and ∆( S ) F (cid:48) = F ( S ⊗ S ) . This implies that the twist of Example 2.4.2 iv.) and the skew-symmetrizationof the Jordanian twist of Example 2.4.2 v.) are equivalent, since the secondChevalley-Eilenberg cohomology of the underlying Lie algebra is trivial (c.f.[48] Ex. 5.7);iv.) The quantization of classical r -matrices is closely related to the quantizationof Lie bialgebras. In [49] Thm 6.1, the authors give a functor constructionto quantize any quasi-triangular Lie bialgebra. In particular, this constructionprovides a universal R -matrix for every classical r -matrix;v.) There is a more general notion of classical dynamical r -matrices, which admita quantization for reductive Lie algebras. This was proven in [33] by makinguse of formality theory; Inspired by Remark 3.2.4 i.) and Proposition 2.5.2 we state the following defini-tions. The idea is to induce a star product on a manifold M from a Drinfel’d twiston U g via a Lie algebra action g → Γ ∞ ( T M ). Definition 3.2.5 ([20] Twist Star Product) . Let ( M, π ) be a Poisson manifold.i.) A (formal) Drinfel’d twist on U g is a twist F = 1 ⊗ O ( (cid:126) ) ∈ ( U g ⊗ U g )[[ (cid:126) ]] on U g [[ (cid:126) ]] .ii.) A symmetry g of M is a Lie algebra, acting on M by derivations, i.e. a Liealgebra anti-homomorphism φ : g → Γ ∞ ( T M ) .iii.) A twist star product on ( M, π ) is a star product (cid:63) on ( M, π ) together with asymmetry g of M and a formal Drinfel’d twist F on U g such that f (cid:63) g = ( F − (cid:66) f ) · ( F − (cid:66) g ) (3.18)2 Chapter 3 for all f, g ∈ C ∞ ( M ) , where (cid:66) denotes the Hopf algebra action induced by thesymmetry. By Proposition 2.5.2, for any symmetry g on a manifold M and any (formal)Drinfel’d twist F on U g there is a twist star product f (cid:63) F g = ( F − (cid:66) f ) · ( F − (cid:66) g ) on M with Poisson bracket given by the skew-symmetrization of the first order of F − .It is the strategy of the rest of the section to investigate such Poisson brackets andfind obstructions for their existence, leading to obstructions for twist star products.Before continuing, we first give an example of a twist star product. Example 3.2.6. Consider R with coordinates ( x, y ) . The -torus T is the quotientof R under the identification ( x, y ) ∼ ( x + 1 , y ) ∼ ( x, y + 1) . It is a symplecticmanifold with respect to the Poisson bracket { f, g } = ∂f∂x ∂g∂y − ∂f∂y ∂g∂x for all f, g ∈ C ∞ ( T ) . The Lie algebra R endowed with the zero Lie bracket is asymmetry of T with respect to the usual action of coordinate vector fields ∂∂x and ∂∂y . Consequently, U g can be identified with the symmetric algebra S • g of g . Then,the Moyal-Weyl product ( f (cid:63) g )( x, y ) = exp (cid:18) (cid:126) ∂∂x ∂∂y (cid:19) f ( x, v ) g ( u, y ) | ( u,v )=( x,y ) (3.19) on T , where f, g ∈ C ∞ ( T ) , is a twist star product corresponding to the (formal)Drinfel’d twist F = exp (cid:18) − (cid:126) ∂∂x ⊗ ∂∂y (cid:19) ∈ ( U g ⊗ U g )[[ (cid:126) ]] (3.20) on U g . Similarly, the Moyal-Weyl product on R n given in Example 3.1.5 i.) is atwist star product with respect to the Drinfel’d twist (3.20). A twist star product puts quite strict requirements on the underlying manifold.As a first instance of this, we prove that the Poisson bracket corresponding to atwist star product is induced by a classical r -matrix (c.f. [20] Lem. 2.7). Lemma 3.2.7. Let (cid:63) F be a twist star product on a Poisson manifold ( M, {· , ·} ) .Then { f, g } = ( r (cid:66) f ) · ( r (cid:66) g ) (3.21) for all f, g ∈ C ∞ ( M ) , where r = r ∧ r denotes the r -matrix corresponding to F , · the pointwise product of functions and (cid:66) the induced Hopf algebra action.Proof. Since (cid:63) F is a twist star product ∞ (cid:88) n =0 (cid:126) n ( F n (cid:66) f ) · ( F n (cid:66) g ) = ( F − (cid:66) f ) · ( F − (cid:66) g ) = f (cid:63) F g = ∞ (cid:88) n =0 (cid:126) n C n ( f, g )holds for all f, g ∈ C ∞ ( M ), where C n are the bidifferential operators correspondingto (cid:63) F and F − = (cid:80) ∞ n =0 (cid:126) n F n ⊗ F n . In particular, C ( f, g ) = ( F (cid:66) f ) · ( F (cid:66) g ) andbstructions of Twist Star Products 63the Poisson bracket is { f, g } = C ( f, g ) − C ( g, f )=( F (cid:66) f ) · ( F (cid:66) g ) − ( F (cid:66) g ) · ( F (cid:66) f )= µ (( F ⊗ F − F ⊗ F ) (cid:66) ( f ⊗ g ))= µ ( r (cid:66) ( f ⊗ g ))=( r (cid:66) f ) · ( r (cid:66) g ) , where we used that the pointwise product µ is commutative.More generally, the same statement holds true if a formal deformation is braidedcommutative (see [20] Prop. 2.10). Proposition 3.2.8. Let (cid:63) be a star product on a Poisson manifold ( M, {· , ·} ) suchthat ( C ∞ ( M )[[ (cid:126) ]] , (cid:63) ) is braided commutative with respect to a universal R -matrix R = 1 ⊗ (cid:126) r + O ( (cid:126) ) . Then r = r ∧ r is a classical r -matrix on g and { f, g } = ( r (cid:66) f ) · ( r (cid:66) g ) (3.22) for all f, g ∈ C ∞ ( M ) .Proof. Let f, g ∈ C ∞ ( M ). For f (cid:63) g = (cid:80) ∞ n =0 (cid:126) n C n ( f, g ) we obtain (cid:126) { f, g } + O ( (cid:126) ) = f (cid:63) g − g (cid:63) f = f (cid:63) g − ( R − (cid:66) f ) (cid:63) ( R − (cid:66) g )= (cid:126) ( C ( f, g ) − C ( f, g ) + ( r (cid:66) f ) · ( r (cid:66) g )) + O ( (cid:126) )= (cid:126) ( r (cid:66) f ) · ( r (cid:66) g ) + O ( (cid:126) ) , which implies the statement. Note that we used that the first order in (cid:126) of R − is − r . In particular r is skew-symmetric since the Poisson bracket is.If g is a finite-dimensional Lie algebra over C with basis e , . . . , e n ∈ g , any r -matrix r on g reads r = (cid:88) i Let ( M, {· , ·} ) be a connected compact symplectic manifold al-lowing for a twist star product (cid:63) F , where F ∈ ( U g ⊗ U g )[[ (cid:126) ]] . Then M is ahomogeneous G -space.Proof. By Palais’ theorem [90], a Lie algebra action integrates to a Lie group actionif all corresponding fundamental vector fields have complete flow. This is the case,since M is compact (c.f. [75] Thm. 12.12). In particular, we can apply Proposi-tion 3.2.10 to obtain the result. Example 3.2.12 ([20] Ex. 3.9) . The connected orientable Riemann surfaces T ( g ) ofgenus g > are not homogeneous. This follows from theorems [82, 83] of Mostow,which say that connected compact homogeneous spaces have non-negative Euler char-acteristic. Of course the Euler characteristic of T ( g ) is χ ( T ( g )) = 2(1 − g ) . Onthe other hand, the canonical symplectic structure on T ( g ) , which we discussed inExample 3.1.2, admits a star product quantization according to Proposition 3.1.6.By Corollary 3.2.11 we conclude that those star products can not be induced by aDrinfel’d twist on a universal enveloping algebra. bstructions of Twist Star Products 65In Example 3.2.6 we recognized that the Moyal-Weyl product on the symplectic2-torus T (1) = T is a twist star product. The question remains if there are twiststar products on the symplectic 2-sphere T (0) = S . We address this question inthe rest of this section. Lemma 3.2.13 ([20] Prop. 3.6) . Let Φ : G × M → M be a Lie group action withcorresponding Lie algebra action φ : g → Γ ∞ ( T M ) . Then the following statementshold.i.) If Φ is transitive, the induced Lie group action Ψ : G/ ker Φ ⊗ M (cid:51) ([ g ] , p ) (cid:55)→ Φ( g, p ) ∈ M is transitive and effective.ii.) If r ∈ Λ g is a classical r -matrix, so is [ r ] ∈ Λ g / ker φ .iii.) If φ ( r ) := (cid:80) i Corollary 3.2.14. If there is a twist star product on a connected compact symplecticmanifold M , there exists a non-degenerate classical r -matrix on a Lie algebra g suchthat the corresponding connected and simply connected Lie group G acts transitivelyand effectively on M .Proof. By Corollary 3.2.11 there is a classical r -matrix ˜ r ∈ Λ ˜ g on a Lie algebra˜ g such that the corresponding connected and simply connected Lie group ˜ G actstransitively on M . Denote this group action by ˜Φ and the corresponding Lie al-gebra action by ˜ φ . Performing the quotient G (cid:48) = ˜ G/ ker ˜Φ we obtain a Lie groupaction Φ (cid:48) : G (cid:48) × M → M , which is well-defined, transitive and effective accordingto Lemma 3.2.13, such that the induced r -matrix r ∈ Λ g (cid:48) induces the Poissonstructure on M . This r -matrix is non-degenerate in the Lie subalgebra g r ⊆ g (cid:48) according to Lemma 3.2.9. The Lie group action of the corresponding Lie group G r is still transitive and effective, which can be checked by repeating the proof ofProposition 3.2.10 for g r . This concludes the proof of the corollary.Finally we are able to conclude the main obstruction of this section.6 Chapter 3 Theorem 3.2.15 ([20] Cor. 3.12) . There is no twist star product deforming thesymplectic -sphere.Proof. Assume there is a twist star product on symplectic S . Then there is anon-degenerate classical r -matrix r on a Lie algebra g such that G acts transitivelyand effectively on S , according to Corollary 3.2.14. In particular, g is semisimpleaccording to Onishchik [87, 88, 89]. This gives a contradiction since there are nonon-degenerate r -matrices on semisimple Lie algebras (c.f. [50] Prop. 5.2). A weaker notion of equivalence of algebras than isomorphism is given by Moritaequivalence. It identifies two algebras if their categories of representations are equiv-alent. There are several properties which are preserved under Morita equivalence,like e.g. algebraic K -Theory (c.f. [92]). Those invariants are said to be Moritainvariants . We will see that commutativity is not one of them. Following [32] Sec. 3and [27] Sec. 2. we recall some basic concepts of Morita theory, which we apply tostar product algebras and in particular to twist star product algebras in Section 3.4.For a general introduction to Morita equivalence of rings we refer to [73].Fix an associative unital algebra A over a commutative unital ring k in thefollowing. As usual, its category of representations if denoted by A M . If B isanother associative unital algebra isomorphic to A , it follows that A M and B M are equivalent categories. This means that there are two functors F : A M → B M and G : B M → A M , as well as two natural isomorphisms id A M → G ◦ F and F ◦ G → id B M . The following example shows that being isomorphic is not a necessarycondition for algebras to have equivalent categories of modules. Example 3.3.1. Fix a natural number n > . The set M n ( A ) of n × n -matriceswith entries in A is itself an associative unital algebra with product given by matrixmultiplication and unit being the n × n -matrix n with units on the diagonal and zerosin every other entry. An object in M n ( A ) M can be identified with n left A -modules M i , such that (cid:80) ni =1 a ji · m i ∈ M j for all a ji ∈ A and m i ∈ M i , where ≤ j ≤ n .It follows that M i = M for all ≤ i ≤ n . This means that any left M n ( A ) -module is of the form M n for a left A -module M . Similarly one proves that any left M n ( A ) -homomorphism Φ : M n → N n is of the form Φ = φ n for a left A -modulehomomorphism φ : M → N . It is easy to prove that the assignments M (cid:55)→ M n and φ (cid:55)→ φ n define an invertible functor with inverse given by the projection to the firstcomponent. This proves that A M and M n ( A ) M are equivalent categories. However, A and M n ( A ) are not isomorphic as algebras in general, since M n ( A ) might benoncommutative even if A is commutative. Motivated from this example we state the following definition. Definition 3.3.2. Two associative unital algebras A and B are said to be Moritaequivalent if A M and B M are equivalent categories. It follows from the previous discussion that two isomorphic algebras are Moritaequivalent and A is Morita equivalent to M n ( A ) for every n > 0. There are severalcharacterizations of Morita equivalence. For instance, we note that the functorfrom Example 3.3.1, which assigns to any left A -module M the M n ( A )-modulebstructions of Twist Star Products 67 M n and to any left A -module homomorphism φ : M → N the left M n ( A )-modulehomomorphism φ n : M n → N n , can be represented as the tensor product with the M n ( A )- A -bimodule M n ( A ) A n A . The latter is defined as A n , which is a left M n ( A )-module via matrix-vector multiplication and a right A -module by component-wisemultiplication from the right. Clearly the actions commute. Furthermore, for anyleft A -module M the tensor product M n ( A ) A n A ⊗ A M over A is isomorphic to theleft M n ( A )-module M n . The left M n ( A )-module isomorphism is given by M n (cid:51) m ... m n (cid:55)→ ⊗ A m + . . . + ⊗ A m n ∈ M n ( A ) A n A ⊗ A M , with inverse M n ( A ) A n A ⊗ A M (cid:51) a ... a n ⊗ A m (cid:55)→ a · m ... a n · m ∈ M n . One easily proves that these assignments respect the left M n ( A )-module actions. Itturns out that this is not a specific ramification of the Morita equivalence of A and M n ( A ) but rather a general construction underlying every Morita equivalence. Proposition 3.3.3 ([27] Cor. 2.4) . Two associative unital algebras A and B areMorita equivalent if and only if there is a B - A -bimodule B E A and an A - B -bimodule A E B such that A E B ⊗ B B E A ∼ = A A A and B E A ⊗ A A E B ∼ = B B B (3.24) as bimodules. The bimodules B E A and A E B are said to be Morita equivalence bimodules . Wewant to point out that there is an interpretation of Morita equivalence bimodulesas invertible morphisms in the following category: objects are defined as associativeunital k -algebras and morphisms are isomorphism classes of bimodules. Namely,let A , B and C be algebras and fix an A - B -bimodule A E B and a B - C -bimodule B E C .Then ⊗ B is associative up to isomorphism, with A A A and B B B functioning as unitmorphisms. A morphism A E B is invertible if and only if there is a morphism B E A such that (3.24) holds, i.e. if and only if it is a Morita equivalence bimodule. Thisconstitutes a (large) groupoid Pic with objects being associative unital algebras and(invertible) morphisms being Morita equivalence bimodules, called Picard groupoid .The set of A - B Morita equivalence bimodules is denoted by Pic( A , B ), while theset of self-Morita equivalence classes of A , the Picard group , is denoted by Pic( A ).The Morita equivalence class of an associative unital algebra A is defined to be theorbit of A in Pic and Pic( A ) measures in how many ways A is Morita equivalentto another algebra B in its orbit. We refer to [12] for more information on thisinterpretation of Morita equivalence.There is another characterization of Morita equivalence, affirming the similarityto Example 3.3.1. Recall that an element P ∈ M n ( A ) is said to be an idempotent if P = P and it is said to be full if the span of elements of the form M P N ∈ M n ( A )for M, N ∈ M n ( A ) equals M n ( A ).8 Chapter 3 Theorem 3.3.4. Two associative unital k -algebras A and B are Morita equivalentif and only if one of the following statements holds.i.) there is an equivalence B M → A M of categories;ii.) there is an A - B -bimodule A E B such that F A E B : B M (cid:51) B E (cid:55)→ A E B ⊗ B B E ∈ A M is an equivalence of categories;iii.) there is an A - B -bimodule A E B , which is finitely generated projective as left A -module and right B -module such that A ∼ = End B ( A E B ) and B ∼ = End A ( A E B ) are isomorphisms of algebras, where the first endomorphisms are right B -linearand the latter left A -linear;iv.) there is an n > and a full idempotent P ∈ M n ( A ) such that B ∼ = End A ( P A n ) = P M n ( A ) P is an isomorphism of algebras; This follows from [27] Thm. 2.6 and Thm. 2.8. In a next step we examine thatthe center of an algebra is a Morita invariant, where we proceed as in [32] Sec. 3.1. Corollary 3.3.5. If A and B are Morita equivalent, there is an isomorphism Z ( A ) ∼ = Z ( B ) of algebras. In particular, two commutative algebras are Morita equivalent ifand only if they are isomorphic. In other words, for every element of the Picard group Pic( A ) there is an au-tomorphism of the center of A . This determines a map h : Pic( A ) → Aut( Z ( A )).On the other hand, every automorphism of A can be interpreted as a self-Moritaequivalence bimodule, defining a map j : Aut( A ) → Pic( A ). Proposition 3.3.6. There are two group homomorphisms j : Aut( A ) → Pic( A ) and h : Pic( A ) → Aut( Z ( A )) . If A is commutative h ◦ j = id Pic( A ) and Pic( A ) = Aut( A ) (cid:110) ker h . In the above proposition, the action of an automorphism Φ ∈ Aut( A ) on a self-Morita equivalence bimodule E ∈ ker( h ) in the semidirect product Aut( A ) (cid:110) ker h ,is defined as the self-Morita equivalence bimodule E Φ , with left and right A -actionsgiven by a · e · b = Φ( a ) · e · Φ( b ) for all a, b ∈ A and e ∈ E .Since we are mainly interested in the algebra of smooth functions on a manifold,we are going to discuss its Morita equivalence classes and Picard group in detail inthe next example (c.f. [32] Ex. 3.5).bstructions of Twist Star Products 69 Example 3.3.7. Consider the associative unital algebra A = C ∞ ( M ) of smoothcomplex-valued functions on a smooth manifold M . If E → M is a smooth complexvector bundle, its space Γ ∞ ( E ) of smooth sections is a Morita equivalence bimodulebetween Γ ∞ (End( E )) and A . Moreover, every finitely generated projective A -moduleis of that form according to the Serre-Swan theorem (c.f. [84] Thm. 11.32). Inparticular, we recover the Morita equivalence of A and M n ( A ) for any n > , byemploying the trivial bundle E = C n × M → M . The kernel of h equals the group Pic( M ) of isomorphism classes of smooth complex line bundles on M . Furtherremark that there is an isomorphism c : Pic( M ) → H ( M, Z ) , given by the Chernclass map (see [65] Sec. 3.8) and Aut( A ) = Diff( M ) . Summing up we obtain Pic( C ∞ ( M )) = Diff( M ) (cid:110) H ( M, Z ) , where Diff( M ) acts on H ( M, Z ) via pull-back. In the next section we observe that the deformation theory is another Moritainvariant. This will have particular consequences for twist star products. We are mainly interested in Morita equivalence of star product algebras (see [108]for a review on this topic). An obvious question is, if Morita equivalence bimodulescan be deformed relative to a deformation of the corresponding algebras. Evenmore, if A and B are Morita equivalent algebras and A a formal deformation of A ,is there a formal deformation B of B , such that A and B are Morita equivalent?Following [27] Sec. 2.3 and [30], we give a positive answer to this question. Wefurther refer to [29, 31]. Afterwards we focus on formal deformations of Moritaequivalence bimodules of twist star product algebras, reviewing the results of [38].Fix an associative unital algebra A and a left A -module M for the moment.Assume that there is a formal deformation (cid:63) of A . The natural question arises, ifthere exists a left A = ( A [[ (cid:126) ]] , (cid:63) )-module structure • on the K [[ (cid:126) ]]-module M [[ (cid:126) ]],such that a • m = a · m + O ( (cid:126) ) for all a ∈ A and m ∈ M , where · denotes the left A -module action on M . If there are K -bilinear maps λ r : A × M → M such that a • m = a · m + (cid:88) r> (cid:126) r λ r ( a, m ) , we call M = ( M [[ (cid:126) ]] , • ) a formal deformation of ( M , · ) with respect to A . Twoformal deformations M = ( M [[ (cid:126) ]] , • ) and M (cid:48) = ( M [[ (cid:126) ]] , • (cid:48) ) of ( M , · ) with respectto A are said to be equivalent if there are K -linear maps T r : M → M such that T = id M + (cid:80) r> (cid:126) r T r extends to an A -module isomorphism T : M → M (cid:48) . Lemma 3.4.1 (c.f. [53]) . Let A = ( A [[ (cid:126) ]] , (cid:63) ) be a formal deformation of A . Then M n ( A ) ∼ = M n ( A )[[ (cid:126) ]] as K [[ (cid:126) ]] -modules and M n ( A ) is a formal deformation of M n ( A ) . Furthermore, if P ∈ M n ( A ) is an idempotent, P = 12 + (cid:18) P − (cid:19) (cid:63) (cid:63) (cid:112) P (cid:63) P − P ) (3.25) defines an idempotent on M n ( A ) such that P = P + O ( (cid:126) ) and P is full if and onlyif P is full. A -module A E there is afinitely generated projective left A -module A E deforming A E , which is unique up toequivalence. This answers the question we stated at the introduction of this section,leading to the following theorem (c.f. [27] Prop. 2.21). Theorem 3.4.2. Let A and B be two Morita equivalent algebras and A = ( A [[ (cid:126) ]] , (cid:63) ) a formal deformation of A . Then, there is a formal deformation B = ( B [[ (cid:126) ]] , (cid:63) (cid:48) ) of B such that A and B are Morita equivalent. Furthermore, there is a bijection Def( A ) ∼ = Der( B ) given by the Morita equivalence bimodules which deform A E B . Applying this to Example 3.3.7 we obtain a result on the action of the Picardgroup of a symplectic manifold on equivalence classes of symplectic star products. Corollary 3.4.3 ([31] Thm. 3.1) . Let ( M, ω ) be a symplectic manifold. The actionof the Picard group Pic( M ) on star products on ( M, ω ) is given by [ (cid:63) ] (cid:55)→ [ (cid:63) ] + 2 πic ( L ) , (3.26) where c ( L ) is the first Chern class of the corresponding line bundle L → M . Inparticular, the obtained star product (cid:63) (cid:48) is equivalent to (cid:63) if and only if the first Chernclass of the line bundle L is trivial. A generalization of this corollary to Poisson manifold is proven in [28] Thm. 3.11.The authors have to employ Kontsevich’s quantization map [70] and the curvature2-form of the line bundle in addition.In the following lines we include a Lie group symmetry into the picture. Thestrategy is to use a Drinfel’d twist on the universal enveloping algebra of the corre-sponding Lie algebra to deform the Morita equivalence bimodule corresponding to anequivariant line bundle in the end. Consider a G -equivariant line bundle pr : L → M for a Lie group G . This means that there are G -actions on L and M , respectively,such that g (cid:66) : L p → L g (cid:66) p for all g ∈ G and p ∈ M , where L p = pr − ( { p } ) ⊆ L is the fiber of p . In other words,we require the Lie group action to be linear on fibers. In particular, this impliespr( g (cid:66) q ) = g (cid:66) pr( q )for all g ∈ G and q ∈ L . Example 3.4.4. Let n > and consider the complex projective space CP n , whichis defined as the set of all orbits of the Lie group action Φ : C × × C n +1 × (cid:51) ( λ, z ) (cid:55)→ λ · z ∈ C n +1 × , where C × = C \ { } . Since Φ is free and proper, CP n can be structured as a smoothmanifold. On CP n there is the tautological line bundle L = { ( (cid:96), z ) ∈ CP n × C n +1 | z ∈ (cid:96) } (3.27) with projection π : L (cid:51) ( (cid:96), z ) (cid:55)→ (cid:96) ∈ CP n , where z ∈ (cid:96) means that z is an element ofthe orbit (cid:96) . It is equivariant with respect to the GL n +1 ( C ) -action given by matrix-vector multiplication. In fact, GL n +1 ( C ) × C n +1 → C n +1 descends to a Lie group bstructions of Twist Star Products 71 action on CP n , since it commutes with Φ . Then, the projection π intertwines thediagonal action on L and CP n , i.e. π ( A (cid:66) ( (cid:96), z )) = π (( A (cid:66) (cid:96) ) , ( A (cid:66) z )) = A (cid:66) (cid:96) = A (cid:66) ( π ( (cid:96), z )) for all A ∈ GL n +1 ( C ) and ( (cid:96), z ) ∈ L . This proves that π : L → CP n is indeed GL n +1 ( C ) -equivariant. An equivariant line bundle naturally induces Lie group actions on the smoothfunctions on the manifold and the smooth sections of the line bundle. Lemma 3.4.5. Let pr : L → M be a G -equivariant line bundle. Then ( g (cid:66) f )( p ) = f ( g − (cid:66) p ) , ( g (cid:66) s )( p ) = g (cid:66) ( s ( g − (cid:66) p )) , (3.28) where g ∈ G , f ∈ C ∞ ( M ) , p ∈ M and s ∈ Γ ∞ ( L ) , define G -actions on C ∞ ( M ) and Γ ∞ ( L ) . Moreover the G -actions respect the C ∞ ( M ) -bimodule structure of Γ ∞ ( L ) ,i.e. g (cid:66) ( f · s ) = ( g (cid:66) f ) · ( g (cid:66) s ) (3.29) for all g ∈ G , f ∈ C ∞ ( M ) and s ∈ Γ ∞ ( L ) . In other words, G acts by group-likeelements.Proof. Consider a section s ∈ Γ ∞ ( L ) and an element g ∈ G . Then g (cid:66) s ∈ Γ ∞ ( L )sincepr(( g (cid:66) s )( p )) = pr( g (cid:66) s ( g − (cid:66) p )) = g (cid:66) pr( s ( g − (cid:66) p )) = g (cid:66) ( g − (cid:66) p ) = p for all p ∈ M . It is an easy exercise to verify that (3.28) are G -actions. In fact( e (cid:66) f )( p ) = f ( p ), ( e (cid:66) s )( p ) = e (cid:66) s ( e − (cid:66) p ) = s ( p ),( g (cid:66) ( h (cid:66) f ))( p ) = f (( h − g − ) (cid:66) p ) = f (( gh ) − (cid:66) p ) = (( gh ) (cid:66) f )( p )and ( g (cid:66) ( h (cid:66) s ))( p ) = ( gh ) (cid:66) s (( h − g − ) (cid:66) p ) = (( gh ) (cid:66) s )( p ) for all g, h ∈ G , p ∈ M , f ∈ C ∞ ( M ) and s ∈ Γ ∞ ( L ). Furthermore, eq.(3.29) follows since( g (cid:66) ( f · s ))( p ) = g (cid:66) (( f · s )( g − (cid:66) p ))= g (cid:66) ( f ( g − (cid:66) p ) s ( g − (cid:66) p ))= f ( g − (cid:66) p ) g (cid:66) ( s ( g − (cid:66) p ))=(( g (cid:66) f ) · ( g (cid:66) s ))( p )for all g ∈ G , p ∈ M f ∈ C ∞ ( M ) and s ∈ Γ ∞ ( L ), where we used that the G -actionsare C -linear.The Lie group actions on functions and sections of the line bundle induce Liealgebra actions of the corresponding Lie algebra. They can be extended to Hopfalgebra actions of the universal enveloping algebra. We prove that those actionsrespect the bimodule structure of sections. Corollary 3.4.6. If pr : L → M is a G -equivariant line bundle, then Γ ∞ ( L ) is a U g -equivariant symmetric C ∞ ( M ) -bimodule, where g is the Lie algebra correspond-ing to G . Proof. By Lemma 3.4.5 there is a Lie group action G × Γ ∞ ( L ) → Γ ∞ ( L ). Recallthat the corresponding Lie algebra action g → End C (Γ ∞ ( L )) is defined by ξ (cid:66) s = dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( t · ξ ) (cid:66) s (3.30)for all ξ ∈ g and s ∈ Γ ∞ ( L ). Note that the triangle on the right hand side of(3.30) denotes the G -action. This extends uniquely to a left Hopf algebra action U g ⊗ Γ ∞ ( L ) → Γ ∞ ( L ). It remains to prove that the U g -action respects the C ∞ ( M )-module action. This is the case, since ξ (cid:66) ( f · s ) = dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( t · ξ ) (cid:66) ( f · s )= dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 (cid:18) (exp( t · ξ ) (cid:66) f ) · (exp( t · ξ ) (cid:66) s ) (cid:19) = (cid:18) dd t (exp( t · ξ ) (cid:66) f ) (cid:19) · (exp( t · ξ ) (cid:66) s ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 + (exp( t · ξ ) (cid:66) f ) · (cid:18) dd t (exp( t · ξ ) (cid:66) s ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t =0 =( ξ (cid:66) f ) · s + f · ( ξ (cid:66) s )=( ξ (1) (cid:66) f ) · ( ξ (2) (cid:66) s )for all ξ ∈ g , f ∈ C ∞ ( M ) and s ∈ Γ ∞ ( L ), where we used (3.29). It is sufficient toprove equivariance on primitive elements.After those general considerations we return to obstructions for twist star prod-ucts. Before proving the main theorem of this section we argue that any equivariantline bundle on a manifold can be twist deformed into a self Morita equivalence bi-module of the twisted functions if there is a Drinfel’d twist on the correspondinguniversal enveloping algebra. Lemma 3.4.7. Let L → M be a G -equivariant line bundle and (cid:63) F be a twist starproduct on M with Drinfel’d twist based on U g . Then there is an algebra isomor-phism ψ : ( C ∞ ( M )[[ (cid:126) ]] , (cid:63) F ) → End ( C ∞ ( M )[[ (cid:126) ]] ,(cid:63) F ) (Γ ∞ ( L )[[ (cid:126) ]] , · F ) (3.31) given by the twisted left C ∞ ( M ) -module action on Γ ∞ ( L ) .Proof. ψ is an algebra homomorphism since f · F ( g · F s ) = ( f (cid:63) F g ) · F s for all f, g ∈ C ∞ ( M ) and s ∈ Γ ∞ ( L ) by the 2-cocycle property. It is an isomorphism sinceΓ ∞ ( L ) is a self Morita equivalence bimodule for the undeformed algebra. NamelyΓ ∞ (End( L )) ∼ = C ∞ ( M ) since End( L ) ∼ = M × C is the trivial line bundle, whichimplies that ( C ∞ ( M ) , · ) → End ( C ∞ ( M ) , · ) (Γ ∞ ( L ) , · )is an algebra isomorphism. It follows that ψ is invertible in the (cid:126) -adic topology since ψ ( f )( s ) = f · s + O ( (cid:126) ) for all f ∈ C ∞ ( M ) and s ∈ Γ ∞ ( L ).Now it is clear how to obtain an obstruction in the symplectic setting: if the linebundle has non-trivial Chern class it follows that its twist deformation is a Moritaequivalence bimodule between the twisted functions and another star product whichis not isomorphic to the twist star product. This is a contradiction to Lemma 3.4.7,which proves that the induced star product has to be the twisted product itself.bstructions of Twist Star Products 73 Theorem 3.4.8 ([38] Thm. 11) . Let ( M, ω ) be a symplectic manifold such that M is a G -space in addition. The following two properties are mutually exclusive.i.) There is a G -equivariant smooth complex line bundle on M with non-trivialChern class;ii.) There is a twist star product on ( M, ω ) for a Drinfel’d twist based on U g [[ (cid:126) ]] ,where g is the Lie algebra corresponding to G ;Proof. Assume that i. ) and ii. ) both hold on the same symplectic manifold and G -space ( M, ω ). In particular, there are two algebra isomorphisms( C ∞ ( M )[[ (cid:126) ]] , (cid:63) F ) ψ −→ End ( C ∞ ( M )[[ (cid:126) ]] ,(cid:63) F ) (Γ ∞ ( L )[[ (cid:126) ]] , · F ) → ( C ∞ ( M )[[ (cid:126) ]] , (cid:63) (cid:48) ) , the first according to Lemma 3.4.7 and the second according to Theorem 3.4.2 andTheorem 3.3.4 iii. ). More precisely, (cid:63) F denotes the twist star product on ( M, ω ) thatexists by assumption ii. ), while (cid:63) (cid:48) denotes the star product which exists correspond-ing to the deformation (cid:63) F of C ∞ ( M ) and · F of Γ ∞ ( L ). According to Corollary 3.4.3this implies c ( L ) = 0 in contradiction to assumption i. ) and we conclude the proofof the theorem.Note that Theorem 3.4.8 only gives obstructions for Drinfel’d twists on U g if g is the Lie algebra corresponding to the symmetry G of the line bundle. Neverthelessthis might rule out a huge class of candidates, like it is depicted in the followingexample. Example 3.4.9. Consider the tautological line bundle L on CP n discussed in Ex-ample 3.4.4. It is G = GL n +1 ( C ) -equivariant and according to Corollary 3.4.6 thisimplies that the sections Γ ∞ ( L ) of L are a g = gl n +1 ( C ) -equivariant C ∞ ( CP n ) -bimodule. It is well-known that c ( L ) (cid:54) = 0 and that the Fubini-Study -form ω FS isa symplectic structure on CP n . According to Theorem 3.4.8 there is no twist starproduct on ( CP n , ω FS ) with a twist based on U g . On the other hand, according toProposition 3.1.6, there are star product quantizing ( CP n , ω FS ) . A particular classof them, given by Berezin-Toeplitz quantization, is described in [98]. hapter 4Braided Cartan Calculi After recalling the well-known theory of quasi-triangular Hopf algebras and theirrepresentations we proceed by elaborating some original results of the author. Themain statement, i.e. the construction of the braided Cartan calculus on an arbitrarybraided commutative algebra, can be found in [111] Section 3, while we add moredetails, proofs and lemmas in the following sections. To give some insight on howseveral formulas were deduced as natural generalizations, we start by recalling theconstruction of the Cartan calculus on a commutative algebra over a commutativeunital ring in Section 4.1 (c.f. [72, 97]). Note that Henri Cartan laid down the foun-dation of the Cartan calculus on a manifold in [34] and with it all of its fundamentalingredients. The study of multivector fields on a manifold and the generalization ofthe Lie bracket of vector fields to a Gerstenhaber bracket (c.f. [59]) goes back toearly works [85, 99, 100] of Schouten and Nijenhuis. In the context of noncommuta-tive geometry [36] the Schouten-Nijenhuis bracket is discussed in [46]. The notionof Cartan calculi on noncommutative algebras goes back to Woronowicz. In [112]he generalized the de Rham differential of a smooth manifold to a noncommutativesetting, focusing on its algebraic properties. Namely, a first order differential calcu-lus over a Hopf algebra A is given as a bimodule Γ over the algebra, together witha k -linear map d : A → Γ, which satisfies a Leibniz rule d( a · b ) = d( a ) · b + a · d( b )for all a, b ∈ A . In this context, Γ represents the bimodule of differential forms on A . Furthermore, he assumes that Γ = A · d A is generated by A and d. If the firstorder calculus is bicovariant , i.e. if (cid:80) k a k d b k = 0 implies (cid:80) k ∆( a k )(id ⊗ d)∆( b k ) = 0and (cid:80) k ∆( a k )(d ⊗ id)∆( b k ) = 0, it admits an extension to the exterior algebra of Γ,which we identify with higher order differential forms. Noncommutative differentialcalculi based on derivations rather than generalizations of differential forms are dis-cussed in [45, 46, 101, 102]. It is a necessity to employ modules over the center of anoncommutative algebra in this case, since derivations are only a central bimodule.Following the approach of [9, 10], we construct a braided derivation based calculusvia bimodules over the whole algebra for the huge class of braided commutativealgebras, in accordance with [112]. In the Sections 4.2-4.4 we repeat the construc-tion of Section 4.1 in the category of equivariant braided symmetric bimodules of abraided commutative algebra. Namely, we shape the braided Gerstenhaber algebraof braided multivector fields with the braided Schouten-Nijenhuis bracket in Sec-tion 4.2, while we construct the braided Graßmann algebra of braided differentialforms in Section 4.3. In particular we define a braided analogue of the de Rham dif-ferential in the latter section via a generalization of the Chevalley-Eilenberg formula.74raided Cartan Calculi 75Then, in Section 4.4, our arrangements culminate in the definition of the braidedCartan calculus. The corresponding braided Lie derivative, insertion and de Rhamdifferential are related by graded braided commutators and resemble the formulasknown from differential geometry. It would be interesting to compare the braidedCartan calculus to the noncommutative calculus [105, 106] and generalize our con-struction to Lie-Rinehart pairs (see [66]). As an application we prove in Section 4.5that the notion of covariant derivative naturally generalizes to the braided Cartancalculus. In particular we compute the curvature and torsion of an equivariant co-variant derivative, we prove that a given equivariant covariant derivative on thealgebra extends to braided differential forms and multivector fields and we give anexistence and uniqueness theorem for an equivariant Levi-Civita covariant derivativeof a given non-degenerate equivariant metric. Covariant derivatives on noncommu-tative algebras are also discussed in e.g. [2, 3, 4, 6, 10, 17, 18, 45, 55, 74, 91]. Ingeneral one has to distinguish between left and right covariant derivatives. However,we prove that these notions coincide in the equivariant case. Finally, in Section 4.6we clarify that Drinfel’d twist gauge equivalence is compatible with the constructionof the braided Cartan calculus and the notion of equivariant covariant derivative.For a given twist we describe a deformation of the data of a braided Cartan calcu-lus and prove that it is isomorphic to the braided Cartan calculus with respect tothe twisted algebra and twisted triangular structure. This recovers the well-knowntwisted Cartan calculus and integrates it in the setting of braided Cartan calculi.Furthermore, the twist deformation of an equivariant covariant derivative can beviewed as an equivariant covariant derivative with respect to the twisted universal R -matrix on the twisted algebra via the same isomorphism. Recalling differential geometry, the Cartan calculus on a smooth manifold M isbased on the commutative algebra C ∞ ( M ) of smooth functions. The Gerstenhaberalgebra ( X • ( M ) , ∧ , (cid:74) · , · (cid:75) ) of multivector fields and the Graßmann algebra (Ω • ( M ) , ∧ )of differential forms are graded symmetric C ∞ ( M )-bimodules. Using the de Rhamdifferential d and the insertion i X of vector fields X ∈ X ( M ) into the first slot of adifferential form, one defines the Lie derivative L X = [i X , d], the bracket denotingthe commutator of endomorphisms. On factorizing multivector fields X = X ∧ · · · ∧ X k ∈ X k ( M ) one defines i X = i X · · · i X k and the Lie derivative by L X = i X d − ( − k − di X . Linear extension leads to homogeneous maps d : Ω • ( M ) → Ω • +1 ( M ), i X : Ω • ( M ) → Ω •− k ( M ) and L X : Ω • ( M ) → Ω •− ( k − ( M ), where X ∈ X k ( M ). In particular we candefine the Lie derivative L X = [i X , d] of any multivector field X ∈ X • ( M ) using thegraded commutator [ · , · ]. One proves that i : X • ( M ) × Ω • ( M ) → Ω • ( M ) is C ∞ ( M )-bilinear. If X ∈ X ( M ) we obtain derivations i X and L X of (Ω • ( M ) , ∧ ). It is theaim of this subsection to reduce the Cartan calculus to its algebraic properties andreconstruct them for any commutative algebra. In the next subsections this will begeneralized to braided commutative algebras.6 Chapter 4Fix a commutative algebra A in the following. An endomorphism Φ : A → A issaid to be a derivation of A if Φ( ab ) = Φ( a ) b + a Φ( b ) for all a, b ∈ A . We denotethe k -module of all derivations of A by Der( A ). Lemma 4.1.1. The derivations Der( A ) of A form a Lie algebra with Lie bracketgiven by the commutator [ · , · ] of endomorphisms. Furthermore, Der( A ) is a sym-metric A -bimodule with left and right A -module actions defined on X ∈ Der( A ) by ( a · X )( b ) = aX ( b ) = ( X · a )( b ) for all a, b ∈ A .Proof. For any a ∈ A and X ∈ Der( A ), a · X is a derivation of A , since( a · X )( bc ) = aX ( bc ) = a ( X ( b ) c + bX ( c )) = (( a · X )( b )) c + b ( a · X )( c )holds for all b, c ∈ A by the commutativity of A . Therefore we obtain well-defined A -module actions by the associativity of A . They are symmetric by definition. Theendomorphisms of A are a Lie algebra with respect to the commutator. It remainsto prove that [ · , · ] closes in Der( A ). Let X, Y ∈ Der( A ). Then[ X, Y ]( ab ) = X ( Y ( ab )) − Y ( X ( ab ))= X ( Y ( a ) b + aY ( b )) − Y ( X ( a ) b + aX ( b ))= X ( Y ( a )) b + Y ( a ) X ( b ) + X ( a ) Y ( b ) + aX ( Y ( b )) − Y ( X ( a )) b − X ( a ) Y ( b ) − Y ( a ) X ( b ) − aY ( X ( b ))=( X ( Y ( a )) − Y ( X ( a ))) b + a ( X ( Y ( b )) − Y ( X ( b )))=([ X, Y ]( a )) b + a ([ X, Y ]( b ))for all a, b ∈ A , which means that [ X, Y ] ∈ Der( A ).Next we consider the tensor algebra T • Der( A ) = A ⊕ Der( A ) ⊕ (Der( A ) ⊗ A Der( A )) ⊕ · · · of the symmetric A -bimodule Der( A ). It is a graded noncommutative algebra withmultiplication given by the tensor product ⊗ A over A . The left and right A -actions,defined on factorizing elements X = X ⊗ A · · · ⊗ A X k ∈ T k Der( A ) and a ∈ A by a · X = ( a · X ) ⊗ A · · · ⊗ A X k = X · a, structure T • Der( A ) as a symmetric A -bimodule. The quotient of T • Der( A ) with theideal generated by expressions X ⊗ A Y − ( − k(cid:96) Y ⊗ A X , where X ∈ T k Der( A ) and Y ∈ T (cid:96) Der( A ), is the Graßmann algebra or exterior algebra Λ • Der( A ) of Der( A ).The induced product, the wedge product , is denoted by ∧ . Since the ideal respectsthe structure of the tensor algebra it follows that Λ • Der( A ) is a graded algebra and asymmetric A -bimodule. In particular, there are symmetric left and right A -moduleactions such that on factorizing elements X = X ∧ · · · ∧ X k ∈ Λ k Der( A ) a · X = a ∧ X = ( a · X ) ∧ · · · ∧ X k = X ∧ a = X · a holds for all a ∈ A . Note that Λ • Der( A ) is graded commutative in addition, i.e. X ∧ Y = ( − k(cid:96) Y ∧ X raided Cartan Calculi 77for all X ∈ Λ k Der( A ) and Y ∈ Λ (cid:96) Der( A ). In the following we write X • ( A ) instead ofΛ • Der( A ) and call it the multivector fields of A . We define the Schouten-Nijenhuisbracket (cid:74) · , · (cid:75) : X • ( A ) × X • ( A ) → X • ( A ) on factorizing elements X = X ∧ · · · ∧ X k ∈ X k ( A ) and Y = Y ∧ · · · ∧ X (cid:96) ∈ X (cid:96) ( A ) by (cid:74) X, Y (cid:75) = k (cid:88) i =1 (cid:96) (cid:88) j =1 ( − i + j [ X i , X j ] ∧ X ∧ · · · ∧ (cid:99) X i ∧ · · · X k ∧ Y ∧ · · · ∧ (cid:98) Y j ∧ · · · Y (cid:96) (4.1)if k, (cid:96) > 0, where (cid:99) X i and (cid:98) Y j are omitted in the above wedge product. We further set (cid:74) a, b (cid:75) = 0 for a, b ∈ A and (cid:74) X, a (cid:75) = X ( a ) = − (cid:74) a, X (cid:75) for all X ∈ X ( A ) and a ∈ A and extend (cid:74) · , · (cid:75) k -bilinearly. Proposition 4.1.2. The Schouten-Nijenhuis bracket structures X • ( A ) as a Ger-stenhaber algebra. Namely, (cid:74) · , · (cid:75) : X k ( A ) × X (cid:96) ( A ) → X k + (cid:96) − ( A ) is a graded (withrespect to the degree shifted by ) Lie bracket, i.e. it is graded skew-symmetric (cid:74) Y, X (cid:75) = − ( − ( k − (cid:96) − (cid:74) X, Y (cid:75) and satisfies the graded Jacobi identity (cid:74) X, (cid:74) Y, Z (cid:75)(cid:75) = (cid:74)(cid:74) X, Y (cid:75) , Z (cid:75) + ( − ( k − (cid:96) − (cid:74) Y, (cid:74) X, Z (cid:75)(cid:75) , such that the graded Leibniz rule (cid:74) X, Y ∧ Z (cid:75) = (cid:74) X, Y (cid:75) ∧ Z + ( − ( k − (cid:96) Y ∧ (cid:74) X, Z (cid:75) holds in addition, where X ∈ X k ( A ) , Y ∈ X (cid:96) ( A ) and Z ∈ X • ( A ) . It is the uniqueGerstenhaber bracket on X • ( A ) such that (cid:74) X, a (cid:75) = X ( a ) and (cid:74) X, Y (cid:75) = [ X, Y ] for all a ∈ A and X, Y ∈ X ( A ) .Proof. By counting degrees we see that (cid:74) · , · (cid:75) : X k ( A ) × X (cid:96) ( A ) → X k + (cid:96) − ( A ). Thismeans that (cid:74) · , · (cid:75) is homogeneous with respect to the degree shifted by 1. Then,using the defining formula (4.1) it is an exercise to verify that (cid:74) · , · (cid:75) is a Gerstenhaberbracket. In fact it is sufficient to prove this in degree 0 and 1 since afterwards aninductive argument, using the graded Leibniz rule, implies that the identities arevalid in any degree. Remark that this is how (4.1) was engineered: one defines (cid:74) · , · (cid:75) in degree 0 and 1, extends (cid:74) X, · (cid:75) : X • ( A ) → X • ( A ) as a derivation of the wedgeproduct for all X ∈ X ( A ) and imposes the graded Leibniz rule and graded skew-symmetry afterwards. Similarly one extends (cid:74) a, · (cid:75) : X • ( A ) → X •− ( A ) as a gradedderivation for all a ∈ A . In particularly this clarifies the uniqueness by controllingthe values of the Gerstenhaber bracket in degree 0 and 1.A preliminary stage to differential forms is given by the dual space Ω ( A ) =Hom A (Der( A ) , A ) corresponding to Der( A ), i.e. by the A -linear maps Der( A ) → A .They form a symmetric A -bimodule with left and right A -action given for a ∈ A and ω ∈ Ω ( A ) by ( a · ω )( X ) = a · ω ( X ) = ( ω · a )( X )8 Chapter 4for all X ∈ Der( A ). The corresponding Graßmann algebra is denoted by (Ω • ( A ) , ∧ ).There is a non-degenerate A -bilinear dual pairing (cid:104)· , ·(cid:105) : Ω ( A ) ⊗ Der( A ) → A definedby (cid:104) ω, X (cid:105) = ω ( X ) for all ω ∈ Ω ( A ) and X ∈ Der( A ). We can also view thisas inserting a derivation X into the functional ω ∈ Ω ( A ). In this picture it iscustomized to write i X ω = (cid:104) ω, X (cid:105) . Extending i X : Ω • ( A ) → Ω •− ( A ) as a gradedderivation of degree − − 1. On factorizing multivector fields X = X ∧ · · · ∧ X k ∈ X k ( A ) we seti X = i X · · · i X k . This determines the A -bilinear insertion of multivector fields i : X • ( A ) × Ω • ( A ) → Ω • ( A ), such that i X : Ω • ( A ) → Ω •− k ( A ) is homogeneous of degree − k for any X ∈ X k ( A ). We further define a k -linear map d : Ω • ( A ) → Ω • +1 ( A ) on Ω • ( A ) by settingi X (d a ) = X ( a ), (d α )( X, Y ) = X (i Y α ) − Y (i X α ) − i [ X,Y ] α on a ∈ A and α ∈ Ω ( A ), for all X, Y ∈ X ( A ) and extending d to higher factorizingelements as a graded derivation of degree 1, i.e. by imposingd( ω ∧ η ) = d ω ∧ η + ( − k ω ∧ d η for all ω ∈ Ω k ( A ) and η ∈ Ω • ( A ). Alternatively one can directly axiomatise the Chevalley-Eilenberg formula (d ω )( X , . . . , X k ) = k (cid:88) i =0 ( − i X i ( ω ( X , · · · , (cid:99) X i , · · · , X k ))+ (cid:88) i The derivation d of degree is a differential on (Ω • ( A ) , ∧ ) .Proof. The homogeneity is clear. Let X, Y, Z ∈ X ( A ). Then(d a )( X, Y ) = X (i Y (d a )) − Y (i X (d a )) − i [ X,Y ] (d a )= X ( Y ( a )) − Y ( X ( a )) − [ X, Y ]( a )=0for all a ∈ A by the definition of the Lie bracket of vector fields. Furthermore(d ω )( X, Y, Z ) = X ((d ω )( Y, Z )) − Y ((d ω )( X, Z )) + Z ((d ω )( X, Y )) − (d ω )([ X, Y ] , Z ) + (d ω )([ X, Z ] , Y ) − (d ω )([ Y, Z ] , X )= X (cid:18) Y (i Z ω ) − Z (i Y ω ) − i [ Y,Z ] ω (cid:19) − Y (cid:18) X (i Z ω ) − Z (i X ω ) − i [ X,Z ] ω (cid:19) + Z (cid:18) X (i Y ω ) − Y (i X ω ) − i [ X,Y ] ω (cid:19) raided Cartan Calculi 79 − (cid:18) [ X, Y ](i Z ω ) − Z (i [ X,Y ] ω ) − i [[ X,Y ] ,Z ] ω (cid:19) + [ X, Z ](i Y ω ) − Y (i [ X,Z ] ω ) − i [[ X,Z ] ,Y ] ω − (cid:18) [ Y, Z ](i X ω ) − X (i [ Y,Z ] ω ) − i [[ Y,Z ] ,X ] ω (cid:19) =0for all ω ∈ Ω ( A ), where we used the Jacobi identity of the Lie bracket of vectorfields. Since d is a graded derivation it is sufficient to verify d ω = 0 for ω ∈ Ω k ( A )with k < = 0.Differential forms on A are generated by A and the image of the differential dvia the wedge product. Definition 4.1.4. The smallest differential graded subalgebra Ω • ( A ) ⊆ Ω • ( A ) suchthat A ⊆ Ω • ( A ) is said to be the Graßmann algebra of differential forms on A . Since the former operations on Ω • ( A ) respect the wedge product ∧ , they alsoclose in Ω • ( A ). Lemma 4.1.5. The differential forms (Ω • ( A ) , ∧ ) are a graded commutative algebraand a symmetric A -bimodule. Every element ω ∈ Ω k ( A ) is a finite sum of expres-sions of the form a d a ∧ · · · ∧ d a k , where a , . . . , a k ∈ A . The restrictions of theinsertion of multivector fields and differential to Ω • ( A ) are well-defined. The last operation missing to describe the Cartan calculus is the Lie derivative .It is defined on any X ∈ X • ( A ) by L X = [i X , d] , where the bracket denotes the graded commutator. If X ∈ X k ( A ), we obtain ahomogeneous map L X : Ω • ( A ) → Ω •− ( k − ( A ) of degree − ( k − k = 1, L X isa derivation with respect to the wedge product. The first statement holds becausei X is homogeneous of degree − k and d is homogeneous of degree 1. For the secondstatement let X ∈ X ( A ). Then L X ( ω ∧ η ) =i X d( ω ∧ η ) − ( − ( − · di X ( ω ∧ η )=i X (d ω ∧ η + ( − k ω ∧ d η ) + d(i X ω ∧ η + ( − k ω ∧ i X η )= L X ω ∧ η + ( − k +1 d ω ∧ i X η + ( − k i X ω ∧ d η + ( − k ω ∧ i X d η + ( − k − i X ω ∧ d η + ( − k d ω ∧ i X η + ( − k ω ∧ di X η = L X ω ∧ η + ω ∧ L X η for all ω ∈ Ω k ( A ) and η ∈ Ω • ( A ), using that i X and d are graded derivations of thewedge product of degree − Lemma 4.1.6. One has L a ω = − (d a ) ∧ ω and L X ∧ Y = i X L Y + ( − (cid:96) L X i Y for all a ∈ A , X ∈ X • ( A ) , Y ∈ X (cid:96) ( A ) and ω ∈ Ω • ( A ) . If X, Y ∈ X ( A )[ L X , i Y ] = i [ X,Y ] holds. Proof. By the definition of the Lie derivative and since d is a graded derivation ofdegree 1 L a ω =[i a , d] ω =i a d ω − ( − · ( − di a ω = a ∧ d ω − d( a ∧ ω )= a ∧ d ω − (d a ∧ ω + ( − a ∧ d ω )= − d a ∧ ω follows. The relation L X ∧ Y =[i X ∧ Y , d] = [i X i Y , d] = i X [i Y , d] + ( − ( − (cid:96) ) · [i X , d]i Y =i X L Y + ( − (cid:96) L X i Y is obtained from the graded Leibniz rule of the graded commutator. The missingformula trivially holds on differential forms of degree 0, while for ω ∈ Ω ( A ) oneobtains [ L X , i Y ] ω = L X i Y ω − ( − · i Y L X ω =(i X d + di X )i Y ω − i Y (i X d + di X ) ω = X (i Y ω ) + 0 − (d ω )( X, Y ) − Y (i X ω )=i [ X,Y ] ω for all X, Y ∈ X ( A ). Since [ L X , i Y ] is a graded derivation this is all we have toprove.It follows the main theorem of this subsection, where we relate the operationsgiven by the Lie derivative, the insertion and the de Rham differential via gradedcommutators. It describes the Cartan calculus on A . Theorem 4.1.7 (Cartan Calculus) . Let A be a commutative algebra. Then [ L X , L Y ] = L (cid:74) X,Y (cid:75) , [ L X , i Y ] =i (cid:74) X,Y (cid:75) , [ L X , d] =0 , [i X , i Y ] =0 , [i X , d] = L X , [d , d] =0 hold for all X, Y ∈ X • ( A ) .Proof. We are going to prove all formulas in reversed order. First, [d , d] = 2d = 0follows since d is a differential, which we proved in Lemma 4.1.3. The next formula,[i X , d] = L X , holds by definition of the Lie derivative for all X ∈ X • ( A ). Let X ∈ X k ( A ) and Y ∈ X (cid:96) ( A ). Then[i X , i Y ] = i X i Y − ( − k · (cid:96) i Y i X = i X ∧ Y − ( − k · (cid:96) i Y ∧ X = 0by the graded commutativity of the wedge product. For the next equation we utilizethe graded Jacobi identity of the graded commutator to conclude[ L X , d] = [[i X , d] , d] = [i X , [d , d]] + ( − · [[i X , d] , d] = 0 − [ L X , d]raided Cartan Calculi 81for all X ∈ X • ( A ), which implies [ L X , d] = 0. Recall that for any a ∈ A , X ∈ X ( A )and any homogeneous element Y = Y ∧ · · · ∧ Y (cid:96) ∈ X (cid:96) ( A ) (cid:74) a, Y (cid:75) = (cid:96) (cid:88) j =1 ( − j Y j ( a ) Y ∧ · · · ∧ (cid:98) Y j ∧ · · · ∧ Y (cid:96) and (cid:74) X, Y (cid:75) = (cid:88) j =1 Y ∧ · · · ∧ Y j − ∧ [ X, Y j ] ∧ Y j +1 ∧ · · · ∧ Y (cid:96) hold. Together with Lemma 4.1.6 this implies for all ω ∈ Ω • ( A )i (cid:74) a,Y (cid:75) ω = (cid:96) (cid:88) j =1 ( − j i Y j ( a ) Y ∧···∧ (cid:99) Y j ∧··· Y (cid:96) ω = − d a ∧ i Y ω + ( − (cid:96) (cid:18) (cid:96) (cid:88) j =1 ( − (cid:96) − j i Y j (d a )i Y · · · (cid:99) i Y j · · · i Y (cid:96) ω + ( − (cid:96) d a ∧ i Y ω (cid:19) = − d a ∧ i Y ω + ( − (cid:96) (cid:18) (cid:96) (cid:88) j =2 ( − (cid:96) − j i Y j (d a )i Y · · · (cid:99) i Y j · · · i Y (cid:96) ω + ( − (cid:96) − i Y (d a ∧ i Y · · · i Y (cid:96) ω ) (cid:19) = · · · = − d a ∧ i Y ω + ( − (cid:96) (cid:18) i Y (cid:96) (d a )i Y · · · i Y (cid:96) − ω − i Y · · · i Y (cid:96) − (d a ∧ i Y (cid:96) ω ) (cid:19) = − d a ∧ i Y ω + ( − (cid:96) i Y · · · i Y (cid:96) − (cid:18) i Y (cid:96) (d a ) ∧ ω − d a ∧ i Y (cid:96) ω (cid:19) = − d a ∧ i Y ω + ( − (cid:96) i Y (d a ∧ ω )= L a i Y ω − ( − (cid:96) i Y L a ω =[ L a , i Y ] ω, and [ L X , i Y ] =[ L X , i Y ]i Y ∧···∧ Y (cid:96) + ( − · i Y [ L X , i Y ∧···∧ Y (cid:96) ]=i [ X,Y ] i Y ∧···∧ Y (cid:96) + i Y ([ L X , i Y ]i Y ∧···∧ Y (cid:96) + i Y [ L X , i Y ∧···∧ Y (cid:96) ])= · · · = (cid:96) (cid:88) j =1 i Y ∧···∧ Y j − ∧ [ X,Y j ] ∧ Y j +1 ∧···∧ Y (cid:96) =i (cid:74) X,Y (cid:75) , where we used the graded Leibniz rule of the graded commutator and Lemma 4.1.6.This proves [ L X , i Y ] = i (cid:74) X,Y (cid:75) for all X ∈ X k ( A ) with k < Y ∈ X • ( A ). Weassume now inductively that this formula holds for a fixed k > 0. Let X ∈ X k ( A ), Y ∈ X ( A ) and Z ∈ X (cid:96) ( A ) for an arbitrary (cid:96) ∈ N . Then[ L X ∧ Y , i Z ] =[i X L Y − L X i Y , i Z ]2 Chapter 4=i X [ L Y , i Z ] + ( − · (cid:96) [i X , i Z ] L Y − L X [i Y , i Z ] − ( − · (cid:96) [ L X , i Z ]i Y =i X i (cid:74) Y,Z (cid:75) + 0 − − (cid:96) − i (cid:74) X,Z (cid:75) i Y =i (cid:74) X ∧ Y,Z (cid:75) , where we utilized the graded Leibniz rules of (cid:74) · , · (cid:75) and [ · , · ] as well as Lemma 4.1.6.We conclude that [ L X , i Y ] = i (cid:74) X,Y (cid:75) holds for all X, Y ∈ X • ( A ). For the remainingformula we note that[ L X , L Y ] =[ L X , [i Y , d]]=[[ L X , i Y ] , d] + ( − ( k − · (cid:96) [i Y , [ L X , d]]=[i (cid:74) X,Y (cid:75) , d] + 0= L (cid:74) X,Y (cid:75) holds for all X ∈ X k ( A ) and Y ∈ X (cid:96) ( A ). This concludes the proof of the theorem.The motivating example is of course the Cartan calculus on A = C ∞ ( M ) for asmooth manifold M . In the next sections we are going to repeat the construction ofthe Cartan calculus, however, in a more general setting. The commutative algebrais replaced by a braided commutative one. Consistently, the braided symmetry hasto be transferred to every involved object and morphism. In the words of Section 2.5we have to work in the symmetric braided monoidal category of equivariant braidedsymmetric bimodules. In this section we intend to motivate why vector fields on braided commutativealgebras should be represented by braided derivations rather than by usual deriva-tions. The braided derivations are an equivariant braided symmetric bimodule withrespect to the adjoint Hopf algebra action and a braided Lie algebra with braidedLie bracket given by the braided commutator. In this sense the braided deriva-tions are the ”correct” generalization of vector fields in the category of equivariantbraided symmetric bimodules. Keeping track of the braided symmetry we constructthe braided Graßmann algebra of braided derivations and furthermore extend thebraided commutator via a graded braided Leibniz rule to higher order multivectorfields. The latter turns out to be a braided Gerstenhaber bracket, structuring thebraided multivector fields as a braided Gerstenhaber algebra.Fix a triangular Hopf algebra ( H, R ) and a braided commutative algebra A .Recall that this means that A is an associative unital left H -module algebra suchthat b · a = ( R − (cid:66) a ) · ( R − (cid:66) b ) for all a, b ∈ A . It is well known that the endomorphisms End k ( A ) of A can be structured as a left H -module algebra via the adjoint action ( ξ (cid:66) Φ)( a ) = ξ (1) (cid:66) (Φ( S ( ξ (2) ) (cid:66) a ))for all a ∈ A , where Φ ∈ End k ( A ). The action on End k ( A ) is well-defined sincethe H -action on A is k -linear, it respects the product, which is the concatenation,raided Cartan Calculi 83because(( ξ (1) (cid:66) Φ)( ξ (2) (cid:66) Ψ))( a ) = ξ (1) (cid:66) (Φ(( S ( ξ (2) ) ξ (3) ) (cid:66) (Ψ( S ( ξ (4) ) (cid:66) a ))))= ξ (1) (cid:66) (Φ(Ψ( S ( ξ (2) ) (cid:66) a )))=( ξ (cid:66) (Φ ◦ Ψ))( a )for all Φ , Ψ ∈ End k ( A ) and a ∈ A and it respects the unit endomorphism id A , since( ξ (cid:66) id A )( a ) = ξ (1) (cid:66) (id A ( S ( ξ (2) ) (cid:66) a )) = ( ξ (1) S ( ξ (2) )) (cid:66) a = (cid:15) ( ξ )id A ( a )for all a ∈ A and ξ ∈ H . It strikes that we did not utilize the triangular structureof H or the braided commutativity of A at all. In fact, it follows by the samecomputation that End k ( A ) is a module algebra for any Hopf algebra and any modulealgebra A . However, the triangular structure matters if one wants to restrict the H -module action of endomorphisms to derivations. This is not possible in generalunless H is cocommutative. Moreover, the Lie bracket of derivations is not H -equivariant in general unless H is cocommutative. Also, for all a, b, c ∈ A and X ∈ Der( A )( a · X )( b · c ) = ( a · X )( b ) · c + ( R − (cid:66) b ) · (( R − (cid:66) a ) · X )( c )further suggests that Der( A ) is not an interesting object in this setting since it is nota left A -module for the canonical module action in general unless A is commutativeand H is cocommutative, i.e. unless R = 1 ⊗ 1. To fix all of these problems wedefine braided derivations as the k -linear endomorphisms X : A → A of A whichsatisfy X ( a · b ) = X ( a ) · b + ( R − (cid:66) a ) · ( R − (cid:66) X )( b ) (4.2)for all a, b ∈ A . The k -module of braided derivations on A is denoted by Der R ( A ). Lemma 4.2.1. The braided derivations on A form a braided Lie algebra with respectto the braided commutator of endomorphisms [ X, Y ] R = XY − ( R − (cid:66) Y )( R − (cid:66) X ) , where X, Y ∈ Der R ( A ) . Furthermore, Der R ( A ) is an H -equivariant braided sym-metric A -bimodule with respect to the adjoint Hopf algebra action and the left andright A -module actions defined by ( a · X )( b ) = a · X ( b ) and ( X · a )( b ) = X ( R − (cid:66) b ) · ( R − (cid:66) a ) (4.3) for all a, b ∈ A and X ∈ Der R ( A ) .Proof. Let ξ ∈ H , a, b, c ∈ A and X, Y, Z ∈ Der R ( A ). We split the proof of thelemma in two parts.i.) (Der R ( A ) , [ · , · ] R ) is a braided Lie algebra: first of all, the adjoint H -actionis well-defined on Der R ( A ), since( ξ (cid:66) X )( a · b ) = ξ (1) (cid:66) (cid:18) X (( S ( ξ (2) ) (1) (cid:66) a ) · ( S ( ξ (2) ) (1) (cid:66) a )) (cid:19) = ξ (1) (cid:66) (cid:18) X ( S ( ξ (3) ) (cid:66) a ) · ( S ( ξ (2) ) (cid:66) b )4 Chapter 4+ (( R − S ( ξ (3) )) (cid:66) a )( R − (cid:66) X )( S ( ξ (2) ) (cid:66) b ) (cid:19) =( ξ (1) (cid:66) X )(( ξ (2) S ( ξ (5) )) (cid:66) a ) · (( ξ (3) S ( ξ (4) )) (cid:66) b )+ (( ξ (1) R − S ( ξ (5) )) (cid:66) a )(( ξ (2) R − ) (cid:66) X )(( ξ (3) S ( ξ (4) )) (cid:66) b )=( ξ (1) (cid:66) X )(( ξ (2) S ( ξ (3) )) (cid:66) a ) · b + (( R − ξ (2) S ( ξ (3) )) (cid:66) a )(( R − ξ (1) ) (cid:66) X )( b )=( ξ (cid:66) X )( a ) · b + ( R − (cid:66) a )( R − (cid:66) ( ξ (cid:66) X ))( b )by the quasi-cocommutativity of ∆ and the anti-coalgebra property of S . Itremains to prove that [ X, Y ] R ∈ Der R ( A ), since the braided commutator ofendomorphisms is clearly a braided Lie bracket. Using the hexagon relationswe see that the difference of X ( Y ( a · b )) = X ( Y ( a ) · b + ( R − (cid:66) a )( R − (cid:66) Y )( b ))= X ( Y ( a )) · b + ( R − (cid:66) ( Y ( a ))) · ( R − (cid:66) X )( b )+ X ( R − (cid:66) a ) · ( R − (cid:66) Y )( b ))+ (( R (cid:48) − R − ) (cid:66) a ) · ( R (cid:48) − (cid:66) X )(( R − (cid:66) Y )( b ))and( R − (cid:66) Y )(( R − (cid:66) X )( a · b )) =( R − (cid:66) Y )(( R − (cid:66) X )( a ) · b + ( R (cid:48) − (cid:66) a ) · (( R (cid:48) − R − ) (cid:66) X )( b ))=( R − (cid:66) Y )(( R − (cid:66) X )( a )) · b + ( R (cid:48) − (cid:66) ( R − (cid:66) X )( a )) · (( R (cid:48) − R − ) (cid:66) Y )( b )+ ( R − (cid:66) Y )( R (cid:48) − (cid:66) a ) · (( R (cid:48) − R − ) (cid:66) X )( b )+ (( R (cid:48)(cid:48) − R (cid:48) − ) (cid:66) a ) · (( R (cid:48)(cid:48) − R − ) (cid:66) Y )((( R (cid:48) − R − ) (cid:66) X )( b ))equals [ X, Y ] R ( a ) · b + ( R (cid:66) a ) · ( R − (cid:66) [ X, Y ] R )( b ) , where the second and third terms cancel.ii.) Der R ( A ) is an H -equivariant braided symmetric A -bimodule: the k -linear maps in (4.3) are H -equivariant, because( ξ (cid:66) ( a · X ))( b ) = ξ (1) (cid:66) (( a · X )( S ( ξ (2) ) (cid:66) b ))= ξ (1) (cid:66) ( a · X ( S ( ξ (2) ) (cid:66) b ))=( ξ (1) (cid:66) a ) · ( ξ (2) (cid:66) X )(( ξ (3) S ( ξ (4) )) (cid:66) b )=(( ξ (1) (cid:66) a ) · ( ξ (2) (cid:66) X ))( b )and ( ξ (cid:66) ( X · a ))( b ) = ξ (1) (cid:66) (( X · a )( S ( ξ (2) ) (cid:66) b ))= ξ (1) (cid:66) ( X (( R − S ( ξ (2) )) (cid:66) b ) · ( R − (cid:66) a ))=( ξ (1) (cid:66) X )(( ξ (2) R − S ( ξ (4) )) (cid:66) b ) · (( ξ (3) R − ) (cid:66) a )=( ξ (1) (cid:66) X )(( R − ξ (3) S ( ξ (4) )) (cid:66) b ) · (( R − ξ (2) ) (cid:66) a )=(( ξ (1) (cid:66) X ) · ( ξ (2) (cid:66) a ))( b ) , raided Cartan Calculi 85using the adjoint H -module action on k -linear endomorphisms. Next, we notethat a · X and X · a are in fact braided derivations, since( a · X )( b · c ) = a · X ( b · c )= a · ( X ( b ) · c + ( R − (cid:66) b ) · ( R − (cid:66) X )( c ))=( a · X )( b ) · c + (( R (cid:48) − R − ) (cid:66) b ) · ( R (cid:48) − (cid:66) a ) · ( R − (cid:66) X )( c )=( a · X )( b ) · c + ( R − (cid:66) b ) · ( R − (cid:66) ( a · X ))( c )and( X · a )( b · c ) = X (( R − (cid:66) b ) · ( R − (cid:66) c )) · ( R − (cid:66) a )= (cid:18) X ( R − (cid:66) b ) · ( R − (cid:66) c )+ (( R (cid:48) − R − ) (cid:66) b ) · ( R (cid:48) − (cid:66) X )( R − (cid:66) c ) (cid:19) · ( R − (cid:66) a )= X ( R − (cid:66) b ) · (( R (cid:48) − R − ) (cid:66) a ) · (( R (cid:48) − R − ) (cid:66) c )+ (( R (cid:48) − R − ) (cid:66) b ) · ( R (cid:48) − (cid:66) X )( R (cid:48)(cid:48) − (cid:66) c ) · (( R (cid:48)(cid:48) − R − ) (cid:66) a )= X ( R − (cid:66) b ) · (( R (cid:48) − R (cid:48)(cid:48) − R − ) (cid:66) a ) · (( R (cid:48) − R (cid:48)(cid:48) − ) (cid:66) c )+ (( R (cid:48) − R − ) (cid:66) b ) · (( R (cid:48) − (cid:66) X ) · ( R − (cid:66) a ))( c )=( X · a )( b ) · c + ( R − (cid:66) b ) · ( R − (cid:66) ( X · a ))( c ) . One has ( a · ( b · X ))( c ) = a · b · X ( c ) = (( a · b ) · X )( c ) and(( X · a ) · b )( c ) =( X · a )( R − (cid:66) c ) · ( R − (cid:66) b )= X (( R (cid:48) − R − ) (cid:66) c ) · ( R (cid:48) − (cid:66) a ) · ( R − (cid:66) b )= X ( R − (cid:66) c ) · ( R − (cid:66) a ) · ( R − (cid:66) b )= X ( R − (cid:66) c ) · ( R − (cid:66) ( a · b ))=( X · ( a · b ))( c )showing that (4.3) define A -module actions. They commute because(( a · X ) · b )( c ) =( a · X )( R − (cid:66) c ) · ( R − (cid:66) b )= a · X ( R − (cid:66) c ) · ( R − (cid:66) b )= a · ( X · b )( c )=( a · ( X · b ))( c )and they are braided symmetric, since( X · a )( b ) = X ( R − (cid:66) b ) · ( R − (cid:66) a )=(( R (cid:48) − R − ) (cid:66) a ) · ( R (cid:48) − (cid:66) X )(( R (cid:48) − R − ) (cid:66) b )=(( R (cid:48)(cid:48) − R (cid:48) − R − ) (cid:66) a ) · ( R (cid:48)(cid:48) − (cid:66) X )(( R (cid:48) − R − ) (cid:66) b )=(( R − (cid:66) a ) · ( R − (cid:66) X ))( b ) . This concludes the proof of the lemma.6 Chapter 4Since Der R ( A ) is an A -bimodule we can build the tensor algebraT • Der R ( A ) = A ⊕ Der R ( A ) ⊕ (Der R ( A ) ⊗ A Der R ( A )) ⊕ · · · of Der R ( A ) with respect to the tensor product ⊗ A over A . It is a braided symmet-ric H -equivariant A -bimodule with module actions defined on factorizing elements X ⊗ A · · · ⊗ A X k ∈ T k Der R ( A ) by ξ (cid:66) ( X ⊗ A · · · ⊗ A X k ) =( ξ (1) (cid:66) X ) ⊗ A · · · ⊗ A ( ξ ( k ) (cid:66) X k ) , (4.4) a · ( X ⊗ A · · · ⊗ A X k ) =( a · X ) ⊗ A · · · ⊗ A X k , (4.5)( X ⊗ A · · · ⊗ A X k ) · a = X ⊗ A · · · ⊗ A ( X k · a ) (4.6)for all ξ ∈ H and a ∈ A . Note however that (T • Der R ( A ) , ⊗ A ) is not (graded)braided commutative in general. There is an ideal I in (T • Der R ( A ) , ⊗ A ) generatedby elements X ⊗ A · · · ⊗ A X k ∈ T k Der R ( A ) which equal X ⊗ A · · · ⊗ A X i − ⊗ A (cid:18) R (cid:48) − (cid:66) (cid:18) ( R − (cid:66) X j ) ⊗ A ( R − (cid:66) ( X i +1 ⊗ A · · · ⊗ A X j − )) (cid:19)(cid:19) ⊗ A ( R (cid:48) − (cid:66) X i ) ⊗ A X j +1 ⊗ A · · · ⊗ A X k for a pair ( i, j ) such that 1 ≤ i < j ≤ k . We illustrate this for small tensorpowers: if k = 2 the elements of I are those X ⊗ A Y ∈ T Der R ( A ) satisfying X ⊗ A Y = ( R − (cid:66) Y ) ⊗ A ( R − (cid:66) X ), if k = 3 elements X ⊗ A Y ⊗ A Z ∈ T Der R ( A )have to equal ( R − (cid:66) Y ) ⊗ A ( R − (cid:66) X ) ⊗ A Z , X ⊗ A ( R − (cid:66) Z ) ⊗ A ( R − (cid:66) Y ) or(( R (cid:48) − R − ) (cid:66) Z ) ⊗ A (( R (cid:48) − R − ) (cid:66) Y ) ⊗ A ( R (cid:48) − (cid:66) X )in order to belong to I . According to Lemma B.2 from the appendix, the H -module action (4.4) and the A -bimodule actions (4.5) and (4.6) respect the ideal I .We denote the quotient T • Der R ( A ) /I by X •R ( A ) and call them braided multivectorfields on A . The induced product ∧ R is said to be the braided wedge product . ByProposition B.3 we conclude the following statement. Proposition 4.2.2. The braided multivector fields ( X •R ( A ) , ∧ R ) are a braided Graß-mann algebra. Namely, X •R ( A ) is a braided symmetric H -equivariant A -bimodulesuch that H (cid:66) X k R ( A ) ⊆ X k R ( A ) and ∧ R : X k R ( A ) × X (cid:96) R ( A ) → X k + (cid:96) R ( A ) is associative, H -equivariant and graded braided commutative, i.e. Y ∧ R X = ( − k · (cid:96) ( R − (cid:66) X ) ∧ R ( R − (cid:66) Y ) for all X ∈ X k R ( A ) and Y ∈ X (cid:96) R ( A ) . We are defining a k -bilinear operation (cid:74) · , · (cid:75) R : X k R ( A ) × X (cid:96) R ( A ) → X k + (cid:96) − R ( A ) inthe following. If a, b ∈ A we set (cid:74) a, b (cid:75) R = 0. For a ∈ A and a factorizing element X = X ∧ R · · · ∧ R X k ∈ X k R ( A ) where k > 0, we define (cid:74) X, a (cid:75) R = k (cid:88) i =1 ( − k − i X ∧ R · · · ∧ R X i − ∧ R ( X i ( R − (cid:66) a )) ∧ R (cid:18) R − (cid:66) (cid:18) X i +1 ∧ R · · · ∧ R X k (cid:19)(cid:19) raided Cartan Calculi 87and (cid:74) a, X (cid:75) R = k (cid:88) i =1 ( − i (cid:18) R − (cid:66) (cid:18) X ∧ R · · · ∧ R X i − (cid:19)(cid:19) ∧ R (( R − (cid:66) X i )( R − (cid:66) a )) ∧ R X i +1 ∧ R · · · ∧ R X k . Furthermore, on factorizing elements X = X ∧ R · · · ∧ R X k ∈ X k R ( A ) and Y = Y ∧ R · · · ∧ R Y (cid:96) ∈ X (cid:96) R ( A ), where k, (cid:96) > 0, we define (cid:74) X, Y (cid:75) R = k (cid:88) i =1 (cid:96) (cid:88) j =1 ( − i + j [ R − (cid:66) X i , R (cid:48) − (cid:66) Y j ] R ∧ R (cid:18) R (cid:48) − (cid:66) (cid:18)(cid:18) R − (cid:66) ( X ∧ R · · · ∧ R X i − ) (cid:19) ∧ R (cid:99) X i ∧ R X i +1 ∧ R · · · ∧ R X k Y ∧ R · · · ∧ R Y j − (cid:19)(cid:19) ∧ R (cid:98) Y j ∧ R Y j +1 ∧ R · · · ∧ R Y (cid:96) , where [ · , · ] R denotes the braided commutator. The operation (cid:74) · , · (cid:75) R is said to be the braided Schouten-Nijenhuis bracket . In practice the bracket is determined on X k R ( A )for k < graded braided Leibniz rules (cid:74) X, Y ∧ R Z (cid:75) R = (cid:74) X, Y (cid:75) R ∧ R Z + ( − ( k − · (cid:96) ( R − (cid:66) Y ) ∧ R (cid:74) R − (cid:66) X, Z (cid:75) R and (cid:74) X ∧ R Y, Z (cid:75) R = X ∧ R (cid:74) Y, Z (cid:75) R + ( − (cid:96) · ( m − (cid:74) X, R − (cid:66) Z (cid:75) R ∧ R ( R − (cid:66) Y )for all X ∈ X k R ( A ), Y ∈ X (cid:96) R ( A ) and Z ∈ X m R ( A ). In those low orders one obtains (cid:74) a, b (cid:75) R =0 , (cid:74) a, X (cid:75) R = − ( R − (cid:66) X )( R − (cid:66) a ) , (cid:74) X, a (cid:75) R = X ( a ) , (cid:74) X, Y (cid:75) R =[ X, Y ] R for all a, b ∈ A and X, Y ∈ X R ( A ). As a consequence, we conclude the followingresult. Proposition 4.2.3. The braided Schouten-Nijenhuis bracket (cid:74) · , · (cid:75) R structures thebraided Graßmann algebra ( X •R ( A ) , ∧ R ) of braided multivector fields as a braidedGerstenhaber algebra. Namely, (cid:74) · , · (cid:75) R : X k R ( A ) × X (cid:96) R ( A ) → X k + (cid:96) − R ( A ) is graded withrespect to the degree shifted by , H -equivariant, graded braided skewsymmetric, i.e. (cid:74) Y, X (cid:75) R = − ( − ( k − · ( (cid:96) − (cid:74) R − (cid:66) X, R − (cid:66) Y (cid:75) R , satisfies the graded braided Jacobi identity (cid:74) X, (cid:74) Y, Z (cid:75) R (cid:75) R = (cid:74)(cid:74) X, Y (cid:75) R , Z (cid:75) R + ( − ( k − · ( (cid:96) − (cid:74) R − (cid:66) Y, (cid:74) R − (cid:66) X, Z (cid:75) R (cid:75) R and the graded braided Leibniz rule (cid:74) X, Y ∧ R Z (cid:75) R = (cid:74) X, Y (cid:75) R ∧ R Z + ( − ( k − · (cid:96) ( R − (cid:66) Y ) ∧ R (cid:74) R − (cid:66) X, Z (cid:75) R , where X ∈ X k R ( A ) , Y ∈ X (cid:96) R ( A ) and Z ∈ X •R ( A ) . Furthermore, (cid:74) · , · (cid:75) R is the uniquebraided Gerstenhaber bracket on ( X •R ( A ) , ∧ R ) such that (cid:74) X, a (cid:75) R = X ( a ) and (cid:74) X, Y (cid:75) R = [ X, Y ] R hold for all a ∈ A and X, Y ∈ X R ( A ) . Considering X R ( A ) as a module over A , its dual space consists of k -linear maps X R ( A ) which are right A -linear in addition. It can be structured as an equivariantbraided symmetric A -bimodule and consequently its braided exterior algebra is awell-defined braided Graßmann algebra. The duality with braided multivector fieldsbecomes explicit if one considers the braided insertion of braided multivector fields.We prove that the insertion is an equivariant map, which is left A -linear in the firstand right A -linear in the second argument. It is a morphism in the category ofequivariant braided symmetric bimodules. In analogy to the de Rham differentialwe define a differential in low degrees and extend it as a braided derivation of thebraided wedge product to higher orders. Since this differential turns out to beequivariant it is actually a derivation and a braided derivation at the same time.Afterwards we define the braided differential forms to be the smallest differentialgraded subalgebra with respect to the previous differential, which shelters A (c.f.[45]). In other words, the braided differential forms are generated by A and thedifferential.Consider the k -module Ω R ( A ) of k -linear maps ω : Der R ( A ) → A such that ω ( X · a ) = ω ( X ) · a for all a ∈ A and X ∈ Der R ( A ). Lemma 4.3.1. Ω R ( A ) is an H -equivariant braided symmetric A -bimodule with re-spect to the H -adjoint action and left and right A -module actions given by ( a · ω )( X ) = a · ω ( X ) and ( ω · a )( X ) = ω ( R − (cid:66) X ) · ( R − (cid:66) a ) for all a ∈ A and X ∈ Der R ( A ) .Proof. Let a, b, c ∈ A , ω ∈ Ω R ( A ) and X ∈ Der R ( A ). First of all, the H -moduleaction and the A -module actions are well-defined, since( ξ (cid:66) ω )( X · a ) = ξ (1) (cid:66) ( ω (( S ( ξ (2) ) (1) (cid:66) X ) · ( S ( ξ (2) ) (2) (cid:66) a )))= ξ (1) (cid:66) ( ω ( S ( ξ (3) ) (cid:66) X ) · ( S ( ξ (2) ) (cid:66) a ))=( ξ (1) (cid:66) ω ( S ( ξ (4) ) (cid:66) X )) · (( ξ (2) S ( ξ (3) )) (cid:66) a )=(( ξ (1) (cid:66) ω )(( ξ (2) S ( ξ (3) )) (cid:66) X )) · a =(( ξ (cid:66) ω )( X )) · a, ( a · ω )( X · b ) = a · ω ( X · b ) = a · ω ( X ) · b = ( a · ω )( X ) · b and( ω · a )( X · b ) = ω (( R − (cid:66) X ) · ( R − (cid:66) b )) · ( R − (cid:66) a )= ω ( R − (cid:66) X ) · (( R (cid:48) − R − ) (cid:66) a ) · (( R (cid:48) − R − ) (cid:66) b )= ω ( R − (cid:66) X ) · (( R (cid:48) − R (cid:48)(cid:48) − R − ) (cid:66) a ) · (( R (cid:48) − R (cid:48)(cid:48) − ) (cid:66) b )= ω ( R − (cid:66) X ) · ( R − (cid:66) a ) · b =( ω · a )( X ) · b hold by the hexagon relations and the bialgebra anti-homomorphism properties of S .By the associativity of the product on A and the hexagon relations those assignmentsare left and right A -module actions, respectively. They commute since (( a · ω ) · b )( c ) =( a · ω )( R − (cid:66) c ) · ( R − (cid:66) b ) = ( a · ( ω · b ))( c ). Furthermore( ξ (cid:66) ( a · ω · b ))( c ) = ξ (1) (cid:66) (( a · ω · b )( S ( ξ (2) ) (cid:66) c ))raided Cartan Calculi 89=( ξ (1) (cid:66) a ) · ( ξ (2) (cid:66) ω )(( ξ (3) R − S ( ξ (5) )) (cid:66) c ) · (( ξ (4) R − ) (cid:66) b )=( ξ (1) (cid:66) a ) · ( ξ (2) (cid:66) ω )(( R − ξ (4) S ( ξ (5) )) (cid:66) c ) · (( R − ξ (3) ) (cid:66) b )=(( ξ (1) (cid:66) a ) · ( ξ (2) (cid:66) ω ) · ( ξ (3) (cid:66) b ))( c )proves that Ω R ( A ) is an H -equivariant A -bimodule. It is braided symmetric because( ω · a )( b ) = ω ( R − (cid:66) b ) · ( R − (cid:66) a )=(( R (cid:48) − R − ) (cid:66) a ) · ( R (cid:48) − (cid:66) ω )(( R (cid:48) − R − ) (cid:66) b )=( R − (cid:66) a ) · ( R − (cid:66) ω )( b )=(( R − (cid:66) a ) · ( R − (cid:66) ω ))( b ) . This concludes the proof.It follows from Proposition B.3 and Lemma B.5 that the braided exterior al-gebra Ω •R ( A ) of Ω R ( A ) is an H -equivariant braided symmetric A -bimodule and agraded braided commutative algebra. In the following lines we show that it is alsocompatible with the braided evaluation. For two elements ω, η ∈ Ω R ( A ) we definea k -bilinear map ω ∧ R η : Der R ( A ) × Der R ( A ) → A by( ω ∧ R η )( X, Y ) = ω ( R − (cid:66) X ) · ( R − (cid:66) η )( Y ) − ω ( R − (cid:66) Y ) · ( R − (cid:66) ( η ( X )))for all X, Y ∈ Der R ( A ). One proves that − ( ω ∧ R η )( R − (cid:66) Y, R − (cid:66) X ) = ( ω ∧ R η )( X, Y ) = − (( R − (cid:66) η ) ∧ R ( R − (cid:66) ω ))( X, Y )and that ( ω ∧ R η )( X, Y · a ) =(( ω ∧ R η )( X, Y )) · a, ( ω ∧ R η )( a · X, Y ) =( R − (cid:66) a ) · (( R − (cid:66) ( ω ∧ R η ))( X, Y )) ,ξ (cid:66) (( ω ∧ R η )( X, Y )) =(( ξ (1) (cid:66) ω ) ∧ R ( ξ (2) (cid:66) η ))( ξ (3) (cid:66) X, ξ (4) (cid:66) Y )hold for all ξ ∈ H , ω, η ∈ Ω R ( A ), a ∈ A and X, Y ∈ Der R ( A ). The evaluations ofthe H -action and A -module actions read( ξ (cid:66) ( ω ∧ R η ))( X, Y ) = ξ (1) (cid:66) (( ω ∧ R η )( S ( ξ (3) ) (cid:66) X, S ( ξ (2) ) (cid:66) Y )) , ( a · ( ω ∧ R η ))( X, Y ) = a · (( ω ∧ R η )( X, Y )) , (( ω ∧ R η ) · a )( X, Y ) =(( ω ∧ R η )( R − (cid:66) X, R − (cid:66) Y )) · ( R − (cid:66) a ) . Inductively one defines the evaluation of higher wedge products. Explicitly, theevaluated module actions on factorizing elements ω ∧ R . . . ∧ R ω k ∈ Ω k R ( A ) read( ξ (cid:66) ( ω ∧ R . . . ∧ R ω k ))( X , . . . , X k )= ξ (1) (cid:66) (( ω ∧ R . . . ∧ R ω k )( S ( ξ ( k +1) ) (cid:66) X , . . . , S ( ξ (2) ) (cid:66) X k )) , ( a · ( ω ∧ R . . . ∧ R ω k ))( X , . . . , X k ) = a · (( ω ∧ R . . . ∧ R ω k )( X , . . . , X k ))and (( ω ∧ R . . . ∧ R ω k ) · a )( X , . . . , X k )=(( ω ∧ R . . . ∧ R ω k )( R − (cid:66) X , . . . , R − k ) (cid:66) X k )) · ( R − (cid:66) a ) . X , . . . , X ∈ Der R ( A ), a ∈ A and ξ ∈ H . It is useful to further define theinsertion i R X : Ω •R ( A ) → Ω •− R ( A ) of an element X ∈ Der R ( A ) into the last slot of anelement ω ∈ Ω k R ( A ) byi R X ω = ( − k − ( R − (cid:66) ω )( · , . . . , · , R − (cid:66) X ) . More general, we inductively definei R X ∧ R Y = i R X i R Y for all X, Y ∈ X •R ( A ). Lemma 4.3.2. (Ω •R ( A ) , ∧ R ) is a graded braided commutative associative unital al-gebra and an H -equivariant braided symmetric A -bimodule. The insertion i R : X k R ( A ) ⊗ Ω •R ( A ) → Ω •− k R ( A ) of braided multivector field is H -equivariant such that i R X is a right A -linear andbraided left A -linear homogeneous map of degree − k for all X ∈ X k R ( A ) . Further-more, i R X is left A -linear and braided right A -linear in X . For k = 1 we obtain agraded braided derivation i R X of degree − .Proof. We already proved that Ω •R ( A ) is an H -equivariant braided symmetric A -bimodule and that it is compatible with braided evaluation. We prove that i R X is abraided graded derivation of the wedge product for X ∈ Der R ( A ). Let ω, η ∈ Ω R ( A ).Then i R X ( ω ∧ R η ) =( − − (( R − (cid:66) ω ) ∧ R ( R − (cid:66) η ))( · , R − (cid:66) X )= − ( R − (cid:66) ω )( R − (cid:66) η )( R − (cid:66) X )+ ( R − (cid:66) ω )(( R (cid:48) − R − ) (cid:66) X )(( R (cid:48) − R − ) (cid:66) η )=i R X ( ω ) ∧ R η + ( − · ( R − (cid:66) ω ) ∧ R i RR − (cid:66) X η. In particular this implies ξ (cid:66) (i R X ( ω ∧ R η )) = i R ξ (1) (cid:66) X (( ξ (2) (cid:66) ω ) ∧ R ( ξ (3) (cid:66) η )) for all ξ ∈ H . Inductively, one showsi R X ( ω ∧ R η ) = (i R X ω ) ∧ R η + ( − k ( R − (cid:66) ω ) ∧ R i RR − (cid:66) X η and ξ (cid:66) (i R X η ) = i R ξ (1) (cid:66) X ( ξ (2) (cid:66) η ) for all ξ ∈ H , X ∈ Der R ( A ), ω ∈ Ω k R ( A ) and η ∈ Ω •R ( A ). For factorizing elements X ∧ R X ∈ X R ( A ) this implies ξ (cid:66) i R X ∧ R X ω = ξ (cid:66) (i R X i R X ω ) = i R ξ (1) (cid:66) X i R ξ (2) (cid:66) X ( ξ (3) (cid:66) ω ) = i R ξ (1) (cid:66) ( X ∧ R X ) ( ξ (2) (cid:66) ω )for all ξ ∈ H and ω ∈ Ω •R ( A ) and inductively one obtains ξ (cid:66) (i R X ω ) = i R ξ (1) (cid:66) X ( ξ (2) (cid:66) ω )for any X ∈ Ω •R ( A ). It is easy to verify that i R X inherits the linearity propertiesi R a · X ω = a · (i R X ω ) , i R X · a ω =(i R X ( R − (cid:66) ω )) · ( R − (cid:66) a ) , i R X ( ω · a ) =(i R X ω ) · a, i R X ( a · ω ) =( R − (cid:66) a ) · (i RR − (cid:66) X ω )for all X ∈ X •R ( A ), a ∈ A and ω ∈ Ω •R ( A ).raided Cartan Calculi 91One defines a k -linear map d : Ω •R ( A ) → Ω • +1 R ( A ) on a ∈ A by i R X (d a ) = X ( a )for all X ∈ Der R ( A ), on ω ∈ Ω R ( A ) by(d ω )( X, Y ) = ( R − (cid:66) X )(( R − (cid:66) ω )( Y )) − ( R − (cid:66) Y )( R − (cid:66) ( ω ( X ))) − ω ([ X, Y ] R )for all X, Y ∈ Der R ( A ) and extends d to higher wedge powers by demanding it tobe a graded derivation with respect to ∧ R , i.e.d( ω ∧ R ω ) = (d ω ) ∧ R ω + ( − k ω ∧ R (d ω )for ω ∈ Ω k R ( A ) and ω ∈ Ω •R ( A ). One can also directly define d ω ∈ Ω k +1 R ( A ) forany ω ∈ Ω k R ( A ) by(d ω )( X , . . . , X k ) = k (cid:88) i =0 ( − i ( R − (cid:66) X i ) (cid:18) ( R − (cid:66) ω ) (cid:18) R − (cid:66) X , . . . , R − i +1) (cid:66) X i − , (cid:99) X i , X i +1 , . . . , X k (cid:19)(cid:19) + (cid:88) i The map d : Ω •R ( A ) → Ω • +1 R ( A ) is a differential, i.e. d = 0 .Proof. It is sufficient to prove d = 0 on Ω k R ( A ) for k < 2, since d is a gradedbraided derivation. Let X, Y, Z ∈ Der R ( A ). For a ∈ A we obtain(d a )( X, Y ) =( R − (cid:66) X )(( R − (cid:66) (d a ))( Y )) − ( R − (cid:66) Y )( R − (cid:66) ((d a )( X ))) − (d a )([ X, Y ] R )=( R − (cid:66) X )(( R (cid:48) − (cid:66) Y )(( R (cid:48) − R − ) (cid:66) a )) − ( R − (cid:66) Y )( R − (cid:66) (( R (cid:48) − (cid:66) X )( R (cid:48) − (cid:66) a ))) − ( R − (cid:66) [ X, Y ] R )( R − (cid:66) a )=( R − (cid:66) X )(( R − (cid:66) Y )( R − (cid:66) a )) − (( R (cid:48)(cid:48) − R − ) (cid:66) Y )((( R (cid:48)(cid:48) − R (cid:48) − ) (cid:66) X )(( R − R (cid:48) − ) (cid:66) a )) − [ R − (cid:66) X, R − (cid:66) Y ] R ( R − (cid:66) a )=( R − (cid:66) X )(( R − (cid:66) Y )( R − (cid:66) a )) − (( R (cid:48)(cid:48) − R − ) (cid:66) Y )((( R (cid:48)(cid:48) − R − ) (cid:66) X )( R − (cid:66) a ))2 Chapter 4 − [ R − (cid:66) X, R − (cid:66) Y ] R ( R − (cid:66) a )=0by the definition of the braided commutator. For ω ∈ Ω R ( A )(d ω )( X, Y, Z ) =( R − (cid:66) X )((d( R − (cid:66) ω ))( Y, Z )) − ( R − (cid:66) Y )((d( R − (cid:66) ω ))( R − (cid:66) X, Z ))+ ( R − (cid:66) Z )((d( R − (cid:66) ω ))( R − (cid:66) X, R − (cid:66) Y )) − d ω ([ X, Y ] R , Z ) + d ω ([ X, R − (cid:66) Z ] R , R − (cid:66) Y ) − d ω ([ R − (cid:66) Y, R − (cid:66) Z ] R , R − (cid:66) X )=( R − (cid:66) X ) (cid:18) ( R (cid:48) − (cid:66) Y )(( R (cid:48) − R − ) (cid:66) ω )( Z ) − ( R (cid:48) − (cid:66) Z )(( R (cid:48) − R − ) (cid:66) ω )( R (cid:48) − (cid:66) Y ) − ( R − (cid:66) ω )([ Y, Z ] R ) (cid:19) − ( R − (cid:66) Y ) (cid:18) (( R (cid:48) − R − ) (cid:66) X )(( R (cid:48) − R − ) (cid:66) ω )( Z ) − ( R (cid:48) − (cid:66) Z )(( R (cid:48) − R − ) (cid:66) ω )(( R (cid:48) − R − ) (cid:66) X ) − ( R − (cid:66) ω )([ R − (cid:66) X, Z ] R ) (cid:19) + ( R − (cid:66) Z ) (cid:18) (( R (cid:48) − R − ) (cid:66) X )(( R (cid:48) − R − ) (cid:66) ω )( R − (cid:66) Y ) − (( R (cid:48) − R − ) (cid:66) Y )(( R (cid:48) − R − ) (cid:66) ω )(( R (cid:48) − R − ) (cid:66) X ) − ( R − (cid:66) ω )( R − (cid:66) [ X, Y ] R ) (cid:19) − ( R − (cid:66) [ X, Y ] R )( R − (cid:66) ω )( Z )+ ( R − (cid:66) Z )( R − (cid:66) ω )( R − (cid:66) [ X, Y ] R )+ ω ([[ X, Y ] R , Z ] R )+ ( R (cid:48) − (cid:66) [ X, R − (cid:66) Z ] R )( R (cid:48) − (cid:66) ω )( R − (cid:66) Y ) − (( R (cid:48) − R − ) (cid:66) Y )( R (cid:48) − (cid:66) ω )( R (cid:48) − (cid:66) [ X, R − (cid:66) Z ] R ) − ω ([[ X, R − (cid:66) Z ] R , R − (cid:66) Y ] R ) − (( R (cid:48) − R − ) (cid:66) [ Y, Z ] R )( R (cid:48) − (cid:66) ω )( R − (cid:66) X )+ (( R (cid:48) − R − ) (cid:66) X )( R (cid:48) − (cid:66) ω )(( R (cid:48) − R − ) (cid:66) [ Y, Z ] R )+ ω ([[ R − (cid:66) Y, R − (cid:66) Z ] R , R − (cid:66) X ] R )=0 , where those 18 terms cancel in the following way: 12,15,18 because of the braidedJacobi identity, 1,4,10 and 2,7,13 and 5,8,16 by the definition of the braided com-mutator, while 3,17 and 6,14 and 9,11 simply cancel each other.We define the braided differential forms Ω •R ( A ) on A to be the smallest differ-ential graded subalgebra of Ω •R ( A ) such that A ⊆ Ω •R ( A ). Every element in Ω k R ( A )can be written as a finite sum of elements of the form a d a ∧ R . . . ∧ R d a k , for a i ∈ A .raided Cartan Calculi 93 We enter the main section of this chapter. In the following lines we construct anoncommutative Cartan calculus for any braided commutative algebra A . Buildingon Section 4.2 and Section 4.3, we define the braided Lie derivative as the gradedbraided commutator of the braided insertion and the braided de Rham differential. Itis a well-defined morphism in the category of equivariant braided symmetric bimod-ules and it completes the data of the braided Cartan calculus. Using graded braidedcommutators and the braided Schouten-Nijenhuis bracket we relate the braided Liederivative, the braided insertion and the braided de Rham differential to each other.The result is a generalization of the usual relations of the Cartan calculus in thecategory of equivariant braided symmetric bimodules. Since there is no choice in-volved in our construction, as in differential geometry, it seems justified to call theresulting data the braided Cartan calculus of A . It is a noncommutative Cartancalculus with the difference that we are not restricted to incorporate modules of thecenter of A but are free to work with modules over the whole algebra instead.The graded braided commutator of two homogeneous maps Φ , Ψ : G • → G • ofdegree k and (cid:96) between braided Graßmann algebras is defined by[Φ , Ψ] R = Φ ◦ Ψ − ( − k(cid:96) ( R − (cid:66) Ψ) ◦ ( R − (cid:66) Φ) . If Φ or Ψ is equivariant, the graded braided commutator coincides with the gradedcommutator. If Φ , Ψ : X •R ( A ) ⊗ G • → G • are two H -equivariant maps such thatΦ X , Ψ Y : G • → G • are homogeneous of degree k and (cid:96) for any X ∈ X k R ( A ) and Y ∈ X (cid:96) R ( A ), respectively, the graded braided commutator of Φ X and Ψ Y reads[Φ X , Ψ Y ] R = Φ X Ψ Y − ( − k(cid:96) Ψ R − (cid:66) Y Φ R − (cid:66) X . For any X ∈ X k R ( A ) we define the braided Lie derivative L R X : Ω •R ( A ) → Ω •− ( k − R ( A )by L R X = [i R X , d] R . It is a homogeneous map of degree − ( k − 1) and a braidedderivation of Ω •R ( A ) for k = 1. Moreover, it is easy to check that L R : X •R ( A ) ⊗ Ω •R ( A ) → Ω •R ( A )is H -equivariant. We prove an auxiliary lemma in analogy to Lemma 4.1.6. Lemma 4.4.1. One has L R a ω = − (d a ) ∧ R ω and L R X ∧ R Y = i R X L R Y + ( − (cid:96) L R X i R Y for all a ∈ A , ω ∈ Ω •R ( A ) , X ∈ X •R ( A ) and Y ∈ X (cid:96) R ( A ) . If X, Y ∈ X R ( A )[ L R X , i R Y ] R = i R [ X,Y ] R holds.Proof. By the very definition of the braided Lie derivative L R a ω =i R a d ω − ( − · d(i R a ω ) = a ∧ R d ω − ((d a ) ∧ R ω + ( − a ∧ R d ω )= − (d a ) ∧ R ω L R X ∧ R Y =[i R X ∧ R Y , d] R = [i R X i R Y , d] R = i R X [i R Y , d] R + ( − − · (cid:96) [i R X , d] R i R Y =i R X L R Y + ( − (cid:96) L R X i R Y . The missing formula trivially holds on braided differential forms of degree 0, whilefor ω ∈ Ω R ( A )[ L R X , i R Y ] R ω = L R X i R Y ω − ( − · i RR − (cid:66) Y L RR − (cid:66) X ω =(i R X d + di R X )i R Y ω − i RR − (cid:66) Y (i RR − (cid:66) X d + di RR − (cid:66) X ) ω = X (i R Y ω ) + 0 + (d(( R (cid:48)(cid:48) − R (cid:48) − ) (cid:66) ω ))(( R (cid:48)(cid:48) − R − ) (cid:66) Y, ( R (cid:48) − R − ) (cid:66) X ) − ( R − (cid:66) Y )(i RR − (cid:66) X ω )=i R [ X,Y ] ω for all X, Y ∈ X R ( A ). Since [ L R X , i R Y ] R is a graded braided derivation this is sufficientto conclude the proof of the lemma.Now we are prepared to prove the main theorem of this section. It assigns to anybraided commutative left H -module algebra A a noncommutative Cartan calculus,which we call the braided Cartan calculus of A in the following. Theorem 4.4.2 (Braided Cartan calculus) . Let A be a braided commutative left H -module algebra and consider the braided differential forms (Ω •R ( A ) , ∧ R , d) andbraided multivector fields ( X •R ( A ) , ∧ R , (cid:74) · , · (cid:75) R ) on A . The homogeneous maps L R X : Ω •R ( A ) → Ω •− ( k − R ( A ) and i R X : Ω •R ( A ) → Ω •− k R ( A ) , where X ∈ X k R ( A ) , and d : Ω •R ( A ) → Ω • +1 R ( A ) satisfy [ L R X , L R Y ] R = L R (cid:74) X,Y (cid:75) R , [ L R X , i R Y ] R =i R (cid:74) X,Y (cid:75) R , [ L R X , d] R =0 , [i R X , i R Y ] R =0 , [i R X , d] R = L R X , [d , d] R =0 , (4.7) for all X, Y ∈ X •R ( A ) .Proof. We are going to prove the above formulas in reversed order. Since d is adifferential it follows that [d , d] R = 2d = 0. Recall that there is no braidingappearing here since d is equivariant. By the definition of the braided Lie derivative[i R X , d] R = L R X holds for all X ∈ X •R ( A ). Let X ∈ X k R ( A ) and Y ∈ X (cid:96) R ( A ). Then[i R X , i R Y ] R = i R X i R Y − ( − k(cid:96) i RR − (cid:66) Y i RR − (cid:66) X = i R X ∧ R Y − ( − k(cid:96) ( R − (cid:66) Y ) ∧ R ( R − (cid:66) X ) = 0follows by the very definition of i R X ∧ R Y = i R X i R Y . Using the graded braided Jacobiidentity of the graded braided commutator we obtain[[i R X , d] R , d] R = [i R X , [d , d] R ] R + ( − · [[i R X , d] R , d] R = − [[i R X , d] R , d] R for all X ∈ X •R ( A ), which implies [ L R X , d] R = 0. Again, there is no braidingappearing since d is equivariant. Recall that the braided Schouten-Nijenhuis bracketraided Cartan Calculi 95of a homogeneous element Y = Y ∧ R · · ·∧ R Y (cid:96) ∈ X (cid:96) R ( A ) with a ∈ A and X ∈ X R ( A )read (cid:74) a, Y (cid:75) R = (cid:96) (cid:88) j =1 ( − j +1 ( R − (cid:66) Y ) ∧ R · · · ∧ R ( R − j − (cid:66) Y j − ) ∧ R (cid:74) R − (cid:66) a, Y j (cid:75) R ∧ R Y j +1 ∧ R · · · ∧ R Y (cid:96) and (cid:74) X, Y (cid:75) R = (cid:96) (cid:88) j =1 ( R − (cid:66) Y ) ∧ R · · · ∧ R ( R − j − (cid:66) Y j − ) ∧ R [ R − (cid:66) X, Y j ] R ∧ R Y j +1 ∧ R · · · ∧ R Y (cid:96) , respectively. If (cid:96) = 1 we obtain[ L R a , i R Y ] R ω =( L R a i R Y − ( − ( − · i RR − (cid:66) Y L RR − (cid:66) a ) ω = − d a ∧ R i R Y ω − i RR − (cid:66) Y (d( R − (cid:66) a ) ∧ R ω )= − d a ∧ R i R Y ω − ( R − (cid:66) Y )( R − (cid:66) a ) ∧ R ω + d(( R (cid:48) − R − ) (cid:66) a ) ∧ R i R ( R (cid:48)− R − ) (cid:66) Y ω =i R (cid:74) a,Y (cid:75) R ω for all ω ∈ Ω •R ( A ) by Lemma 4.4.1. Using the graded braided Leibniz rule thisextends to any (cid:96) > 1, namely[ L R a , i R Y ∧ R ···∧ R Y (cid:96) ] R =[ L R a , i R Y ] R i R Y ∧ R ···∧ R Y (cid:96) + ( − ( − · i RR − (cid:66) Y [ L RR − (cid:66) a , i R Y ∧ R ···∧ R Y (cid:96) ]=i R (cid:74) a,Y (cid:75) R ∧ R Y ∧ R ···∧ R Y (cid:96) − i RR − (cid:66) Y [ L RR − (cid:66) a , i R Y ∧ R ···∧ R Y (cid:96) ]= · · · = i R (cid:74) a,Y (cid:75) R . Again by Lemma 4.4.1 we know that [ L R X , i R Y ] R = i R [ X,Y ] R holds for (cid:96) = 1 and X ∈ X R ( A ). Using the graded braided Leibniz rule this extends to all Y ∈ X •R ( A ).Assume now that [ L R X , i R Z ] R = i R (cid:74) X,Z (cid:75) R holds for all X ∈ X k R ( A ) and Z ∈ X •R ( A ) fora fixed k > 0. Then, for all X ∈ X k R ( A ), Y ∈ X R ( A ) and Z ∈ X m R ( A ) it follows that[ L R X ∧ R Y , i R Z ] R =[i R X L R Y − L R X i R Y , i R Z ] R =i R X [ L R Y , i R Z ] R + [i R X , i RR − (cid:66) Z ] R L RR − (cid:66) Y − L R X [i R Y , i R Z ] R − ( − m [ L R X , i RR − (cid:66) Z ] R i RR − (cid:66) Y =i R X [ L R Y , i R Z ] R − ( − m [ L R X , i RR − (cid:66) Z ] R i RR − (cid:66) Y =i R X i R (cid:74) Y,Z (cid:75) R − ( − m i R (cid:74) X, R − (cid:66) Z (cid:75) R i R ( R − (cid:66) Y ) =i R X ∧ R (cid:74) Y,Z (cid:75) R + ( − m − i R (cid:74) X, R − (cid:66) Z (cid:75) R ∧ R ( R − (cid:66) Y ) =i R (cid:74) X ∧ R Y,Z (cid:75) R for all X ∈ X k R ( A ), Y ∈ X R ( A ) and Z ∈ X m R ( A ) using Lemma 4.4.1. By induction[ L R X , i R Y ] R = i R (cid:74) X,Y (cid:75) R holds for all X, Y ∈ X •R ( A ). The remaining formula is verified6 Chapter 4via [ L R X , L R Y ] R =[ L R X , [i R Y , d] R ] R =[[ L R X , i R Y ] R , d] R + ( − ( k − (cid:96) [i RR − (cid:66) Y , [ L RR − (cid:66) X , d] R ] R =[i R (cid:74) X,Y (cid:75) R , d] R + 0= L R (cid:74) X,Y (cid:75) R for all X ∈ X k R ( A ) and Y ∈ X (cid:96) R ( A ). This concludes the proof of the theorem.In particular, the Cartan calculus on a smooth manifold M is a braided Cartancalculus with respect to the trivial action of any cocommutative Hopf algebra and R = 1 ⊗ 1. The equations (4.7) reduce to the usual formulas of the classical Cartancalculus in this case. In Section 4.6 we study another class of examples of braidedCartan calculi, namely twisted Cartan calculi. Having the braided Cartan calculus for any braided commutative algebra A at handit is nearby to ask if other fundamental constructions of differential geometry canbe generalized to this setting. Focusing on the algebraic properties of covariantderivatives, namely their linearity in the first argument and that they satisfy aLeibniz rule with respect to the second argument, we define equivariant covariantderivatives on equivariant braided symmetric bimodules. Note that there are severalnotions of covariant derivatives in the context of noncommutative algebras in theliterature, as already mentioned in the introduction to this chapter. While in thegeneral noncommutative case one has to distinguish between left and right covariantderivatives these two notions coincide for braided commutative algebras if one alsorequires equivariance of the covariant derivative. We further introduce curvature andtorsion of an equivariant covariant derivative and prove that they have the expectedsymmetry properties. Given an equivariant covariant derivative on A we constructan equivariant covariant derivative on braided differential 1-forms by employing thedual pairing and in a next step we extend both to braided multivector fields anddifferential forms. We introduce equivariant metrics on A and prove the existenceand uniqueness of an equivariant Levi-Civita covariant derivative with respect to afixed non-degenerate equivariant metric. Let ( H, R ) be a triangular Hopf algebraand A a braided commutative left H -module algebra in the following. Definition 4.5.1 (Equivariant covariant derivative) . Consider an H -equivariantbraided symmetric A -bimodule M . An H -equivariant map ∇ R : X R ( A ) ⊗ M → M is said to be an equivariant covariant derivative on M (with respect to R ), if for all a ∈ A , X ∈ X R ( A ) and s ∈ M one has ∇ R a · X s = a · ( ∇ R X s ) and ∇ R X ( a · s ) = ( L R X a ) · s + ( R − (cid:66) a ) · ( ∇ RR − (cid:66) X s ) . (4.8) The curvature of an equivariant covariant derivative ∇ R on M is defined by R ∇ R ( X, Y ) = ∇ R X ∇ R Y − ∇ RR − (cid:66) Y ∇ RR − (cid:66) X − ∇ R [ X,Y ] R (4.9)raided Cartan Calculi 97 for X, Y ∈ X R ( A ) . If M = X R ( A ) we can further define the torsion of ∇ R by Tor ∇ R ( X, Y ) = ∇ R X Y − ∇ RR − (cid:66) Y ( R − (cid:66) X ) − [ X, Y ] R , (4.10) for all X, Y ∈ X R ( A ) . An equivariant covariant derivative ∇ R is flat if R ∇ R = 0 and torsion-free if Tor ∇ R = 0 . Recall that the H -equivariance of an equivariant covariant derivative ∇ R on M reads ξ (cid:66) ( ∇ R X s ) = ∇ R ξ (1) (cid:66) X ( ξ (2) (cid:66) s ) on elements ξ ∈ H , X ∈ X R ( A ) and s ∈ M . Inthe next lemma we are going to investigate the linearity properties and symmetriesof the curvature and torsion. In short, we prove that R ∇ R descends to a map X R ( R ) ⊗ M → M and, in the case it is defined, Tor ∇ R to a map X R ( A ) → X R ( A ). Lemma 4.5.2. Let ∇ R be an equivariant covariant derivative on an H -equivariantbraided symmetric A -bimodule M . Then ∇ R X · a s = ∇ R X ( R − (cid:66) s ) · ( R − (cid:66) a ) , ∇ R X ( s · a ) =( ∇ R X s ) · a + ( R − (cid:66) s ) · ( L RR − (cid:66) X a ) hold for all a ∈ A , X ∈ X R ( A ) and s ∈ M . Furthermore, R ∇ R ( Y, X ) = − R ∇ R ( R − (cid:66) X, R − (cid:66) Y ) ,R ∇ R ( a · X, Y · b ) s = a · R ∇ R ( X, Y )( R − (cid:66) s ) · ( R − (cid:66) b ) ,R ∇ R ( X · a, Y ) s = R ∇ R ( X, R − (cid:66) Y )( R − (cid:66) s ) · ( R − (cid:66) a )= R ∇ R ( X, a · Y ) s for all a, b ∈ A , X, Y ∈ X R ( A ) and s ∈ M . In the case M = X R ( A ) we obtain Tor ∇ R ( Y, X ) = − Tor ∇ R ( R − (cid:66) X, R − (cid:66) Y ) , Tor ∇ R ( a · X, Y · b ) = a · Tor ∇ R ( X, Y ) · b, Tor ∇ R ( X · a, Y ) =Tor ∇ R ( X, R − (cid:66) Y ) · ( R − (cid:66) a )=Tor ∇ R ( X, a · Y ) for all a, b ∈ A and X, Y ∈ X R ( A ) .Proof. Fix elements a ∈ A , X, Y ∈ X R ( A ) and s ∈ M . Then ∇ R X · a s = ∇ R ( R − (cid:66) a ) · ( R − (cid:66) X ) s =( R − (cid:66) a ) · ( ∇ RR − (cid:66) X s )=( R (cid:48) − (cid:66) ( ∇ RR − (cid:66) X s )) · (( R (cid:48) − R − ) (cid:66) a )= ∇ R ( R (cid:48)− R − ) (cid:66) X ( R (cid:48) − (cid:66) s ) · (( R (cid:48) − R − ) (cid:66) a )= ∇ R ( R (cid:48)− R − ) (cid:66) X ( R (cid:48)(cid:48) − (cid:66) s ) · (( R (cid:48)(cid:48) − R (cid:48) − R − ) (cid:66) a )= ∇ R X ( R − (cid:66) s ) · ( R − (cid:66) a )and ∇ R X ( s · a ) = ∇ R X (( R − (cid:66) a ) · ( R − (cid:66) s ))8 Chapter 4= L R X ( R − (cid:66) a ) · ( R − (cid:66) s ) + (( R (cid:48) − R − ) (cid:66) a ) · ∇ RR (cid:48)− (cid:66) X ( R − (cid:66) s )=(( R (cid:48) − R − ) (cid:66) s ) · L RR (cid:48)− (cid:66) X (( R (cid:48) − R − ) (cid:66) a )+ ∇ R ( R (cid:48)(cid:48)− R (cid:48)− ) (cid:66) X (( R (cid:48)(cid:48) − R − ) (cid:66) s ) · (( R (cid:48)(cid:48) − R (cid:48) − R − ) (cid:66) a )=(( R (cid:48)(cid:48) − R (cid:48) − R − ) (cid:66) s ) · L RR (cid:48)(cid:48)− (cid:66) X (( R (cid:48) − R − ) (cid:66) a )+ ∇ R ( R (cid:48)(cid:48)− R (cid:48)− ) (cid:66) X (( R (cid:48)(cid:48)(cid:48) − R − ) (cid:66) s ) · (( R (cid:48)(cid:48)(cid:48) − R (cid:48)(cid:48) − R (cid:48) − R − ) (cid:66) a )= ∇ R X s · a + ( R − (cid:66) s ) · L RR − (cid:66) X a, proving the first two equations. By the braided skew-symmetry of [ · , · ] R we observethat R ∇ R ( R − (cid:66) X, R − (cid:66) Y ) = ∇ RR − (cid:66) X ∇ RR − (cid:66) Y − ∇ R ( R (cid:48)− R − ) (cid:66) Y ∇ R ( R (cid:48)− R − ) (cid:66) X − ∇ R [ R − (cid:66) X, R − (cid:66) Y ] R = − ( ∇ R Y ∇ R X − ∇ RR − (cid:66) X ∇ RR − (cid:66) Y − ∇ R [ Y,X ] R )= − R ∇ R ( Y, X )and using the equivariancy of ∇ R it follows that R ∇ R ( a · X, Y ) = ∇ R a · X ∇ R Y − ∇ RR − (cid:66) Y ∇ RR − (cid:66) ( a · X ) − ∇ R [ a · X,Y ] R = a · ∇ R X ∇ R Y − ∇ RR − (cid:66) Y (( R − (cid:66) a ) · ∇ RR − (cid:66) X ) − ∇ R a · [ X,Y ] R +((( R (cid:48)− R − ) (cid:66) Y )( R (cid:48)− (cid:66) a )) · ( R − X ) = a · ∇ R X ∇ R Y − L RR − (cid:66) Y ( R − (cid:66) a ) · ∇ RR − (cid:66) X − (( R (cid:48) − R − ) (cid:66) a ) · ∇ R ( R (cid:48)− R − ) (cid:66) Y ∇ RR − (cid:66) X − ∇ R a · [ X,Y ] R + ((( R (cid:48) − R − ) (cid:66) Y )( R (cid:48) − (cid:66) a )) · ∇ RR − X = a · ∇ R X ∇ R Y − (( R (cid:48) − R (cid:48)(cid:48) − ) (cid:66) a ) · ∇ R ( R (cid:48)− R (cid:48)(cid:48)− R − ) (cid:66) Y ∇ RR − (cid:66) X − a · ∇ R [ X,Y ] R = a · R ∇ R ( X, Y )and R ∇ R ( X, Y · a ) s = ∇ R X ∇ R Y · a s − ∇ RR − (cid:66) ( Y · a ) ∇ RR − (cid:66) X s − ∇ R [ X,Y · a ] R s = ∇ R X ( ∇ R Y ( R − (cid:66) s ) · ( R − (cid:66) a )) − ∇ RR − (cid:66) Y ∇ R ( R (cid:48)− R − ) (cid:66) X ( R (cid:48) − (cid:66) s ) · (( R (cid:48) − R − ) (cid:66) a ) − ∇ R [ X,Y ] R · a +( R − (cid:66) Y ) · (( R − (cid:66) X )( a )) s = ∇ R X ∇ R Y ( R − (cid:66) s ) · ( R − (cid:66) a )+ ∇ RR (cid:48)− (cid:66) Y (( R (cid:48) − R − ) (cid:66) s ) · L RR (cid:48)− (cid:66) X ( R − (cid:66) a ) − ∇ RR − (cid:66) Y ∇ RR − (cid:66) X ( R (cid:48) − (cid:66) s ) · ( R (cid:48) − (cid:66) a ) − ∇ R [ X,Y ] R · a s − ∇ R ( R − (cid:66) Y ) · (( R − (cid:66) X )( a )) s raided Cartan Calculi 99= ∇ R X ∇ R Y ( R − (cid:66) s ) · ( R − (cid:66) a )+ ∇ RR (cid:48)− (cid:66) Y (( R (cid:48)(cid:48) − R − ) (cid:66) s ) · L R ( R (cid:48)(cid:48)− R (cid:48)− ) (cid:66) X ( R − (cid:66) a ) − ∇ RR − (cid:66) Y ∇ RR − (cid:66) X ( R (cid:48) − (cid:66) s ) · ( R (cid:48) − (cid:66) a ) − ∇ R [ X,Y ] R ( R − (cid:66) s ) · ( R − (cid:66) a ) − ∇ RR − (cid:66) Y ( R (cid:48) − (cid:66) s ) · ((( R (cid:48) − R − ) (cid:66) X )( R (cid:48) − (cid:66) a ))= R ∇ R ( X, Y )( R − (cid:66) s ) · ( R − (cid:66) a )hold. Furthermore R ∇ R ( X · a, Y ) s = ∇ R X · a ∇ R Y s − ∇ RR − (cid:66) Y ∇ RR − (cid:66) ( X · a ) s − ∇ R [ X · a,Y ] R s = ∇ R X ∇ RR − (cid:66) Y ( R − (cid:66) s ) · ( R − (cid:66) a ) − ∇ RR − (cid:66) Y ( ∇ RR − (cid:66) X ( R (cid:48) − (cid:66) s ) · (( R (cid:48) − R − ) (cid:66) a )) − ∇ R [ X, R − (cid:66) Y ] R · ( R − (cid:66) a ) − X · (( R − (cid:66) Y )( R − (cid:66) a )) s = ∇ R X ∇ RR − (cid:66) Y ( R − (cid:66) s ) · ( R − (cid:66) a ) − ∇ RR − (cid:66) Y ∇ RR − (cid:66) X ( R (cid:48) − (cid:66) s ) · (( R (cid:48) − R − ) (cid:66) a ) − ∇ R ( R (cid:48)(cid:48)− R − ) (cid:66) X (( R (cid:48)(cid:48) − R (cid:48) − ) (cid:66) s ) · L R ( R (cid:48)(cid:48)− R − ) (cid:66) Y (( R (cid:48) − R − ) (cid:66) a ) − ∇ R [ X, R − (cid:66) Y ] R · ( R − (cid:66) a ) s + ∇ R X ( R (cid:48) − (cid:66) s ) · ((( R (cid:48) − R − ) (cid:66) Y )(( R (cid:48) − R − ) (cid:66) a ))= ∇ R X ∇ RR − (cid:66) Y ( R (cid:48) − (cid:66) s ) · (( R (cid:48) − R − ) (cid:66) a ) − ∇ R ( R (cid:48)(cid:48)− R − ) (cid:66) Y ∇ RR (cid:48)(cid:48)− (cid:66) X ( R (cid:48) − (cid:66) s ) · (( R (cid:48) − R − ) (cid:66) a ) − ∇ R [ X, R − (cid:66) Y ] R ( R (cid:48) − (cid:66) s ) · (( R (cid:48) − R − ) (cid:66) a )= R ∇ R ( X, R − (cid:66) Y )( R − (cid:66) s ) · ( R − (cid:66) a )= R ∇ R ( X, ( R − (cid:66) Y ) · (( R (cid:48) − R − ) (cid:66) a ))(( R (cid:48) − R − ) (cid:66) s )= R ∇ R ( X, a · Y ) s. It remains to discuss the properties of torsion. We obtainTor ∇ R ( R − (cid:66) X, R − (cid:66) Y ) = ∇ RR − (cid:66) X ( R − (cid:66) Y ) − ∇ R ( R (cid:48)− R − ) (cid:66) Y (( R (cid:48) − R − ) (cid:66) X ) − [ R − (cid:66) X, R − (cid:66) Y ] R = − ∇ R Y X + ∇ RR − (cid:66) X ( R − (cid:66) Y ) + [ Y, X ] R = − Tor ∇ R ( Y, X ) , as well asTor ∇ R ( a · X, Y · b ) = ∇ R a · X ( Y · b ) − ∇ RR − (cid:66) ( Y · b ) ( R − (cid:66) ( a · X )) − [ a · X, Y · b ] R = a · ∇ R X Y · b + a · ( R − (cid:66) Y ) · ( R − (cid:66) X )( b )00 Chapter 4 − ∇ RR − Y (( R (cid:48) − R − ) (cid:66) ( a · X )) · (( R (cid:48) − R − ) (cid:66) b ) − a · [ X, Y · b ] R + (( R (cid:48) − R − ) (cid:66) ( Y · b ))( R (cid:48) − (cid:66) a ) · ( R − (cid:66) X )= a · ∇ R X Y · b + a · ( R − (cid:66) Y ) · ( R − (cid:66) X )( b ) − ( R − (cid:66) Y )(( R (cid:48) − R − ) (cid:66) a ) · (( R (cid:48) − R − ) (cid:66) X ) · (( R (cid:48) − R − ) (cid:66) b ) − (( R (cid:48)(cid:48) − R (cid:48) − R − ) (cid:66) a ) · ∇ R ( R (cid:48)(cid:48)− R − ) (cid:66) Y (( R (cid:48) − R − ) (cid:66) X ) · (( R (cid:48) − R − ) (cid:66) b ) − a · [ X, Y ] R · b − a · ( R − (cid:66) Y ) · ( R − (cid:66) X )( a )+ (( R (cid:48) − R − ) (cid:66) ( Y · b ))( R (cid:48) − (cid:66) a ) · ( R − (cid:66) X )= a · ∇ R X Y · b − a · [ X, Y ] R · b − (( R (cid:48)(cid:48) − R (cid:48) − R − ) (cid:66) a ) · ∇ R ( R (cid:48)(cid:48)− R − ) (cid:66) Y (( R (cid:48)(cid:48)(cid:48) − R − ) (cid:66) X ) · (( R (cid:48)(cid:48)(cid:48) − R (cid:48) − R − ) (cid:66) b ) − ( R − (cid:66) Y )(( R (cid:48) − R − ) (cid:66) a ) · (( R (cid:48) − R − ) (cid:66) X ) · (( R (cid:48) − R − ) (cid:66) b )+ (( R (cid:48) − R − ) (cid:66) Y )(( R (cid:48)(cid:48) − R (cid:48) − ) (cid:66) a ) · (( R (cid:48)(cid:48) − R (cid:48) − R − ) (cid:66) b ) · ( R − (cid:66) X )= a · ∇ R X Y · b − a · [ X, Y ] R · b − (( R (cid:48)(cid:48) − R − ) (cid:66) a ) · ∇ R ( R (cid:48)(cid:48)− R − ) (cid:66) Y ( R − (cid:66) X ) · b − ( R − (cid:66) Y )(( R (cid:48) − R − ) (cid:66) a ) · (( R (cid:48) − R − ) (cid:66) X ) · (( R (cid:48) − R − ) (cid:66) b )+ (( R (cid:48) − R − ) (cid:66) Y )(( R (cid:48)(cid:48) − R (cid:48) − ) (cid:66) a ) · (( R (cid:48)(cid:48)(cid:48) − R − ) (cid:66) X ) · (( R (cid:48)(cid:48)(cid:48) − R (cid:48)(cid:48) − R (cid:48) − R − ) (cid:66) b )= a · Tor ∇ R ( X, Y ) · b − ( R − (cid:66) Y )(( R (cid:48) − R − ) (cid:66) a ) · (( R (cid:48)(cid:48) − R − ) (cid:66) X ) · (( R (cid:48)(cid:48) − R (cid:48) − R − ) (cid:66) b )+ (( R (cid:48) − R − ) (cid:66) Y )( R (cid:48) − (cid:66) a ) · ( R − (cid:66) X ) · b = a · Tor ∇ R ( X, Y ) · b andTor ∇ R ( X · a, Y ) = ∇ R X · a Y − ∇ RR − (cid:66) Y (( R − (cid:66) X ) · ( R − (cid:66) a )) − [ X · a, Y ] R = ∇ R X ( R − (cid:66) Y ) · ( R − (cid:66) a ) − ∇ RR − (cid:66) Y ( R − (cid:66) X ) · ( R − (cid:66) a ) − (( R (cid:48) − R − ) (cid:66) X ) · (( R (cid:48) − R − ) (cid:66) Y )( R − (cid:66) a ) − [ X, R − (cid:66) Y ] · ( R − (cid:66) a ) + X · ( R − (cid:66) Y )( R − (cid:66) a )=Tor ∇ R ( X, R − (cid:66) Y ) · ( R − (cid:66) a ) − (( R (cid:48) − R (cid:48)(cid:48) − ) (cid:66) X ) · (( R (cid:48) − R (cid:48)(cid:48) − R − ) (cid:66) Y )( R − (cid:66) a )raided Cartan Calculi 101+ X · ( R − (cid:66) Y )( R − (cid:66) a )=Tor ∇ R ( X, R − (cid:66) Y ) · ( R − (cid:66) a )=Tor ∇ R ( X, ( R − (cid:66) Y ) · ( R − (cid:66) a ))=Tor ∇ R ( X, a · Y ) . This concludes the proof of the lemma.There are natural extensions of an equivariant covariant derivative ∇ R on A tobraided multivector fields and differential forms in analogy to differential geometry.We define the braided dual pairing (cid:104)· , ·(cid:105) R : Ω R ( A ) ⊗ X R ( R ) → A by (cid:104) ω, X (cid:105) R = ω ( X )for all ω ∈ Ω R ( A ) and X ∈ X R ( A ). It is H -equivariant, left A -linear in the firstand right A -linear in the second argument. Proposition 4.5.3. An equivariant covariant derivative ∇ R on A induces an equiv-ariant covariant derivative ˜ ∇ R on Ω R ( A ) via (cid:104) ˜ ∇ R X ω, Y (cid:105) R = L R X (cid:104) ω, Y (cid:105) R − (cid:104)R − (cid:66) ω, ∇ RR − (cid:66) X Y (cid:105) R (4.11) for all X, Y ∈ X R ( A ) and ω ∈ Ω R ( A ) . Moreover, ∇ R and ˜ ∇ R extend as braidedderivations to equivariant covariant derivatives ∇ R : X R ( A ) ⊗ X •R ( A ) → X •R ( A ) and ˜ ∇ R : X R ( A ) ⊗ Ω •R ( A ) → Ω •R ( A ) , respectively. Namely, we inductively set ∇ R X a = X ( a ) = ˜ ∇ R X a , ∇ R X ( Y ∧ R Z ) = ∇ R X Y ∧ R Z + ( R − (cid:66) Y ) ∧ R ∇ RR − (cid:66) X Z and ˜ ∇ R X ( ω ∧ R η ) = ˜ ∇ R X ω ∧ R η + ( R − (cid:66) ω ) ∧ R ˜ ∇ RR − (cid:66) X η for all a ∈ A , X ∈ X R ( A ) , Y, Z ∈ X •R ( A ) and ω, η ∈ Ω •R ( A ) .Proof. Let ξ ∈ H , X ∈ X R ( A ), Y, Z ∈ X •R ( A ), ω ∈ Ω R ( A ) and η, χ ∈ Ω •R ( A ) bearbitrary. Then ˜ ∇ R is well-defined on Ω R ( A ), since L R X (cid:104) ω, Y · a (cid:105) R − (cid:104)R − (cid:66) ω, ∇ RR − (cid:66) X ( Y · a ) (cid:105) R = L R X (cid:104) ω, Y (cid:105) R · a + (cid:104)R − (cid:66) ω, R − (cid:66) Y (cid:105) R · ( R − (cid:66) X )( a ) − (cid:104)R − (cid:66) ω, ∇ RR − (cid:66) X Y (cid:105) R · a − (cid:104)R − (cid:66) ω, ( R (cid:48) − (cid:66) Y ) · (( R (cid:48) − R − ) (cid:66) X )( a ) (cid:105) R =( L R X (cid:104) ω, Y (cid:105) R − (cid:104)R − (cid:66) ω, ∇ RR − (cid:66) X Y (cid:105) R ) · a. It is an equivariant covariant derivative because (cid:104) ˜ ∇ R ξ (1) (cid:66) X ( ξ (2) (cid:66) ω ) , Y (cid:105) R = L R ξ (1) (cid:66) X (cid:104) ξ (2) (cid:66) ω, Y (cid:105) R − (cid:104) ( R − ξ (2) ) (cid:66) ω, ∇ R ( R − ξ (1) ) (cid:66) X Y (cid:105) R = ξ (1) (cid:66) ( L R X (cid:104) ω, S ( ξ (2) ) (cid:66) Y (cid:105) R ) − ξ (1) (cid:66) (cid:104)R − (cid:66) ω, ∇ RR − (cid:66) X ( S ( ξ (2) ) (cid:66) Y ) (cid:105) R = ξ (1) (cid:66) (cid:104) ˜ ∇ R X ω, S ( ξ (2) ) (cid:66) Y (cid:105) R = (cid:104) ξ (cid:66) ( ˜ ∇ R X ω ) , Y (cid:105) R 02 Chapter 4shows that it is H -equivariant, while (cid:104) ˜ ∇ R a · X ω, Y (cid:105) R = L R a · X (cid:104) ω, Y (cid:105) R − (cid:104)R − (cid:66) ω, ∇ R ( R − (cid:66) a ) · ( R − (cid:66) X ) Y (cid:105) R = a · L R X (cid:104) ω, Y (cid:105) R − (( R (cid:48) − R − ) (cid:66) a ) · (cid:104) ( R (cid:48) − R − ) (cid:66) ω, ∇ RR − (cid:66) X Y (cid:105) R = a · L R X (cid:104) ω, Y (cid:105) R − a · (cid:104)R − (cid:66) ω, ∇ RR − (cid:66) X Y (cid:105) R = a · (cid:104) ˜ ∇ R X ω, Y (cid:105) R = (cid:104) a · ˜ ∇ R X ω, Y (cid:105) R and (cid:104) ˜ ∇ R X ( a · ω ) , Y (cid:105) R = L R X (cid:104) a · ω, Y (cid:105) R − (cid:104) ( R − (cid:66) a ) · ( R − (cid:66) ω ) , ∇ RR − (cid:66) X Y (cid:105) R = X ( a ) · (cid:104) ω, Y (cid:105) R + ( R − (cid:66) a ) · L RR − (cid:66) X (cid:104) ω, Y (cid:105) R − ( R − (cid:66) a ) · (cid:104)R (cid:48) − (cid:66) ω, ∇ R ( R (cid:48)− R − ) (cid:66) X Y (cid:105) R = (cid:104) L R X ( a ) · ω + ( R − (cid:66) a ) · ( ˜ ∇ RR − (cid:66) X ω ) , Y (cid:105) R prove that ˜ ∇ R provides the correct linearity properties via the non-degeneracy of thebraided dual pairing. To prove that the extension of ∇ R to X •R ( A ) is well-definedit is sufficient to show that ∇ R X Y ∧ R Z + ( R − (cid:66) Y ) ∧ R ∇ RR − (cid:66) X Z − ( − k · (cid:96) (cid:18) ∇ R X ( R − (cid:66) Z ) ∧ R ( R − (cid:66) Y )+ (( R − R (cid:48) − ) (cid:66) Z ) ∧ R ∇ RR − (cid:66) X ( R (cid:48) − (cid:66) Y ) (cid:19) =0where k and (cid:96) are the degrees of Y and Z , respectively. Starting from k = (cid:96) = 0 thisfollows inductively by the braided commutativity of ∧ R and the equivariance of ∇ R .Assume that ∇ R a · X Y = a · ∇ R X Y and ∇ R X ( a · Y ) = L R X a · Y + ( R − (cid:66) a ) · ∇ RR − (cid:66) X Y for a fixed degree k > Y . Let the degree of Z be 1. Then ∇ R a · X ( Y ∧ R Z ) = ∇ R a · X Y ∧ R Z + ( R − (cid:66) Y ) ∧ R ∇ R ( R − (cid:66) a ) · ( R − (cid:66) X ) Z = a · ∇ R X Y ∧ R Z + (( R (cid:48) − R − ) (cid:66) a ) · (( R (cid:48) − R − ) (cid:66) Y ) ∧ R ∇ RR − (cid:66) X Z = a · ∇ R X Y ∧ R Z + a · ( R − (cid:66) Y ) ∧ R ∇ RR − (cid:66) X Z = a · ( ∇ R X ( Y ∧ R Z ))and ∇ R X ( a · Y ∧ R Z ) = ∇ R X ( a · Y ) ∧ R Z + ( R − (cid:66) a ) · ( R − (cid:66) Y ) ∧ R ∇ RR − (cid:66) X Z = L R X ( a ) · Y ∧ R Z + ( R − (cid:66) a ) · ( ∇ RR − (cid:66) X Y ) ∧ R Z + ( R − (cid:66) a ) · ( R (cid:48) − (cid:66) Y ) ∧ R ∇ R ( R (cid:48)− R − ) (cid:66) X Z = L R X ( a ) · Y ∧ R Z + ( R − (cid:66) a ) · ∇ RR − (cid:66) X ( Y ∧ R Z )raided Cartan Calculi 103show that ∇ R has the correct linearity properties on elements of degree k + 1.Inductively this shows ∇ R is an equivariant covariant derivative on X •R ( A ). Similarlyone proves that ˜ ∇ R is an equivariant covariant derivative on Ω •R ( A ).In Riemannian geometry, covariant derivatives are always consider together witha Riemannian metric. We want to generalize them to the braided symmetric setting. Definition 4.5.4 (Equivariant Metric) . For a triangular Hopf algebra ( H, R ) anda braided commutative left H -module algebra A we define a k -linear map g : X R ( A ) ⊗ A X R ( A ) → A to be an equivariant metric if it is left A -linear in the first argument, H -equivariantand braided symmetric, i.e. if g ( Y, X ) = g ( R − (cid:66) X, R − (cid:66) Y ) for all X, Y ∈ X R ( A ) .An equivariant metric g is said to bei.) non-degenerate if g ( X, Y ) = 0 for all Y ∈ X R ( A ) implies X = 0 ;ii.) strongly non-degenerate if g ( X, X ) (cid:54) = 0 for X (cid:54) = 0 ;iii.) Riemannian if g is strongly non-degenerate such that g ( X, X ) ≥ for all X ∈ X R ( A ) and a partial order ≥ on A ;An equivariant covariant derivative ∇ R : X R ( A ) ⊗ X R ( A ) → X R ( A ) on A is said tobe a metric equivariant covariant derivative with respect to an equivariant metric g if L R X ( g ( Y, Z )) = g ( ∇ R X Y, Z ) + g ( R − (cid:66) Y, ∇ RR − (cid:66) X Z ) (4.12) holds for all X, Y, Z ∈ X R ( A ) . Note that equivariance of a metric is a quite strong requirement. Similar ap-proaches which omit this condition are e.g. [4, 7, 57]. Lemma 4.5.5. Any equivariant metric is braided right A -linear in the first argumentas well as right A -linear and braided left A -linear in the second argument. In the subsequent proposition we prove the existence and uniqueness of a Levi-Civita covariant derivative in braided geometry: for every non-degenerate equivari-ant metric there exists a unique metric torsion-free equivariant covariant derivative.We are even going to prove this for an arbitrary value of the torsion. Note that thenon-degeneracy is crucial in the proof since the equivariant covariant derivative isdefined implicitly in terms of the equivariant metric. Proposition 4.5.6 (Levi-Civita) . Let g be a non-degenerate equivariant metric on A . Then there is a unique metric equivariant covariant derivative on A with fixedtorsion T : X R ( A ) → X R ( A ) .Proof. Fix an equivariant metric g on A . Any equivariant covariant derivative ∇ R on A which is metric with respect to g satisfies2 g ( ∇ R X Y, Z ) = X ( g ( Y, Z )) + ( R − (cid:66) Y )( g ( R − (cid:66) Z, R − (cid:66) X )) − ( R − (cid:66) Z )( g ( R − (cid:66) X, R − (cid:66) Y )) − g ( X, [ Y, Z ] R ) + g ( R − (cid:66) Y, [ R − (cid:66) Z, R − (cid:66) X ] R )+ g ( R − (cid:66) Z, [ R − (cid:66) X, R − (cid:66) Y ] R ) − g ( X, Tor ∇ R ( Y, Z )) − g ( R − (cid:66) Y, Tor ∇ R ( R − (cid:66) X, Z ))+ g ( R − (cid:66) Z, Tor ∇ R ( R − (cid:66) X, R − (cid:66) Y )) (4.13)04 Chapter 4for all X, Y, Z ∈ X R ( A ). To see this, we first note that X ( g ( Y, Z )) = g ( ∇ R X Y, Z ) + g ( R − (cid:66) Y, ∇ RR − (cid:66) X Z )= g ( ∇ R X Y, Z ) + g ( R − (cid:66) Y, ∇ RR (cid:48)− (cid:66) Z (( R (cid:48) − R − ) (cid:66) X ))+ g ( R − (cid:66) Y, [ R − (cid:66) X, Z ] R ) + g ( R − (cid:66) Y, Tor ∇ R ( R − (cid:66) X, Z )) . Then X ( g ( Y, Z )) + ( R − (cid:66) Y )( g ( R − (cid:66) Z, R − (cid:66) X )) − ( R − (cid:66) Z )( g ( R − (cid:66) X, R − (cid:66) Y ))= g ( ∇ R X Y, Z ) + g ( R − (cid:66) Y, ∇ RR (cid:48)− (cid:66) Z (( R (cid:48) − R − ) (cid:66) X ))+ g ( R − (cid:66) Y, [ R − (cid:66) X, Z ] R ) + g ( R − (cid:66) Y, Tor ∇ R ( R − (cid:66) X, Z ))+ g ( ∇ RR (cid:48)− (cid:66) Y ( R (cid:48) − (cid:66) Z ) , R (cid:48) − (cid:66) X )+ g (( R − R (cid:48) − ) (cid:66) Z, ∇ R ( R (cid:48)(cid:48)− R (cid:48)− ) (cid:66) X (( R (cid:48)(cid:48) − R − R (cid:48) − ) (cid:66) Y ))+ g (( R − R (cid:48) − ) (cid:66) Z, [( R − R (cid:48) − ) (cid:66) Y, R (cid:48) − (cid:66) X ] R )+ g (( R − R (cid:48) − (cid:66) Z, Tor ∇ R (( R − R (cid:48) − ) (cid:66) Y, R (cid:48) − (cid:66) X )) − g ( ∇ RR (cid:48)− (cid:66) Z ( R (cid:48) − (cid:66) X ) , R (cid:48) − (cid:66) Y ) − g (( R − R (cid:48)(cid:48) − ) (cid:66) X, ∇ R ( R (cid:48)− R (cid:48)(cid:48)− ) (cid:66) Y (( R (cid:48) − R − R (cid:48)(cid:48) − ) (cid:66) Z )) − g (( R − R (cid:48) − ) (cid:66) X, [( R − R (cid:48) − ) (cid:66) Z, R (cid:48) − (cid:66) Y ] R ) − g (( R − R (cid:48) − ) (cid:66) X, Tor ∇ R (( R − R (cid:48) − ) (cid:66) Z, R (cid:48) − (cid:66) Y ))=2 g ( ∇ R X Y, Z )+ g ( R − (cid:66) Y, [ R − (cid:66) X, Z ] R ) + g ( R − (cid:66) Y, Tor ∇ R ( R − (cid:66) X, Z ))+ g (( R − R (cid:48) − ) (cid:66) Z, [( R − R (cid:48) − ) (cid:66) Y, R (cid:48) − (cid:66) X ] R )+ g (( R − R (cid:48) − (cid:66) Z, Tor ∇ R (( R − R (cid:48) − ) (cid:66) Y, R (cid:48) − (cid:66) X )) − g (( R − R (cid:48) − ) (cid:66) X, [( R − R (cid:48) − ) (cid:66) Z, R (cid:48) − (cid:66) Y ] R ) − g (( R − R (cid:48) − ) (cid:66) X, Tor ∇ R (( R − R (cid:48) − ) (cid:66) Z, R (cid:48) − (cid:66) Y ))gives the proposed formula. Since the equivariant metric is non-degenerate thisproves that a metric equivariant covariant derivative with a fixed torsion is unique.To also provide existence we show that the above formula actually defines a metricequivariant covariant derivative with torsion T . Replacing X in (4.13) by a · X for an arbitrary a ∈ A we observe that the terms including torsion are linear withrespect to a , as well as the first and the fourth term on the right-hand-side. Theadditional non- A -linear terms of the second and third summand cancel with thecorresponding additional terms of the sixth and fifth summand, respectively. Bythe non-degeneracy of g this proves that ∇ R is left A -linear in the first argument.Similarly one demonstrates the braided Leibniz rule in the second argument.The unique torsion-free metric equivariant covariant derivative on ( A , g ) is saidto be the equivariant Levi-Civita covariant derivative .raided Cartan Calculi 105 In this last section we prove that the gauge equivalence induced by a Drinfel’d twistis compatible with the construction of the braided Cartan calculus and the notionof equivariant covariant derivative. Namely, we describe how the Drinfel’d functortogether with the corresponding natural transformation discussed in Section 2.4deforms the involved structure. For a given braided commutative algebra A wedefine the twisted multivector fields and twisted differential forms, as well as thetwisted Gerstenhaber bracket, Lie derivative, insertion and de Rham differential.If the universal R -matrix is trivial this recovers the well-known twisted Cartancalculus (see e.g. [7]). Even in the general case, we prove that the twisted Cartancalculus is isomorphic to the braided Cartan calculus with respect to the twistedalgebra and twisted triangular structure. In particular, this isomorphism (takenfrom [7, 54, 63]) intertwines the braided Lie derivative, insertion and de Rhamdifferential. In a similar fashion we prove that a twisted covariant derivative canbe interpreted as an equivariant covariant derivative on the twisted algebra withrespect to the twisted triangular structure, by applying the same isomorphism. Theconcepts of equivariant metric and equivariant Levi-Civita covariant derivative arerespected as well. We refer to [10] Chap. 3-4 for a similar discussion.Recall that for any Drinfel’d twist F on a triangular Hopf algebra ( H, R ) andany braided commutative left H -module algebra A the twisted product a · F b = ( F − (cid:66) a ) · ( F − (cid:66) b ) , where a, b ∈ A , structures A as a left H F -module algebra which is braided com-mutative with respect to R F = F RF − . As usual we write A F = ( A , · F ). Moregeneral, the Drinfel’d functor Drin F : H A M → H F A F M , which is the identity on morphisms and assigns to any H -equivariant left A -module M the same left H -module, however with left A F -module action a · F m = ( F − (cid:66) a ) · ( F − (cid:66) m ) , where a ∈ A and m ∈ M is an isomorphism of categories. As usual we denote M F = ( M , · F ). On H -equivariant braided symmetric A -bimodules the Drinfel’dfunctor Drin F : ( H A M RA , ⊗ A ) → ( H F A F M R F A F , ⊗ A F )is a braided monoidal equivalence of braided monoidal categories with braidedmonoidal natural transformation given on objects M and M (cid:48) of H A M RA by ϕ M , M (cid:48) : M F ⊗ A F M (cid:48)F (cid:51) ( m ⊗ A F m (cid:48) ) (cid:55)→ ( F − (cid:66) m ) ⊗ A ( F − (cid:66) m (cid:48) ) ∈ ( M ⊗ A M (cid:48) ) F . Fix a Drinfel’d twist F on H in the following. Lemma 4.6.1. The assignment ( X R ( A )) F (cid:51) X (cid:55)→ X F ∈ X R F ( A F ) , 06 Chapter 4 where X F ( a ) = ( F − (cid:66) X )( F − (cid:66) a ) for all a ∈ A F , is an isomorphism of H F -equivariant braided symmetric A F -bimodules. Namely, ξ (cid:66) X F =( ξ (cid:66) X ) F ,a · R F X F =( a · F X ) F ,X F · R F a =( X · F a ) F for all ξ ∈ H , a ∈ A and X ∈ X R ( A ) , where we denoted the A F -module actions on X R F ( A F ) by · R F .Proof. The assignment is well-defined, since for every X ∈ Der R ( A ) one obtains anelement X F ∈ Der R F ( A F ) because X F ( a · F b ) =( F − (cid:66) X )((( F − F (cid:48) − ) (cid:66) a ) · (( F − F (cid:48) − ) (cid:66) b ))=( F − (cid:66) X )(( F − F (cid:48) − ) (cid:66) a ) · (( F − F (cid:48) − ) (cid:66) b )+ (( R − F − F (cid:48) − ) (cid:66) a ) · (( R − F − ) (cid:66) X )(( F − F (cid:48) − ) (cid:66) b )= X F ( a ) · F b + (( R − F − F (cid:48) − ) (cid:66) a ) · (( R − F − F (cid:48) − ) (cid:66) X )( F − (cid:66) b )= X F ( a ) · F b + (( F − R − F (cid:48) − ) (cid:66) a ) · (( F − R − F (cid:48) − ) (cid:66) X )( F − (cid:66) b )= X F ( a ) · F b + (( F − F (cid:48) − R − F ) (cid:66) a ) · (( F − F (cid:48) − R − F ) (cid:66) X )( F − (cid:66) b )= X F ( a ) · F b + (( F − R − F ) (cid:66) a ) · (( F − F (cid:48) − R − F ) (cid:66) X )(( F − F (cid:48) − ) (cid:66) b )= X F ( a ) · F b + ( R − F (cid:66) a ) · F ( R − F (cid:66) X ) F ( b )= X F ( a ) · F b + ( R − F (cid:66) a ) · F ( R − F (cid:66) X F )( b )for all a, b ∈ A , where we used that ( ξ (cid:66) X ) F = ξ (cid:66) X F for all ξ ∈ H . The latteris true because( ξ (cid:66) X F )( a ) = ξ (cid:99) (1) (cid:66) ( X F ( S F ( ξ (cid:99) (2) ) (cid:66) a ))=( F ξ (1) F (cid:48) − ) (cid:66) (( F (cid:48)(cid:48) − (cid:66) X )(( F (cid:48)(cid:48) − βS ( F ξ (2) F (cid:48) − ) β − ) (cid:66) a ))=( F ξ (1) ) (cid:66) (( F (cid:48) − (cid:66) X )(( F (cid:48) − F (cid:48)(cid:48) − βS ( F ξ (2) F (cid:48) − F (cid:48)(cid:48) − ) β − ) (cid:66) a ))=( F ξ (1) ) (cid:66) (( F (cid:48) − (cid:66) X )(( F (cid:48) − S ( F ξ (2) F (cid:48) − ) β − ) (cid:66) a ))=( F ξ (1) ) (cid:66) ( X (( S ( F ξ (2) ) β − ) (cid:66) a ))=(( F ξ ) (cid:66) X )(( F S ( F ) β − ) (cid:66) a )=(( F (cid:48) F ξ ) (cid:66) X ) F (( F (cid:48) F S ( F ) β − ) (cid:66) a )=(( F ξ ) (cid:66) X ) F (( F (cid:48) F S ( F (cid:48) F ) β − ) (cid:66) a )=( ξ (cid:66) X ) F (( ββ − ) (cid:66) a )=( ξ (cid:66) X ) F ( a ) . Furthermore, the left and right A -module actions are respected since( a · F X ) F ( b ) =(( F − F (cid:48) − ) (cid:66) a ) · (( F − F (cid:48) − ) (cid:66) X )( F − (cid:66) b )=( F − (cid:66) a ) · (( F − F (cid:48) − ) (cid:66) X )(( F − F (cid:48) − ) (cid:66) b )= a · F ( X F ( b ))=( a · F X F )( b )raided Cartan Calculi 107and ( X · F a ) F ( b ) =((( F − F (cid:48) − ) (cid:66) X ) · (( F − F (cid:48) − ) (cid:66) a ))( F − (cid:66) b )=( F − (cid:66) X )(( R − F − F (cid:48) − ) (cid:66) b ) · (( R − F − F (cid:48) − ) (cid:66) a )=( F − (cid:66) X )(( F − R − F (cid:48) − ) (cid:66) b ) · (( F − R − F (cid:48) − ) (cid:66) a )=( F − (cid:66) X )(( F − F (cid:48) − R − F ) (cid:66) b ) · (( F − F (cid:48) − R − F ) (cid:66) a )=(( F − F (cid:48) − ) (cid:66) X )(( F − F (cid:48) − R − F ) (cid:66) b ) · (( F − R − F ) (cid:66) a )= X F ( R − F (cid:66) b ) · F ( R − F (cid:66) a )=( X F · F a )( b )hold for all a, b ∈ A . The inverse homomorphism is given byDer R F ( A F ) (cid:51) Ξ (cid:55)→ Ξ F − ∈ Der R ( A ) , where Ξ F − ( a ) = ( F (cid:66) Ξ)( F (cid:66) a ) for all a ∈ A .This proves that we can work with X R ( A ) F instead of X R F ( A F ). Applying theDrinfel’d functor on the wedge product ∧ R : X R ( A ) ⊗ A X R ( A ) → X R ( A )gives Drin F ( ∧ R ) : (cid:0) X R ( A ) ⊗ A X R ( A ) (cid:1) F → X R ( A ) F . So if we want to interpret the image of the wedge product as an actual product on X •R ( A ) F we have to make use of the natural transformation ϕ . We define ∧ F = Drin( ∧ ) ◦ ϕ X R ( A ) , X R ( A ) : X R ( A ) F ⊗ A F X R ( A ) F → X R ( A ) F and call it the twisted wedge product . Furthermore we extend the isomorphism X R ( A ) F → X R F ( A F ) to higher wedge powers as a homomorphism of the twistedwedge product, i.e. ( X ∧ F Y ) F = X F ∧ R F Y F for all X, Y ∈ X •R ( A ) F , where ∧ F = Drin F ( ∧ R ) ◦ ϕ X •R ( A ) , X •R ( A ) . By Lemma 4.6.1this is well-defined. Inductively this leads to an isomorphism X •R ( A ) F → X •R F ( A F )of H F -equivariant braided symmetric A F -bimodules. Also the twisted Schouten-Nijenhuis bracket (cid:74) · , · (cid:75) F : Drin F ( (cid:74) · , · (cid:75) R ) ◦ ϕ X •R ( A ) , X •R ( A ) : X •R ( A ) F ⊗ A F X •R ( A ) F → X •R ( A ) F can be defined. On elements X, Y ∈ X •R ( A ) F the twisted structures read X ∧ F Y = ( F − (cid:66) X ) ∧ R ( F − (cid:66) Y )and (cid:74) X, Y (cid:75) F = (cid:74) F − (cid:66) X, F − (cid:66) Y (cid:75) R , respectively.08 Chapter 4 Proposition 4.6.2. This assignment F : ( X •R ( A ) F , ∧ F , (cid:74) · , · (cid:75) F ) → ( X •R F ( A F ) , ∧ R F , (cid:74) · , · (cid:75) R F ) (4.14) is an isomorphism of braided Gerstenhaber algebras.Proof. First note that( X · F Y ) F ( a ) =(( F − F (cid:48) − ) (cid:66) X )(( F − F (cid:48) − ) (cid:66) Y )( F − (cid:66) a )=( F − (cid:66) X )(( F − F (cid:48) − ) (cid:66) Y )(( F − F (cid:48) − ) (cid:66) a )=( X F · R F Y F )( a )for all X, Y ∈ X R ( A ) F and a ∈ A . Then([ X, Y ] F ) F =([ F − (cid:66) X, F − (cid:66) Y ] R ) F =(( F − (cid:66) X ) · R ( F − (cid:66) Y )) F − ((( R − F − ) (cid:66) Y ) · R (( R − F − ) (cid:66) X )) F =( X · F Y ) F − (( R − F (cid:66) Y ) · F ( R − F (cid:66) X )) F = X F · R F Y F − ( R − F (cid:66) Y ) F · R F ( R − F (cid:66) Y ) F = X F · R F Y F − ( R − F (cid:66) Y F ) · R F ( R − F (cid:66) Y F )=[ X F , Y F ] R F . Using the defining formula (see Section 4.2) of the braided Schouten-Nijenhuisbracket, this implies that ( (cid:74) X, Y (cid:75) F ) F = (cid:74) X F , Y F (cid:75) R F for all X, Y ∈ X •R ( A ) F .Similarly we define an isomorphism F : Ω •R ( A ) F → Ω •R F ( A F ) of H F -equivariantbraided symmetric A F -bimodules. On elements ω ∈ Ω R ( A ) F it reads ω F ( X F ) = ( F − (cid:66) ω )( F − (cid:66) X )for all X ∈ X R ( A ) F . In fact ω F is an element of Ω R F ( A F ) since ω F ( X F · R F a ) = ω F (( X · F a ) F )=( F − (cid:66) ω )((( F − F (cid:48) − ) (cid:66) X ) · R (( F − F (cid:48) − ) (cid:66) a ))=(( F − F (cid:48) − ) (cid:66) ω )(( F − F (cid:48) − ) (cid:66) X ) · ( F − (cid:66) a )=( F (cid:48) − (cid:66) ω )( F (cid:48) − (cid:66) X ) · F a = ω F ( X F ) · F a for all a ∈ A . Furthermore we define( ω ∧ F η ) F = ω F ∧ R F η F for all ω, η ∈ Ω •R ( A ) F . Applying the Drinfel’d functor and the natural transforma-tion ϕ on L R and i R leads to L F : X •R ( A ) F ⊗ F Ω •R ( A ) F → Ω •R ( A ) F , i F : X •R ( A ) F ⊗ F Ω •R ( A ) F → Ω •R ( A ) F , raided Cartan Calculi 109while the de Rham differential becomes d : Ω •R ( A ) F → Ω • +1 R ( A ) F after utilizing theDrinfel’d functor. On elements X ∈ X •R ( A ) F and ω ∈ Ω •R ( A ) F the twisted Liederivative and twisted insertion read L F X ω = L RF − (cid:66) X ( F − (cid:66) ω ) and i F X ω = i RF − (cid:66) X ( F − (cid:66) ω ) , while the de Rham differential remains undeformed. We refer to(Ω •R ( A ) F , ∧ F , L F , i F , d) and ( X •R ( A ) F , ∧ F , (cid:74) · , · (cid:75) F )as the twisted Cartan calculus (with respect to F and R ). Theorem 4.6.3. The twisted Cartan calculus with respect to R and F is isomorphicto the braided Cartan calculus on A F with respect to R F via the isomorphism F . Inparticular ( L F X ω ) F = L R F X F ω F , (i F X ω ) F =i R F X F ω F , (d ω ) F =d ω F for all X ∈ X •R ( A ) F and ω ∈ Ω •R ( A ) F .Proof. For a ∈ A , X, Y ∈ X R ( A ) F and ω ∈ Ω R ( A ) F we obtain(i F X ω ) F =(i RF − (cid:66) X ( F − (cid:66) ω )) F =((( R − F − ) (cid:66) ω )(( R − F − ) (cid:66) X )) F =(( F − R − F ) (cid:66) ω )(( F − R − F ) (cid:66) X )=( R − F (cid:66) ω ) F (( R − F (cid:66) X ) F )=( R − F (cid:66) ω F )( R − F (cid:66) X F )=i R F X F ω F , since F is an isomorphism of H F -equivariant braided symmetric A F -bimodules.Similarly (d a ) F ( X F ) =( F − (cid:66) (d a ))( F − (cid:66) X )=(d( F − (cid:66) a ))( F − (cid:66) X )=(( R − F − ) (cid:66) X )(( R − F − ) (cid:66) a )=(( F − R − F ) (cid:66) X )(( F − R − F ) (cid:66) a )=( R − F (cid:66) X ) F ( R − F (cid:66) a )=( R − F (cid:66) X F )( R − F (cid:66) a )=(d a )( X F ) , follows, and since F respects the braided commutator we obtaind ω F ( X F ∧ R F Y F ) =( R − F (cid:66) X F )(( R − F (cid:66) ω F )( Y F )) − ( R − F (cid:66) Y F )( R − F (cid:66) ( ω F ( X F ))) − ω F ([ X F , Y F ] R F )10 Chapter 4=(( F − R − F ) (cid:66) X )((( F − F (cid:48) − R − F ) (cid:66) ω )(( F − F (cid:48) − ) (cid:66) Y )) − (( F − R − F ) (cid:66) Y )(( F − R − F ) (cid:66) (( F (cid:48) − (cid:66) ω )( F (cid:48) − (cid:66) X ))) − ω F (([ X, Y ] F ) F )=(( F − F (cid:48) − R − F ) (cid:66) X )((( F − F (cid:48) − R − F ) (cid:66) ω )( F − (cid:66) Y )) − (( R − F − ) (cid:66) Y )(( R − F − ) (cid:66) (( F (cid:48) − (cid:66) ω )( F (cid:48) − (cid:66) X ))) − ( F − (cid:66) ω )( F − (cid:66) [ X, Y ] F )=(( F − R − F (cid:48) − ) (cid:66) X )((( F − R − F (cid:48) − ) (cid:66) ω )( F − (cid:66) Y )) − (( R − F − ) (cid:66) Y )( R − (cid:66) ((( F − F (cid:48) − ) (cid:66) ω )(( F − F (cid:48) − ) (cid:66) X ))) − ( F − (cid:66) ω )([( F − F (cid:48) − ) (cid:66) X, ( F − F (cid:48) − ) (cid:66) Y ] R )=(( R − F − F (cid:48) − ) (cid:66) X )((( R − F − ) (cid:66) ω )( F − F (cid:48) − ) (cid:66) Y )) − (( R − F − F (cid:48) − ) (cid:66) Y )( R − (cid:66) (( F − (cid:66) ω )(( F − F (cid:48) − ) (cid:66) X ))) − ( F − (cid:66) ω )([( F − F (cid:48) − ) (cid:66) X, ( F − F (cid:48) − ) (cid:66) Y ] R )=(d ω ) F (( X ∧ F Y ) F )=(d ω ) F ( X F ∧ R F Y F ) . Moreover, since d is equivariant L R F X F ω F =[i R F X F , d] R F ω F =i R F X F d ω F − di R F X F ω F =i R F X F (d ω ) F − d(i F X ω ) F =(i F X d ω ) F − (di F X ω ) F =([i F X , d] R ω ) F =( L F X ω ) F , follows. In degree zero there is nothing to prove and higher degrees are generatedby the lowest two degrees. Since F respects the wedge product this concludes theproof of the theorem.Let M be an H -equivariant braided symmetric A -bimodule and ∇ R : X R ( A ) ⊗M → M an equivariant covariant derivative with respect to R . Define the twistedcovariant derivative as ∇ F = Drin F ( ∇ R ) ◦ ϕ X R ( A ) , M : X R ( A ) F ⊗ F M F → M F . On elements X ∈ X R ( A ) F and s ∈ M F this reads ∇ F X s = ∇ RF − (cid:66) X ( F − (cid:66) s ) . Lemma 4.6.4. ∇ F satisfies ξ (cid:66) ( ∇ F X s ) = ∇ F ξ (1) (cid:66) X ( ξ (2) (cid:66) s ) , ∇ F a · F X s = a · F ∇ F X s and ∇ F X ( a · F s ) = ( L F X a ) · F s + ( R − F (cid:66) a ) · F ( ∇ FR − F (cid:66) X s ) for all a ∈ A , X ∈ X R ( A ) F and s ∈ M F . raided Cartan Calculi 111 Proof. Let ξ ∈ H , a ∈ A , X ∈ X R ( A ) F and s ∈ M F . Then ξ (cid:66) ( ∇ F X s ) = ∇ R ( ξ (1) F − ) (cid:66) X (( ξ (2) F − ) (cid:66) s ) = ∇ F ξ (cid:99) (1) (cid:66) X ( ξ (cid:99) (2) (cid:66) s )shows that ∇ F is H F -equivariant, while also ∇ F a · F X s =(( F − F (cid:48) − ) (cid:66) a ) · ( ∇ R ( F − F (cid:48)− ) (cid:66) X ( F − (cid:66) s ))=( F − (cid:66) a ) · ( ∇ R ( F − F (cid:48)− ) (cid:66) X (( F − F (cid:48) − ) (cid:66) s ))=( F − (cid:66) a ) · ( F − (cid:66) ( ∇ RF (cid:48)− (cid:66) X ( F (cid:48) − (cid:66) s )))= a · F ( ∇ F X s )and ∇ F X ( a · F s ) = ∇ RF − (cid:66) X ((( F − F (cid:48) − ) (cid:66) a ) · (( F − F (cid:48) − ) (cid:66) a ))=( L RF − (cid:66) X (( F − F (cid:48) − ) (cid:66) a )) · (( F − F (cid:48) − ) (cid:66) s )+ (( R − F − F (cid:48) − ) (cid:66) a ) · ( ∇ R ( R − F − ) (cid:66) X (( F − F (cid:48) − ) (cid:66) s ))=( F − (cid:66) ( L RF (cid:48)− (cid:66) X ( F (cid:48) − (cid:66) a ))) · ( F − (cid:66) s )+ (( F − R − F (cid:48) − ) (cid:66) a ) · ( ∇ R ( F − R − F (cid:48)− ) (cid:66) X ( F − (cid:66) s ))=( L F X a ) · F s + (( F − F − R − F ) (cid:66) a ) · ( ∇ R ( F − F − R − F ) (cid:66) X ( F − (cid:66) s ))=( L F X a ) · F s + ( R − F (cid:66) a ) · F ( ∇ FR F (cid:66) X s )hold.Note however that strictly speaking ∇ F is not an equivariant covariant deriva-tive with respect to R F , since it is a map X R ( A ) F ⊗ F M F → M F . Using theisomorphism X R ( A ) F → X R F ( A F ) we are able to view ∇ F as an equivariant co-variant derivative on M F with respect to R F nevertheless. This is done in the nextproposition. Proposition 4.6.5. Let ∇ R be an equivariant covariant derivative with respect to R on an object M in H A M A . Then we can define an equivariant covariant derivative ∇ R F : X R F ( A F ) ⊗ F M F → M F with respect to R F via ∇ R F X F s = ∇ F X s for all X ∈ X R ( A ) F and s ∈ M F . If ∇ R is an equivariant covariant derivative withrespect to R on X R ( A ) there is an equivariant covariant derivative ∇ R F : X R F ( A F ) ⊗ F X R F ( A F ) → X R F ( A F ) with respect to R F defined by ∇ R F X F Y F = ( ∇ F X Y ) F 12 Chapter 4 for all X, Y ∈ X R ( A ) F . The corresponding extensions ∇ R F : X R F ( A F ) ⊗ F X •R F ( A F ) → X •R F ( A F ) and ˜ ∇ R F : X R F ( A F ) ⊗ F Ω •R F ( A F ) → Ω •R F ( A F ) of the latter to braided multivector fields and braided differential forms satisfy ∇ R F X F Y F = ( ∇ F X Y ) F and ˜ ∇ R F X F ω F = ( ˜ ∇ F X ω ) F for all X ∈ X R ( A ) F , Y ∈ X •R ( A ) F and ω ∈ Ω •R ( A ) F . Let g be an equivariant metric on A . Then, the twisted metric g F is defined by g F ( X, Y ) = g ( F − (cid:66) X, F − (cid:66) Y )for all X, Y ∈ X R ( A ). In the next lemma we prove that it actually deserves thisname, i.e. that g F is an equivariant metric. Furthermore we show that the as-signment LC : g (cid:55)→ ∇ R , attributing to a non-degenerate equivariant metric g itscorresponding equivariant Levi-Civita covariant derivative ∇ R , is respected by theDrinfel’d functor, i.e. g ∇ R g F ∇ F LCDrin F Drin F LC commutes. Remark that g F might not be non-degenerate in general. Nonetheless, ∇ F is well-defined as twist deformation of ∇ R . Furthermore it is metric with re-spect to g F and torsion-free, while it might not be the unique equivariant covariantderivative with those properties. However, any torsion-free equivariant covariantderivative ˜ ∇ on A F which is metric with respect to g F is mapped to ∇ R via theinverse twist Drin F − . This follows from the uniqueness of the equivariant Levi-Civita covariant derivative ∇ R on ( A , g = ( g F ) F − ). In other words, ∇ F is theunique equivariant Levi-Civita covariant derivative on ( A F , g F ) up to the kernel of F − (cid:66) : X R ( A ) ⊗ X R ( A ) → X R ( A ) ⊗ X R ( A ). Lemma 4.6.6. For any equivariant metric g on A , the twisted metric g F is anequivariant metric with respect to R F on A F . Moreover, twisting the equivariantLevi-Civita covariant derivative with respect to g leads to a torsion-free equivariantcovariant derivative ∇ F , which is metric with respect to g F . If g F is non-degenerate, ∇ F is the unique equivariant covariant derivative with those properties.Proof. Let X, Y ∈ X R ( A ). The relation L F X ( g F ( Y, Z )) = g F ( ∇ F X Y, Z ) + g F ( R − F (cid:66) Y, ∇ FR − F (cid:66) X Z )follows from L R X g ( Y, Z ) = g ( ∇ R X Y, Z )+ g ( R − (cid:66) Y, ∇ RR − (cid:66) X Z ) and the H -equivarianceof g , ∇ R and L R . The last statement holds since Tor ∇ F = 0 if Tor ∇ R = 0.Notice that Lemma 4.6.6 even holds for metrics which are not equivariant if oneassumes F to consist of affine Killing vector fields (c.f. [4] Sec. 6.2). hapter 5Submanifold Algebras We introduce the notion of submanifold algebra on a braided commutative left H -module algebra A for a triangular Hopf algebra ( H, R ), slightly modifying theapproach of [79] (see [37] for a recent discussion on submanifold algebras). It gener-alizes the concept of closed embedded submanifolds from differential geometry. Ina nutshell a submanifold algebra is given by an algebra ideal C ⊆ A which is closedunder the Hopf algebra action. In particular, the surjective projection pr : A → A / C commutes with the H -action. In the course of this chapter we want to make senseof the following commutative diagramGeometry on A Geometry on A / C Geometry on A F Geometry on A F / C F = (cid:0) A / C (cid:1) F prDrin F Drin F pr , (5.1)where F is a Drinfel’d twist on H . This vague picture should be interpreted in thefollowing way: first, we prove that the geometric data on A / C , namely the braidedCartan calculus, equivariant metrics, covariant derivatives, curvature and torsion,are gained as projections of the corresponding objects in A and secondly, we provethat this projection commutes with the Drinfel’d functor. In Section 5.1 and Sec-tion 5.2 we study the horizontal arrow, projecting the braided Cartan calculus andequivariant covariant derivatives, respectively, while the vertical arrow together withthe commutativity of the diagram are examined in Section 5.3. Note that we haveto accept two axioms in order to receive well-defined projected equivariant metricsand covariant derivatives. However, since those assumptions are quite mild we stillobtain Riemannian geometry on smooth submanifolds as a special case of our the-ory. Finally, in Section 5.4, we give an explicit example of twist deformation ofthe 2-sheet elliptic hyperboloid. Starting from the commutative algebra of smoothfunctions with the pointwise product, the vertical arrows of (5.1) correspond to aquantization and the commutativity of (5.1) implies that twist deformation quan-tization and projection to the quadric surface commute. A different approach toRiemannian geometry on noncommutative submanifolds, based on the choice of afinite-dimensional Lie subalgebra g of Der( A ) and a vector space homomorphism g → M into a right A -bimodule M , is considered in [3].Fix a triangular Hopf algebra ( H, R ) and a braided commutative left H -modulealgebra A for the rest of this chapter. 11314 Chapter 5 For any algebra ideal C the coset space A / C becomes an algebra with unit andproduct induced from A . The elements of A / C are equivalence classes of elementsin A , where a and b are in the same equivalence class if and only if there exists anelement c ∈ C such that a = b + c . Choosing an arbitrary representative a ∈ A we denote the corresponding equivalence class by [ a ] or a + C . This constitutes asurjective projection pr : A → A / C , which assigns any element of A its equivalenceclass. It is easy to verify that pr is an algebra homomorphism. The projection isinjective if and only if C = { } . Lemma 5.1.1. Let C ⊆ A be an algebra ideal such that H (cid:66) C ⊆ C . Then A / C is aleft H -module algebra and braided commutative with respect to the same triangularstructure R .Proof. From general algebra we know that the coset space A / C is an algebra withrespect to the induced multiplication if and only if C is an ideal. The induced left H -action, i.e. ξ (cid:66) pr( a ) = pr( ξ (cid:66) a ), where a ∈ A and ξ ∈ H , is well-defined if andonly if H (cid:66) C ⊆ C . For the same reason the action respects the algebra structureand (cid:66) descends to a left H -module algebra action. Since the braiding is encodedvia the Hopf algebra action A / C is braided commutative.It is intuitive that not all braided vector fields on A can be projected to a braidedvector field on A / C . In the end braided vector fields are braided derivations andin particular endomorphisms of A . They only descend to an endomorphism of A / C if C is a subspace of the kernel. It turns out that this is the only obstruction: let X ∈ Der R ( A ) such that X ( C ) ⊆ C , thenpr( X )(pr( a )) = pr( X ( a )) , (5.2)where a ∈ A , defines a braided derivation pr( X ) on A / C with respect to R . In fact,pr( X ) is well-defined exactly because of the condition X ( C ) ⊆ C and it inherits thebraided derivation property from X . Definition 5.1.2. A braided derivation X ∈ Der R ( A ) is said to be a braided tangentvector field (with respect to C ) if X ( C ) ⊆ C . We denote the k -module of braidedtangent vector fields on A with respect to C by X t ( A ) . The k -submodule of braidedtangent vector fields X ∈ X t ( A ) satisfying X ( A ) ⊆ C is denoted by X ( A ) . Thecorresponding elements are called vanishing vector fields. The braided tangent vector fields are closed under the module actions and thebraided commutator. This is discussed in the next lemma. Note that we are ableto define a left H -action on the image of pr : X t ( A ) → X R ( A / C ) by ξ (cid:66) pr( X ) = pr( ξ (cid:66) X ) (5.3)for all ξ ∈ H and X ∈ X t ( A ). With (5.3) we have a natural candidate for a left H -action on X R ( A / C ). However, since it is only defined on the image of the projectionthis sets a further condition on the ideal C .ubmanifold Algebras 115 Definition 5.1.3. If the k -linear map pr : X t ( A ) → X R ( A / C ) defined in eq.(5.2) issurjective, the braided commutative algebra A / C is said to be a submanifold algebraand C a submanifold ideal. Fix a submanifold ideal C for the rest of this section. We want to stress that byour definition this includes the property H (cid:66) C ⊆ C . Lemma 5.1.4. X t ( A ) is an H -equivariant braided symmetric A -bimodule and abraided Lie algebra, while X ( A ) is an H -equivariant braided symmetric A -sub-bimodule and a braided Lie ideal. Furthermore, pr : X t ( A ) → X R ( A / C ) (5.4) is a homomorphism of H -equivariant braided symmetric A -bimodules and a homo-morphism of braided Lie algebras with kernel X ( A ) .Proof. According to Lemma 4.2.1 X R ( A ) is an H -equivariant braided symmetric A -bimodule and a braided Lie algebra with respect to the braided commutator. So itis sufficient to prove that the module actions and the braided commutator are closedin X t ( A ). Let a, b ∈ A , X, Y ∈ X t ( A ) and ξ ∈ H . Then ( a · X )( C ) = a · X ( C ) ⊆ C ,( X · a )( C ) = X ( R − (cid:66) C ) · ( R − (cid:66) a ) ⊆ C and ( ξ (cid:66) X )( C ) = ξ (1) (cid:66) ( X ( S ( ξ (2) ) (cid:66) C )) ⊆ C , while[ X, Y ] R ( C ) = X ( Y ( C )) − ( R − (cid:66) Y )( R − (cid:66) X )( C ) ⊆ C since C is an ideal and H (cid:66) C ⊆ C . This proves that X t ( A ) is an H -equivariantbraided symmetric A -sub-bimodule and a braided Lie subalgebra of X R ( A ). If X ∈ X ( A ) and Y ∈ X t ( A ) we obtain ( a · X )( A ) = a · X ( A ) ⊆ C ,( X · a )( A ) = X ( R − (cid:66) A ) · ( R − (cid:66) a ) ⊆ C , ( ξ (cid:66) X )( A ) = ξ (1) (cid:66) X ( S ( ξ (2) ) (cid:66) A ) ⊆ C and[ X, Y ] R ( A ) = X ( Y ( A )) − ( R − (cid:66) Y )(( R − (cid:66) X )( A )) ⊆ C for all a ∈ A and ξ ∈ H , since C is an ideal and H (cid:66) C ⊆ C . The projection respectsthe H -module action by definition, while it respects the left and right A -moduleactions since pr( a · X )(pr( b )) =pr( a · X ( b ))=pr( a ) · pr( X ( b ))=(pr( a ) · pr( X ))(pr( b ))and pr( X · a )(pr( b )) =pr(( X · a )( b ))=pr( X ( R − (cid:66) b ) · ( R − (cid:66) a ))=pr( X ( R − (cid:66) b )) · pr(( R − (cid:66) a ))=pr( X )pr(( R − (cid:66) b )) · pr(( R − (cid:66) a ))=pr( X )( R − (cid:66) pr( b )) · ( R − (cid:66) pr( a ))=(pr( X ) · pr( a ))(pr( b ))16 Chapter 5hold. Finally,pr([ X, Y ] R )(pr( a )) =pr([ X, Y ] R ( a ))=pr( X ( Y ( a )) − ( R − (cid:66) Y )( R − (cid:66) X )( a ))=pr( X )(pr( Y ( a ))) − pr( R − (cid:66) Y )(pr(( R − (cid:66) X )( a )))=pr( X )(pr( Y )(pr( a ))) − ( R − (cid:66) pr( Y ))(( R − (cid:66) pr( X ))(pr( a )))=[pr( X ) , pr( Y )] R (pr( a ))proves that pr is a homomorphism of braided Lie algebras. By definition X ( A ) isthe kernel of the projection.In the light of Lemma 5.1.4 we are able to reformulate Definition 5.1.3 in thefollowing way. Definition 5.1.5. Let C ⊆ A be an algebra ideal. Then A / C is a submanifoldalgebra if there is a short exact sequence → X ( A ) → X t ( A ) pr −→ X R ( A / C ) → of H -equivariant braided symmetric A -bimodules and braided Lie algebras. In particular, the braided exterior algebra X • t ( A ) = Λ • X t ( A ) = A ⊕ X t ( A ) ⊕ ( X t ( A ) ∧ R X t ( A )) ⊕ · · · of X t ( A ) is an H -equivariant braided symmetric A -sub-bimodule of X •R ( A ) accord-ing to Proposition B.3. If we extend pr : X t ( A ) → X R ( A / C ) as a homomorphism ofthe braided wedge product, i.e.pr( X ∧ R Y ) = pr( X ) ∧ R pr( Y )for all X, Y ∈ X • t ( A ), we obtain the following result. Proposition 5.1.6. There is a short exact sequence → ker pr → X • t ( A ) pr −→ X •R ( A / C ) → of braided Gerstenhaber algebras. In particular, pr( (cid:74) X, Y (cid:75) R ) = (cid:74) pr( X ) , pr( Y ) (cid:75) R for all X, Y ∈ X • t ( A ) .Proof. This is a consequence of Lemma 5.1.4 since the braided multivector fieldsare generated in degree zero and one. To prove that the projection respects thebraided Schouten-Nijenhuis bracket one recalls the expression of (cid:74) · , · (cid:75) R on factorizingelements (see Section 4.2).For braided differential forms ω = a · d a ∧ R · · · ∧ R d a n ∈ Ω •R ( A ) one definespr( ω ) = pr( a )d(pr( a )) ∧ R · · · ∧ R d(pr( a n ))and as for braided multivector fields we define a left H -action on Ω •R ( A / C ) by ξ (cid:66) pr( ω ) = pr( ξ (cid:66) ω ) for all ξ ∈ H and ω ∈ Ω •R ( A ).ubmanifold Algebras 117 Proposition 5.1.7. There is a short exact sequence of differential graded algebras → ker pr → Ω •R ( A ) pr −→ Ω •R ( A / C ) → , where ker pr = (cid:76) k ≥ ker pr k is defined recursively by ker pr = C and ker pr k +1 = { ω ∈ Ω k +1 R ( A ) | i R X ω ∈ ker pr k for all X ∈ X t ( A ) } for k ≥ .Proof. Since every braided differential form can be written as a finite sum of elements a · d a ∧ R · · · ∧ R d a n , where a , . . . , a n and the projection pr : A → A / C is surjective,it follows that the above sequence is exact. By definition the projection commuteswith the de Rham differential.There are braided Cartan calculi on A and A / C with respect to R according toTheorem 4.4.2. In Proposition 5.1.6 and Proposition 5.1.7 we proved that the pro-jection respects the algebraic structure of X •R ( A ) and Ω •R ( A ). As a natural questionwe ask if also the geometric data of the braided Cartan calculus is intertwined bythe projection. A positive answer is given in the following theorem. Theorem 5.1.8. The braided Cartan calculus on A / C is the projection of the braidedCartan calculus on A . Namely, L R pr( X ) pr( ω ) =pr( L R X ω ) , i R pr( X ) pr( ω ) =pr(i R X ω ) , d(pr( ω )) =pr(d ω ) hold for X ∈ X • t ( A ) and ω ∈ Ω •R ( A ) .Proof. By definition, the projection respects the de Rham differential. In a next stepwe prove that this is also the case for the insertion. Let X ∈ X t ( A ) and ω ∈ Ω R ( A ).We can assume without loss of generality that ω = a d b for a, b ∈ A . Theni R pr( X ) pr( ω ) =i R pr( X ) (pr( a )d(pr( b )))=pr( R − (cid:66) a ) · pr( R − (cid:66) X )(pr( b ))=pr( R − (cid:66) a ) · pr(( R − (cid:66) X )( b ))=pr(( R − (cid:66) a ) · ( R − (cid:66) X )( b ))=pr(i R X ω )holds. Similarly one proves that the insertion of a braided vector field into a higherorder braided differential form is respected by the projection and since i R X ∧ R Y = i R X i R Y for another Y ∈ X t ( A ) the same is true for the insertion of braided multivector fieldsof any degree. Since the braided Lie derivative is the graded braided commutatorof the insertion and the de Rham differential it is also respected by the projection.Furthermore, according to Proposition 5.1.6 and Proposition 5.1.7, pr : X • t ( A ) → X •R ( A / C ) and pr : Ω •R ( A ) → Ω •R ( A / C ) are surjections. This concludes the proof ofthe theorem.18 Chapter 5As a special case we recover that the classical Cartan calculus on a closed embed-ded submanifold ι : N → M of a smooth manifold M is obtained by the pullback ι ∗ : Ω • ( M ) → Ω • ( N ) of differential forms and restriction ι ∗ : X • t ( M ) → X • ( N ) oftangent multivector fields to N . The latter is defined for any X ∈ X t ( M ) as theunique vector field X | N ∈ X ( N ), which is ι -related to X , i.e. T q ι ( X | N ) q = X ι ( q ) forall q ∈ N , where T q ι : T q N → T ι ( q ) M denotes the tangent map (c.f. [75] Lem. 5.39).In particular, L ι ∗ ( X ) ι ∗ ( ω ) = ι ∗ ( L X ω ) , i ι ∗ ( X ) ι ∗ ( ω ) = ι ∗ (i X ω ) and d ι ∗ ( ω ) = ι ∗ (d ω )for all X ∈ X • ( M ) and ω ∈ Ω • ( M ). Fix a submanifold ideal C of A and a strongly non-degenerate equivariant metric g on A in the following. There is a direct sum decomposition X R ( A ) = X t ( A ) ⊕ X n ( A ) , where X n ( A ) are the so-called braided normal vector fields with respect to C and g , defined to be the subspace orthogonal to X t ( A ) with respect to g . Namely, X ∈ X R ( A ) is an element of X n ( A ) if and only if g ( X, Y ) = 0 for all Y ∈ X t ( A ).Then we define pr g : X R ( A ) → X R ( A / C ) as the k -linear map which first projects tothe first subspace in the above decomposition and applies pr : X t ( A ) → X R ( A / C )afterwards. In particular pr g ( X ) = pr( X ) for all X ∈ X t ( A ). Lemma 5.2.1. The braided normal vector fields X n ( A ) are an H -equivariant braidedsymmetric A -sub-bimodule of X R ( A ) and pr g : X R ( A ) → X R ( A / C ) is a homomor-phism of H -equivariant braided symmetric A -bimodules.Proof. Corresponding the first claim, it is sufficient to prove that the module actionsare closed in X n ( A ). Let a, b ∈ A , ξ ∈ H and X ∈ X n ( A ). Then g ( a · X · b, Y ) = a · g ( X, R − (cid:66) Y ) · ( R − (cid:66) b ) = 0and g ( ξ (cid:66) X, Y ) = ξ (1) (cid:66) g ( X, S ( ξ (2) ) (cid:66) Y ) = 0follow for all Y ∈ X t ( A ), since X t ( A ) is an H -equivariant braided symmetric A -sub-bimodule and g is H -equivariant as well as left A -linear and braided right A -linearin the first argument. In particular, this implies that pr g respects the H -action and A -bimodule actions: the tangent and normal parts are closed under the actions andpr : X t ( A ) → X R ( A / C ) is a homomorphism of H -equivariant braided symmetric A -bimodules according to Proposition 5.1.6. This concludes the proof.Note that the definition of braided normal vector fields still makes sense for non-degenerate braided metrics and that also in this case they form an H -equivariantbraided symmetric A -sub-bimodule of X R ( A ). Nonetheless we stick to stronglyubmanifold Algebras 119non-degenerate braided metrics in this section, which is motivated in the followinglines. Recall that the vanishing vector fields X ( A ) were defined as the kernel of theprojection pr : X t ( A ) → X R ( A / C ). Consider the k -linear map g A / C : X R ( A / C ) ⊗ A / C X R ( A / C ) → A / C (5.5)which is determined by g A / C (pr g ( X ) , pr g ( Y )) = pr( g ( X, Y )) for all X, Y ∈ X R ( A ).It is well-defined if the following property holds. Axiom 1: for every X ∈ X ( A ) there are finitely many c i ∈ C and X i ∈ X t ( A )such that X = (cid:88) i c i · X i .This is for example the case if X ( A ) is finitely generated as a C -bimodule. Notethat every linear combination c · X with c ∈ C and X ∈ X t ( A ) defines a vanishingvector field. Moreover, since we assumed g to be strongly non-degenerate it followsthat g A / C is strongly non-degenerate as well, if the following property holds. Axiom 2: if X ∈ X t ( A ), then g ( X, X ) ∈ C implies X ∈ X ( A ).Note that non-degeneracy of g in combination with Axiom 2 is not sufficient toprove non-degeneracy of g A / C . We further would like to point out that in the caseof closed embedded smooth submanifolds both, Axiom 1 and Axiom 2 are satisfiedand that any Riemannian metric is strongly non-degenerate in our sense. Let usperform a rigorous proof of the above discussion. Lemma 5.2.2. Let C be a submanifold ideal of A and g a strongly non-degenerateequivariant metric on A such that Axiom 1 and Axiom 2 are satisfied. Then g A / C is awell-defined strongly non-degenerate equivariant metric on A / C and any equivariantcovariant derivative ∇ R : X R ( A ) ⊗ X R ( A ) → X R ( A ) on A projects to an equivariantcovariant derivative ∇ A / C pr( X ) pr( Y ) = pr g ( ∇ R X Y ) , on A / C with respect to R , where X, Y ∈ X t ( A ) .Proof. As already claimed, Axiom 1 assures g A / C to be well-defined, since g A / C (pr g ( X ) , pr g ( Y )) = g A / C (cid:18) pr g (cid:18) (cid:88) i c i · X i (cid:19) , pr g ( Y ) (cid:19) = pr (cid:18) g (cid:18) (cid:88) i c i · X i , Y (cid:19)(cid:19) =pr (cid:18) (cid:88) i c i · g ( X i , Y ) (cid:124) (cid:123)(cid:122) (cid:125) ∈C (cid:19) = 0and similarly g A / C (pr g ( Y ) , pr g ( X )) = 0 for all X ∈ X ( A ) and Y ∈ X R ( A ). Weprove that g A / C has the correct linearity and symmetry properties. Let a ∈ A , X, Y ∈ X R ( A ) and ξ ∈ H . Then g A / C (pr( a ) · pr g ( X ) , pr g ( Y )) = g A / C (pr g ( a · X ) , pr g ( Y ))=pr( g ( a · X, Y ))=pr( a · g ( X, Y ))20 Chapter 5=pr( a ) · pr( g ( X, Y ))=pr( a ) · g A / C (pr g ( X ) , pr g ( Y )) , g A / C (pr g ( X ) · pr( a ) , pr g ( Y )) = g A / C (pr g ( X · a ) , pr g ( Y ))=pr( g ( X · a, Y ))=pr( g ( X, a · Y ))= g A / C (pr g ( X ) , pr g ( a · Y ))= g A / C (pr g ( X ) , pr( a ) · pr g ( Y ))and ξ (cid:66) g A / C (pr g ( X ) , pr g ( Y )) = ξ (cid:66) pr( g ( X, Y ))=pr( ξ (cid:66) g ( X, Y ))=pr( g ( ξ (1) (cid:66) X, ξ (2) (cid:66) Y ))= g A / C (pr g ( ξ (1) (cid:66) X ) , pr g ( ξ (2) (cid:66) Y ))= g A / C ( ξ (1) (cid:66) pr g ( X ) , ξ (2) (cid:66) pr g ( Y ))hold, proving that g A / C is a k -linear map X R ( A / C ) ⊗ A / C X R ( A / C ) → A / C . It isbraided symmetric since g A / C (pr g ( Y ) , pr g ( X )) =pr( g ( Y, X ))=pr( g ( R − (cid:66) X, R − (cid:66) Y ))= g A / C (pr g ( R − (cid:66) X ) , pr g ( R − (cid:66) Y ))= g A / C ( R − (cid:66) pr g ( X ) , R − (cid:66) pr g ( Y ))for all X, Y ∈ X R ( A ). Using Axiom 2 we prove that g A / C is strongly non-degenerate.Let X ∈ X R ( A / C ) and choose Y ∈ X t ( A ) such that pr( Y ) = X . Then0 = g A / C ( X, X ) = pr( g ( Y, Y ))implies g ( Y, Y ) ∈ C , i.e. Y ∈ X ( A ) by Axiom 2. In other words g A / C ( X, X ) = 0 im-plies X = 0, which is equivalent to the statement that X (cid:54) = 0 implies g A / C ( X, X ) (cid:54) =0, i.e. strong non-degeneracy of g A / C . From Axiom 1 it follows that ∇ A / C is well-defined. In fact, for X = (cid:80) i c i · X i ∈ X ( A ) and Y ∈ X t ( A ) we obtain ∇ A / C pr( X ) pr( Y ) = pr g ( ∇ R X Y ) = pr g (cid:18) (cid:88) i c i · ∇ R X i Y (cid:124) (cid:123)(cid:122) (cid:125) ∈ X ( A ) (cid:19) = 0and ∇ A / C pr( Y ) pr( X ) =pr g (cid:18) (cid:88) i ∇ R Y ( c i · X i ) (cid:19) = (cid:88) i pr g ( ∈C (cid:122) (cid:125)(cid:124) (cid:123) ( L R Y c i ) · X i (cid:124) (cid:123)(cid:122) (cid:125) ∈ X ( A ) + ∈C (cid:122) (cid:125)(cid:124) (cid:123) ( R − (cid:66) c i ) ·∇ RR − (cid:66) X i Y (cid:124) (cid:123)(cid:122) (cid:125) ∈ X ( A ) )=0 , ubmanifold Algebras 121since ∇ R is left A -linear in the first argument and satisfies a braided Leibniz rulein the second argument. Clearly ∇ A / C is k -linear. It is left A / C -linear in the firstargument since ∇ A / C pr( a ) · pr( X ) pr( Y ) = ∇ A / C pr( a · X ) pr( Y ) = pr g ( ∇ R a · X Y ) = pr g ( a · ∇ R X Y )=pr( a ) · pr g ( ∇ R X Y ) = pr( a ) · ( ∇ A / C pr( X ) pr( Y ))and for all X, Y ∈ X t ( A ) and a ∈ A it satisfies a braided Leibniz rule ∇ A / C pr( X ) (pr( a ) · pr( Y )) = ∇ A / C pr( X ) (pr( a · Y ))=pr g ( ∇ R X ( a · Y ))=pr g (( L R X a ) · Y + ( R − (cid:66) a ) · ( ∇ RR − (cid:66) X Y ))=pr( L R X a ) · pr( Y ) + pr( R − (cid:66) a ) · pr g ( ∇ RR − (cid:66) X Y )=( L R pr( X ) pr( a )) · pr( Y ) + ( R − (cid:66) pr( a )) · ( ∇ A / CR − (cid:66) pr( X ) pr( Y ))in the second argument. This concludes the proof of the lemma.The curvature and torsion of a projected equivariant covariant derivative coincidewith the projections of curvature and torsion of the initial equivariant covariantderivative. This underlines how the concepts of braided commutative geometry onsubmanifold algebras A / C can be obtained from the ones on A . Corollary 5.2.3. Under the assumptions of Lemma 5.2.2, the curvature R ∇ A / C andthe torsion Tor ∇ A / C of the projected equivariant covariant derivative ∇ A / C are givenby R ∇ A / C (pr( X ) , pr( Y ))(pr( Z )) = pr g ( R ∇ R ( X, Y ) Z ) and Tor ∇ A / C (pr( X ) , pr( Y )) = pr g (Tor ∇ R ( X, Y )) for all X, Y, Z ∈ X t ( A ) . If furthermore, ∇ R is the equivariant Levi-Civita covariantderivative with respect to g , ∇ A / C is the equivariant Levi-Civita covariant derivativeon A / C with respect to g A / C .Proof. Let X, Y, Z ∈ X t ( A ). According to Lemma 5.2.2 ∇ A / C is an equivariantcovariant derivative on A / C with respect to R . In particular, its curvature andtorsion are well-defined and R ∇ A / C (pr( X ) , pr( Y ))(pr( Z )) = ∇ A / C pr( X ) ∇ A / C pr( Y ) pr( Z ) − ∇ A / CR − (cid:66) pr( Y ) ∇ A / CR − (cid:66) pr( X ) pr( Z ) − ∇ A / C [pr( X ) , pr( Y )] R pr( Z )= ∇ A / C pr( X ) ∇ A / C pr( Y ) pr( Z ) − ∇ A / C pr( R − (cid:66) Y ) ∇ A / C pr( R − (cid:66) X ) pr( Z ) − ∇ A / C pr([ X,Y ] R ) pr( Z )=pr g ( ∇ R X ∇ R Y Z − ∇ RR − (cid:66) Y ∇ RR − (cid:66) X Z − ∇ R [ X,Y ] R Z )=pr g ( R ∇ R ( X, Y ) Z )as well asTor ∇ A / C (pr( X ) , pr( Y )) = ∇ A / C pr( X ) pr( Y ) − ∇ A / CR − (cid:66) pr( Y ) ( R − (cid:66) pr( X ))22 Chapter 5 − [pr( X ) , pr( Y )] R =pr g ( ∇ R X Y − ∇ RR − (cid:66) Y ( R − (cid:66) X ) − [ X, Y ] R )=pr g (Tor ∇ R ( X, Y ))hold. If ∇ R is the Levi-Civita covariant derivative with respect to g , then L R X g ( Y, Z ) = g ( ∇ R X Y, Z ) + g ( R − (cid:66) Y, ∇ RR − (cid:66) X Z ) , where X, Y, Z ∈ X R ( A ), implies L R pr( X ) g A / C (pr( Y ) , pr( Z )) = L R pr( X ) pr( g ( Y, Z ))=pr( L R X g ( Y, Z ))=pr( g ( ∇ R X Y, Z ) + g ( R − (cid:66) Y, ∇ RR − (cid:66) X Z ))= g A / C (pr g ( ∇ R X Y ) , pr( Z ))+ g A / C (pr( R − (cid:66) Y ) , pr g ( ∇ RR − (cid:66) X Z ))= g A / C ( ∇ A / C pr( X ) pr( Y ) , pr( Z ))+ g A / C ( R − (cid:66) pr( Y ) , ∇ A / CR − (cid:66) pr( X ) pr( Z ))for all X, Y, Z ∈ X t ( A ). SinceTor ∇ A / C (pr( X ) , pr( Y )) = pr g (Tor ∇ R ( X, Y )) = pr g (0) = 0 ∇ A / C is the unique equivariant Levi-Civita covariant derivative on A / C with respectto g A / C . This concludes the proof.One extends the projection pr g : X •R ( A ) → X •R ( A / R ) to braided multivectorfields by defining it to coincide with pr on A and to be a homomorphism of thebraided wedge product on higher wedge powers. On braided differential forms weset pr g = pr. Corollary 5.2.4. Under the assumptions of Lemma 5.2.2, the equivariant covari-ant derivatives ∇ A / C : X R ( A / C ) ⊗ X •R ( A / C ) → X •R ( A / C ) and ˜ ∇ A / C : X R ( A / C ) ⊗ Ω •R ( A / C ) → Ω •R ( A / C ) , induced by the projected covariant derivative ∇ A / C on A / C are projected from the covariant derivatives induced by ∇ R . Namely, ∇ A / C pr( X ) pr( Y ) = pr g ( ∇ R X Y ) and ˜ ∇ A / C pr( X ) pr( ω ) = pr( ˜ ∇ R X ω ) (5.6) for all X ∈ X t ( A ) , Y ∈ X • t ( A ) and ω ∈ Ω •R ( A ) .Proof. Let X, Y ∈ X t ( A ) and ω ∈ Ω •R ( A ). Then (cid:104) ˜ ∇ A / C pr( X ) pr( ω ) , pr( Y ) (cid:105) R = L R pr( X ) (cid:104) pr( ω ) , pr( Y ) (cid:105) R − (cid:104)R − (cid:66) pr( ω ) , ∇ A / CR − (cid:66) pr( X ) pr( Y ) (cid:105) R = L R pr( X ) pr( (cid:104) ω, Y (cid:105) R ) − (cid:104) pr( R − (cid:66) ω ) , pr g ( ∇ RR − (cid:66) X Y ) (cid:105) R =pr( L R X (cid:104) ω, Y (cid:105) R − (cid:104)R − (cid:66) ω, ∇ RR − (cid:66) X Y (cid:105) R )=pr( (cid:104) ˜ ∇ R X ω, Y (cid:105) R )= (cid:104) pr( ˜ ∇ R X ω ) , pr( Y ) (cid:105) R ubmanifold Algebras 123implies ˜ ∇ A / C pr( X ) pr( ω ) = pr( ˜ ∇ R X ω ) by the non-degeneracy of (cid:104)· , ·(cid:105) R . Assume now thatwe proved (5.6) for all X ∈ X t ( A ), Y ∈ X kt ( A ) and ω ∈ Ω k R ( A ) for a fixed k > X, Y ∈ X t ( A ), Z ∈ X kt ( A ), ω ∈ Ω R ( A ) and η ∈ Ω k R ( A ). Then ∇ A / C pr( X ) (pr( Y ) ∧ R pr( Z )) =( ∇ A / C pr( X ) pr( Y )) ∧ R pr( Z )+ ( R − (cid:66) Y ) ∧ R ( ∇ A / CR − (cid:66) pr( X ) pr( Z ))=pr g (( ∇ R X Y ) ∧ R Z + ( R − (cid:66) Y ) ∧ R ( ∇ RR − (cid:66) X Z ))=pr g ( ∇ R X ( Y ∧ R Z ))and ˜ ∇ A / C pr( X ) (pr( ω ) ∧ R pr( η )) =( ˜ ∇ A / C pr( X ) pr( ω )) ∧ R pr( η )+ ( R − (cid:66) ω ) ∧ R ( ˜ ∇ A / CR − (cid:66) pr( X ) pr( η ))=pr g (( ˜ ∇ R X ω ) ∧ R η + ( R − (cid:66) ω ) ∧ R ( ˜ ∇ RR − (cid:66) X η ))=pr g ( ˜ ∇ R X ( ω ∧ R η ))prove that (5.6) even holds for all X ∈ X t ( A ), Y ∈ X k +1 t ( A ) and ω ∈ Ω k R ( A ). Byan inductive argument we conclude the proof of the corollary.As a special case, we recover the observation (see e.g. [69] Chap. VII Prop. 3.1)that the Levi-Civita covariant derivative on a Riemannian manifold projects to anyRiemannian submanifold. In the next theorem we prove that the gauge equivalence given by the Drinfel’dfunctor is compatible with the notion of submanifold ideals. In other words, theprojection to submanifold algebras and twisting commutes. In the particular caseof a cocommutative Hopf algebra with trivial triangular structure this means thattwist quantization and projection to the submanifold algebra commute. Theorem 5.3.1. Let C be a submanifold ideal of A . Then, for any twist F on H ,the projection of the twisted Gerstenhaber algebra ( X • t ( A ) F , ∧ F , (cid:74) · , · (cid:75) F ) of braidedmultivector fields on A which are tangent to A / C coincides with the twisted Ger-stenhaber algebra ( X •R ( A / C ) F , ∧ F , (cid:74) · , · (cid:75) F ) on A / C . Moreover, the twisted Cartancalculus on A / C is given by the projection of the twisted Cartan calculus on A .Namely, Ω •R ( A / C ) F = pr(Ω •R ( A ) F ) , L F pr( X ) pr( ω ) = pr( L F X ω ) , i F pr( X ) pr( ω ) = pr(i F X ω ) and d(pr( ω )) = pr(d ω ) for all X ∈ X • t ( A ) and ω ∈ Ω •R ( A ) .Proof. Note that the twisted multivector fields are a braided Gerstenhaber algebrasince the braided multivector fields which are tangent to A / C are an H -submodule24 Chapter 5and a braided symmetric A -sub-bimodule of X •R ( A ). We already noticed thatpr : X • t ( A ) → X • ( A / C ) is surjective. Let X, Y ∈ X • t ( A ) and a ∈ A . Thenpr( X ) ∧ F pr( Y ) = ( F − (cid:66) pr( X )) ∧ R ( F − (cid:66) pr( Y )) = pr( X ∧ F Y ) , and similarly (cid:74) pr( X ) , pr( Y ) (cid:75) F = pr( (cid:74) X, Y (cid:75) F ) and pr( a ) · F pr( X ) = pr( a · F X ) follow.Moreover, L F pr( X ) pr( ω ) = L RF − (cid:66) pr( X ) ( F − (cid:66) pr( ω )) = pr( L F X ω )and i F pr( X ) pr( ω ) = i RF − (cid:66) pr( X ) ( F − (cid:66) pr( ω )) = pr(i F X ω )for all X ∈ X • t ( A ) and ω ∈ Ω •R ( A ) by Theorem 5.1.8.In zero degree X t ( A ) = A . Thus, Theorem 5.3.1 implies that the twisted producton ( A / C ) F pr( a ) · F pr( b ) = pr( a · F b ) , (5.7)where a, b ∈ A , coincides with the projection of the twisted product on A F . Fur-thermore, twisted equivariant covariant derivatives behave well under projection.Fix a strongly non-degenerate equivariant metric g and a submanifold ideal C suchthat Axiom 1 and Axiom 2, defined in the previous section, are satisfied. Also fixan equivariant covariant derivative ∇ R on A . Proposition 5.3.2. For any twist F on H , the projection of the twisted covariantderivative coincides with the twist deformation of the projected equivariant covariantderivative, i.e. ( ∇ A / C ) F pr( X ) pr( Y ) = pr g ( ∇ F X Y ) for all X, Y ∈ X t ( A ) . Similar state-ments hold for the induced (twisted) equivariant covariant derivatives on braideddifferential forms and braided multivector fields.Proof. For all X, Y ∈ X t ( A ) one obtainspr g ( ∇ F X Y ) =pr g ( ∇ RF − (cid:66) X ( F − (cid:66) Y )) = ∇ A / C pr( F − (cid:66) X ) (pr( F − (cid:66) Y ))= ∇ A / CF − (cid:66) pr( X ) ( F − (cid:66) pr( Y )) = ( ∇ A / C ) F pr( X ) pr( Y )and similarly one proves the statements about the induced equivariant covariantderivatives.There are explicit formulas for the curvature and torsion of the twisted covariantderivative on a submanifold algebra in terms of the initial curvature and torsion. Corollary 5.3.3. For all X, Y, Z ∈ X t ( A ) R ( ∇ A / C ) F (pr( X ) , pr( Y ))(pr( Z ))= R ∇ A / C (cid:18) ( F − F (cid:48) − ) (cid:66) pr( X ) , ( F − F (cid:48) − ) (cid:66) pr( Y ) (cid:19) ( F − (cid:66) pr( Z ))=pr (cid:18) R ∇ R (cid:18) ( F − F (cid:48) − ) (cid:66) X, ( F − F (cid:48) − ) (cid:66) Y (cid:19) ( F − (cid:66) Z ) (cid:19) and Tor ( ∇ A / C ) F (pr( X ) , pr( Y )) =Tor ∇ A / C ( F − (cid:66) pr( X ) , F − (cid:66) pr( Y ))=pr (cid:18) Tor ∇ R ( F − (cid:66) X, F − (cid:66) Y ) (cid:19) hold. ubmanifold Algebras 125 Proof. Let X, Y, Z ∈ X t ( A ). Then R ( ∇ A / C ) F (pr( X ) , pr( Y ))(pr( Z ))=( ∇ A / C ) F pr( X ) ( ∇ A / C ) F pr( Y ) pr( Z ) − ( ∇ A / C ) FR − F (cid:66) pr( Y ) ( ∇ A / C ) FR − F (cid:66) pr( X ) pr( Z ) − ( ∇ A / C ) F [pr( X ) , pr( Y )] RF pr( Z )= ∇ A / CF − (cid:66) pr( X ) ∇ A / C ( F − F (cid:48)− ) (cid:66) pr( Y ) (( F − F (cid:48) − ) (cid:66) pr( Z )) − ∇ A / C ( F − R − F ) (cid:66) pr( Y ) ∇ A / C ( F − F (cid:48)− R − F ) (cid:66) pr( X ) (( F − F (cid:48) − ) (cid:66) pr( Z )) − ∇ A / C [ F − (cid:99) (1) (cid:66) pr( X ) , F − (cid:99) (2) (cid:66) pr( Y )] RF ( F − (cid:66) pr( Z ))= ∇ A / CF − (cid:66) pr( X ) ∇ A / C ( F − F (cid:48)− ) (cid:66) pr( Y ) (( F − F (cid:48) − ) (cid:66) pr( Z )) − ∇ A / C ( F − F (cid:48)− R − F ) (cid:66) pr( Y ) ∇ A / C ( F − F (cid:48)− R − F ) (cid:66) pr( X ) ( F − (cid:66) pr( Z )) − ∇ A / C [( F − F (cid:48)− ) (cid:66) pr( X ) , ( F − F (cid:48)− ) (cid:66) pr( Y )] R ( F − (cid:66) pr( Z ))= ∇ A / CF − (cid:66) pr( X ) ∇ A / C ( F − F (cid:48)− ) (cid:66) pr( Y ) (( F − F (cid:48) − ) (cid:66) pr( Z )) − ∇ A / C ( R − F − F (cid:48)− ) (cid:66) pr( Y ) ∇ A / C ( R − F F − F (cid:48)− ) (cid:66) pr( X ) ( F − (cid:66) pr( Z )) − ∇ A / C [( F − F (cid:48)− ) (cid:66) pr( X ) , ( F − F (cid:48)− ) (cid:66) pr( Y )] R ( F − (cid:66) pr( Z ))= R A / C (( F − F (cid:48) − ) (cid:66) pr( X ) , ( F − F (cid:48) − ) (cid:66) pr( Y ))( F − (cid:66) pr( Z ))andTor ( ∇ A / C ) F (pr( X ) , pr( Y )) =( ∇ A / C ) F pr( X ) pr( Y ) − ( ∇ A / C ) FR − F (cid:66) pr( Y ) ( R − F (cid:66) pr( X )) − [pr( X ) , pr( Y )] R F = ∇ A / CF − (cid:66) pr( X ) ( F − (cid:66) pr( Y )) − ∇ A / C ( F − R − F ) (cid:66) pr( Y ) (( F − R − F ) (cid:66) pr( X )) − [ F − (cid:66) pr( X ) , F − (cid:66) pr( Y )] R = ∇ A / CF − (cid:66) pr( X ) ( F − (cid:66) pr( Y )) − ∇ A / C ( R − F − ) (cid:66) pr( Y ) (( R − F − ) (cid:66) pr( X )) − [ F − (cid:66) pr( X ) , F − (cid:66) pr( Y )] R =Tor ∇ A / C ( F − (cid:66) pr( X ) , F − (cid:66) pr( Y ))follow, where we viewed ( ∇ A / C ) F as a covariant derivative with respect to R F (see Proposition 4.6.5) and we identified [ · , · ] R F with the twisted commutator as inProposition 4.6.2.This completes the discussion of the commutative diagram (5.1). We concludethe chapter with the study of an explicit example of twist deformation quantizationon a smooth submanifold.26 Chapter 5 The purpose of this section is to suggest an explicitly construction scheme for twistedCartan calculi and exemplifying this by elaborating one example. The strategy isto consider some classes of submanifolds of R D , find suitable symmetries whichinherit explicit Drinfel’d twists that also respect the submanifolds and project thetwisted Cartan calculi to the submanifolds. This should illustrate the utility ofthe abstract machinery we developed in the previous sections. The reason not toconsider a deformation of the submanifolds from the beginning but rather performinga detour, is that the Cartan calculus on R D is much easier to handle. Furthermore,the submanifolds are given in terms of relations in coordinates of R D . From ourprevious results we know that projection and twist deformation commute, so we areable to first quantize R D and pass to the submanifolds afterwards. Let us quicklyrecall the notion of symmetries for R D . Consider global coordinates x = ( x , . . . , x D )of R D together with the dual basis ( ∂ , . . . , ∂ D ) of vector fields with ∗ -involutions( x i ) ∗ = x i and ∂ ∗ i = − ∂ i . Assume that there is a U g -module ∗ -algebra action (cid:66) : U g ⊗ C ∞ ( R D ) → C ∞ ( R D ) for a complex Lie ∗ -algebra g . The induced U g -actions on X = X i ··· i k ∂ i ∧ . . . ∧ ∂ i k ∈ X k ( R D ) and ω = ω i ··· i k d x i ∧ . . . ∧ d x i k ∈ Ω k ( R D ) are ξ (cid:66) X = ( ξ (1) (cid:66) X i ··· i k )( ξ (2) (cid:66) ∂ i ) ∧ . . . ∧ ( ξ ( k +1) (cid:66) ∂ i k ) ∈ X k ( R D )and ξ (cid:66) ω = ( ξ (1) (cid:66) ω i ··· i k )(d( ξ (2) (cid:66) x i )) ∧ . . . ∧ (d( ξ (2) (cid:66) x i k )) ∈ Ω k ( R D )for all ξ ∈ U g , respectively, where ξ (cid:66) ∂ i ∈ X ( R D ) is defined by L ξ (cid:66) ∂ i f = ξ (1) (cid:66) (cid:0) L ∂ i ( S ( ξ (2) ) (cid:66) f ) (cid:1) ∈ C ∞ ( R D )for f ∈ C ∞ ( R D ). As symmetries g of R D , we choose a finite-dimensional Lie ∗ -subalgebra of X ( R D ) with module algebra action (cid:66) : U g ⊗ C ∞ ( R D ) → C ∞ ( R D ),given by the Lie derivative L . In fact, for all ξ, η ∈ U g one has L ξη f = L ξ ( L η f ), L f = f and L ξ ( f g ) = ( L ξ (1) f )( L ξ (2) g ) , L ξ (cid:15) ( ξ )1 for all f, g ∈ C ∞ ( R D ),which is easily verified on primitive elements. Now, let us turn to the submanifoldswe are interested in. Consider a smooth function F : R D → R having zero as aregular value. According to the regular value theorem (see e.g. [75] Cor. 5.24),the zero set N = F − ( { } ) is a closed embedded submanifold of dimension D − ι : N (cid:44) → R D and the corresponding vanishing idealof functions by C . As an additional assumption we suppose that N is closed underthe ∗ -involution. The surjective projection pr : Ω • ( R D ) → Ω • ( N ) is given by thepullback of ι . On functions it reads pr : C ∞ ( R D ) (cid:51) f (cid:55)→ f + C ∈ C ∞ ( N ). A vectorfield X ∈ X ( R D ) on R D is tangent to N if and only if its action on functions respectsthe vanishing ideal, i.e. if and only if L X C ⊆ C . As usual we write X ∈ X t ( R D )in this case. The Lie ∗ -algebra X t ( R D ) can be projected to X ( N ) by assigning toevery tangent vector field on X ∈ X t ( R D ) the unique ι -related vector field on N (c.f. [75] Lem. 5.39). Geometrically one might think of this ι -related vector fieldas the restriction of X to ι ( N ). On the other hand one might view this projectionas assigning to X its equivalence class consisting of all vector fields on R D thatcoincide with X up to vector fields X ∈ X ( R D ) satisfying L X C ∞ ( R D ) ⊆ C . Moreubmanifold Algebras 127generally, X • t ( R D ) is the Gerstenhaber algebra of multivector fields on R D which aretangent to N and pr : X • t ( R D ) → X • ( N ) denotes the surjective projection with kernel X • ( R D ). Since we are interested in quantizing the submanifold N we have to require g ⊆ X t ( R D ) ⊆ X ( R D ). In other words, we have to choose g such that L g C ⊆ C . If this is achieved, the extension to the universal enveloping algebra automaticallysatisfies L U g C ⊆ C , giving a well-defined U g -module ∗ -algebra action on C ∞ ( R D )that projects to a U g -module ∗ -algebra action on C ∞ ( N ). From now on D = 3.We are going to discuss a twist quantization of the 2-sheet elliptic hyperboloid,which is a quadric surface of R . The cases of the 1-sheet elliptic hyperboloid andthe elliptic cone are entirely similar and likewise all quadric surfaces of R admit atwist quantization (see [57]). Furthermore, we limit our consideration to the Cartancalculus and its twist deformation and only mention that functions, vector fields anddifferential forms on the submanifold are determined by relations that admit twistquantization, such that the latter controll the twisted objects. This point of view isimmersed in [57]. -Sheet Elliptic Hyperboloid Let a, c > N = f − EH ( { } ), where f EH ( x ) = 12 x x + a x ) + c (5.8)is said to be the 2-sheet elliptic hyperboloid in light-like coordinates. It is a closedembedded smooth submanifold of R according to the regular value theorem. It isobtained from the more common normal form f EC ( y ) = (( y ) + a ( y ) − ( y ) ) + c of the defining equation via the coordinate transformation x = y + y , x = y , x = y − y (5.9)from Cartesian coordinates ( y , y , y ). The three vector fields H =2 x ∂ − x ∂ ,E = 1 √ a x ∂ − √ ax ∂ ,E (cid:48) = 1 √ a x ∂ − √ ax ∂ of R satisfy [ H, E ] = 2 E, [ H, E (cid:48) ] = − E (cid:48) and [ E (cid:48) , E ] = H. They span the Lie ∗ -algebra g = so (2 , 1) and provide a basis of the tangent vectorfields X t ( R ), since H ( f ) = E ( f ) = E (cid:48) ( f ) = 0 . Then, according to Example 2.6.4 ii.) F = exp (cid:18) H ⊗ log(1 + i (cid:126) E ) (cid:19) ∈ ( U g ⊗ U g )[[ (cid:126) ]] (5.10)is a unitary Jordanian Drinfel’d twist and Theorem 5.3.1, also in the form of eq.(5.7),implies the following proposition.28 Chapter 5 Proposition 5.4.1. The twist star product f (cid:63) F g = ( F − (cid:66) f )( F − (cid:66) g ) , where f, g ∈ C ∞ ( R ) , induced by the unitary twist (5.10) projects to a twist star product on the -sheetelliptic hyperboloid N , i.e. f (cid:63) F g ∈ C ∞ ( N )[[ (cid:126) ]] for all f, g ∈ C ∞ ( N ) . Note that ( C ∞ ( N )[[ (cid:126) ]] , (cid:63) F ) is a ∗ -algebra with ∗ -involution f ∗ F = S ( β ) (cid:66) f (5.11) for all f ∈ C ∞ ( N ) , where β = F S ( F ) . Furthermore, we obtain twist deformations ( X • ( N ) F , ∧ F , (cid:74) · , · (cid:75) F ) and (Ω • ( N ) F , ∧ F ) as projections from R . There is an explicit description of the twist deformed Hopf ∗ -algebra structureof U g F (compare also to [60]). Using the commutation relations of H, E, E (cid:48) as wellas the series expansionslog(1 + i (cid:126) E ) = − ∞ (cid:88) n =1 ( − i (cid:126) E ) n n , (cid:126) E ) = ∞ (cid:88) n =1 n ( − i (cid:126) E ) n − 11 + i (cid:126) E = ∞ (cid:88) n =0 ( − i (cid:126) E ) n , (5.12)we prove some preliminary equations. Lemma 5.4.2. For all n ≥ E n =( H − n ) E n , (cid:18) H (cid:19) n E = E (cid:18) H (cid:19) n , E (cid:48) n =( H + 2 n ) E (cid:48) n , (cid:18) H (cid:19) n E (cid:48) = E (cid:48) (cid:18) H − (cid:19) n (5.13) hold. Furthermore we obtain log(1 + i (cid:126) E ) n H = H log(1 + i (cid:126) E ) n − (cid:126) n E (cid:126) E log(1 + i (cid:126) E ) n − (5.14) and log(1 + i νE ) n E (cid:48) = E (cid:48) log(1 + i (cid:126) E ) n − i (cid:126) nH log(1 + i (cid:126) E ) n − 11 + i (cid:126) E + n (cid:126) E (1 + i (cid:126) E ) log(1 + i (cid:126) E ) n − + (cid:126) n ( n − E (1 + i (cid:126) E ) log(1 + i (cid:126) E ) n − for all n ≥ .Proof. First remark that E n H = E n − ( H − E = ( H − n ) E n , ubmanifold Algebras 129where n ≥ 0, implieslog(1 + i (cid:126) E ) H = − ∞ (cid:88) n =1 ( − i (cid:126) ) n n E n H = − ∞ (cid:88) n =1 ( − i (cid:126) ) n n ( H − n ) E n = H log(1 + i (cid:126) E ) − (cid:126) E ∞ (cid:88) n =1 ( − i (cid:126) E ) n − = H log(1 + i (cid:126) E ) − (cid:126) E (cid:126) E . Inductively, this leads tolog(1 + i (cid:126) E ) n H = log(1 + i (cid:126) E ) n − (cid:18) H log(1 + i (cid:126) E ) − (cid:126) E (cid:126) E (cid:19) = H log(1 + i (cid:126) E ) n − (cid:126) n E (cid:126) E log(1 + i (cid:126) E ) n − for all n ≥ 0. Furthermore E n E (cid:48) = E n − ( E (cid:48) E − H )= E n − E (cid:48) E + (2( n − − H ) E n − = E (cid:48) E n + (2(( n − 1) + ( n − 2) + · · · + 1) − nH ) E n − = E (cid:48) E n + n ( n − E n − − nHE n − , where we employed the ”little Gauß” (cid:80) nn =1 n = n ( n +1)2 . Thenlog(1 + i (cid:126) E ) E (cid:48) = − ∞ (cid:88) n =1 ( − i (cid:126) ) n n E n E (cid:48) = − ∞ (cid:88) n =1 ( − i (cid:126) ) n n ( E (cid:48) E n + n ( n − E n − − nHE n − )= E (cid:48) log(1 + i (cid:126) E ) − i (cid:126) H 11 + i (cid:126) E + i (cid:126) ∞ (cid:88) n =1 ( − i (cid:126) ) n − ( n − E n − = E (cid:48) log(1 + i (cid:126) E ) − i (cid:126) H 11 + i (cid:126) E + (cid:126) E ∞ (cid:88) n =2 ( − i (cid:126) ) n − ( n − E n − = E (cid:48) log(1 + i (cid:126) E ) − i (cid:126) H 11 + i (cid:126) E + (cid:126) E (1 + i (cid:126) E ) and inductively, for n > (cid:126) E ) n E (cid:48) = log(1 + i (cid:126) E ) n − (cid:18) E (cid:48) log(1 + i (cid:126) E ) − i (cid:126) H 11 + i (cid:126) E + (cid:126) E (1 + i (cid:126) E ) (cid:19) = log(1 + i (cid:126) E ) n − E (cid:48) log(1 + i (cid:126) E )30 Chapter 5 − i (cid:126) log(1 + i (cid:126) E ) n − H 11 + i (cid:126) E + (cid:126) E (1 + i (cid:126) E ) log(1 + i (cid:126) E ) n − = log(1 + i (cid:126) E ) n − E (cid:48) log(1 + i (cid:126) E ) − i (cid:126) (cid:18) H log(1 + i (cid:126) E ) n − − (cid:126) ( n − E (cid:126) E log(1 + i (cid:126) E ) n − (cid:19) 11 + i (cid:126) E + (cid:126) E (1 + i (cid:126) E ) log(1 + i (cid:126) E ) n − = log(1 + i (cid:126) E ) n − E (cid:48) log(1 + i (cid:126) E ) − i (cid:126) H log(1 + i (cid:126) E ) n − 11 + i (cid:126) E + 2 (cid:126) ( n − E (1 + i (cid:126) E ) log(1 + i (cid:126) E ) n − + (cid:126) E (1 + i (cid:126) E ) log(1 + i (cid:126) E ) n − = E (cid:48) log(1 + i (cid:126) E ) n − i (cid:126) nH log(1 + i (cid:126) E ) n − 11 + i (cid:126) E + 2 (cid:126) (( n − 1) + ( n − 2) + · · · + 1) E (1 + i (cid:126) E ) log(1 + i (cid:126) E ) n − + n (cid:126) E (1 + i (cid:126) E ) log(1 + i (cid:126) E ) n − = E (cid:48) log(1 + i (cid:126) E ) n − i (cid:126) nH log(1 + i (cid:126) E ) n − 11 + i (cid:126) E + (cid:126) n ( n − E (1 + i (cid:126) E ) log(1 + i (cid:126) E ) n − + n (cid:126) E (1 + i (cid:126) E ) log(1 + i (cid:126) E ) n − . Lemma 5.4.3. The twisted coproduct and antipode of U g F are given by ∆ F ( H ) =∆( H ) − i (cid:126) (cid:18) H ⊗ E (cid:126) E (cid:19) , ∆ F ( E ) =∆( E ) + i (cid:126) E ⊗ E, ∆ F ( E (cid:48) ) =1 ⊗ E (cid:48) + E (cid:48) ⊗ 11 + i (cid:126) E − i (cid:126) (cid:18) H ⊗ H 11 + i (cid:126) E (cid:19) + (cid:126) (cid:18) H (cid:18) H (cid:19) ⊗ E (1 + i (cid:126) E ) (cid:19) and S F ( H ) = S ( H )(1 + i (cid:126) E ) , ubmanifold Algebras 131 S F ( E ) = S ( E )1 + i (cid:126) E ,S F ( E (cid:48) ) = S ( E (cid:48) )(1 + i (cid:126) E ) − i (cid:126) H + (cid:126) (cid:18) H − (cid:19) HE + i (cid:126) (cid:18) − H (cid:19) HE , respectively.Proof. Note that ( H ⊗ 1) commutes with F . However F (1 ⊗ H ) = ∞ (cid:88) n =0 n ! (cid:18) H (cid:19) n ⊗ log(1 + i (cid:126) E ) n H = ∞ (cid:88) n =0 n ! (cid:18) H (cid:19) n ⊗ (cid:18) H log(1 + i (cid:126) E ) n − (cid:126) n E (cid:126) E log(1 + i (cid:126) E ) n − (cid:19) =(1 ⊗ H ) F − i (cid:126) (cid:18) H ⊗ E (cid:126) E (cid:19) F only commutes up to the second term, proving∆ F ( H ) = ∆( H ) − i (cid:126) (cid:18) H ⊗ E (cid:126) E (cid:19) , since H is primitive, i.e. ∆( H ) = H ⊗ ⊗ H and ∆ F ( H ) = F ∆( H ) F − . Onthe other hand 1 ⊗ E commutes with F , while F ( E ⊗ 1) = ∞ (cid:88) n =0 n ! (cid:18) H (cid:19) n E ⊗ log(1 + i (cid:126) E ) n =( E ⊗ ∞ (cid:88) n =0 n ! (cid:18) H (cid:19) n ⊗ log(1 + i (cid:126) E ) n =( E ⊗ 1) exp (cid:18)(cid:18) H (cid:19) ⊗ log(1 + i (cid:126) E ) (cid:19) . Then∆ F ( E ) =1 ⊗ E + ( E ⊗ 1) exp (cid:18)(cid:18) H (cid:19) ⊗ log(1 + i (cid:126) E ) (cid:19) exp (cid:18) − H ⊗ log(1 + i (cid:126) E ) (cid:19) =1 ⊗ E + ( E ⊗ 1) exp (cid:18)(cid:18) H (cid:19) ⊗ log(1 + i (cid:126) E ) − H ⊗ log(1 + i (cid:126) E ) (cid:19) =1 ⊗ E + ( E ⊗ 1) exp(1 ⊗ log(1 + i (cid:126) E ))=∆( E ) + i (cid:126) E ⊗ E, where we used in the second equation that the exponents commute, leading to atrivial BCH series. Similarly to the last computation we obtain F ( E (cid:48) ⊗ 1) = ( E (cid:48) ⊗ 1) exp (cid:18)(cid:18) H − (cid:19) ⊗ log(1 + i (cid:126) E ) (cid:19) , which implies F ( E (cid:48) ⊗ F − = ( E (cid:48) ⊗ 1) exp( − ⊗ log(1 + i (cid:126) E )) = E (cid:48) ⊗ 11 + i (cid:126) E . 32 Chapter 5On the other hand F (1 ⊗ E (cid:48) ) = ∞ (cid:88) n =0 n ! (cid:18) H (cid:19) n ⊗ log(1 + i (cid:126) E ) n E (cid:48) = ∞ (cid:88) n =0 n ! (cid:18) H (cid:19) n ⊗ (cid:18) E (cid:48) log(1 + i (cid:126) E ) n − i (cid:126) nH log(1 + i (cid:126) E ) n − 11 + i (cid:126) E + n (cid:126) E (1 + i (cid:126) E ) log(1 + i (cid:126) E ) n − + (cid:126) n ( n − E (1 + i (cid:126) E ) log(1 + i (cid:126) E ) n − (cid:19) =(1 ⊗ E (cid:48) ) F − i (cid:126) (cid:18) H ⊗ H 11 + i (cid:126) E (cid:19) F + (cid:126) (cid:18) H ⊗ E (1 + i (cid:126) E ) (cid:19) F + (cid:126) (cid:18) H ⊗ E (1 + i (cid:126) E ) (cid:19) F =(1 ⊗ E (cid:48) ) F − i (cid:126) (cid:18) H ⊗ H 11 + i (cid:126) E (cid:19) F + (cid:126) (cid:18) H (cid:18) H (cid:19) ⊗ E (1 + i (cid:126) E ) (cid:19) F , leading to F (1 ⊗ E (cid:48) ) F − =(1 ⊗ E (cid:48) ) − i (cid:126) (cid:18) H ⊗ H 11 + i (cid:126) E (cid:19) + (cid:126) (cid:18) H (cid:18) H (cid:19) ⊗ E (1 + i (cid:126) E ) (cid:19) . Combining this with F ( E (cid:48) ⊗ F − we obtain the formula for the twisted coproductof E (cid:48) . For the twisted antipode S F note that ξ (cid:99) (1) S F ( ξ (cid:99) (2) ) = (cid:15) ( ξ )1 = S F ( ξ (cid:99) (1) ) ξ (cid:99) (2) forall ξ ∈ U g . Applying this to ξ = H gives0 = (cid:15) ( H )1 = S F ( H (cid:99) (1) ) H (cid:99) (2) = S F ( H ) + H − i (cid:126) S F ( H ) E (cid:126) E , implying S ( H ) = S F ( H ) (cid:18) − i (cid:126) E (cid:126) E (cid:19) = S F ( H ) 11 + i (cid:126) E . Since S ( H ) = − H and 1 + i (cid:126) E is the inverse of (cid:126) E this gives S F ( H ) = S ( H )(1 +i (cid:126) E ). Similarly, S F ( E ) = S ( E )1+i (cid:126) E follows. Finally, for the twisted antipode of E (cid:48) weobtain0 = (cid:15) ( E (cid:48) )1 = S F ( E (cid:48) (cid:99) (1) ) E (cid:48) (cid:99) (2) = E (cid:48) + S F ( E (cid:48) ) 11 + i (cid:126) E − i (cid:126) S F ( H ) H 11 + i (cid:126) E + (cid:126) S F (cid:18) H (cid:18) H (cid:19)(cid:19) E (1 + i (cid:126) E ) , which implies S F ( E (cid:48) ) = S ( E (cid:48) )(1 + i (cid:126) E ) + i (cid:126) S F ( H ) H − (cid:126) S F (cid:18) H (cid:18) H (cid:19)(cid:19) E (cid:126) E ubmanifold Algebras 133= S ( E (cid:48) )(1 + i (cid:126) E ) + i (cid:126) S ( H )(1 + i (cid:126) E ) H − (cid:126) (cid:18) S ( H )(1 + i (cid:126) E )2 + 1 (cid:19) S ( H )(1 + i (cid:126) E ) E (cid:126) E = S ( E (cid:48) )(1 + i (cid:126) E ) − i (cid:126) H (1 + i (cid:126) E ) − (cid:126) HE + (cid:126) (cid:18) S ( H )(1 + i (cid:126) E )2 + 1 (cid:19) HE = S ( E (cid:48) )(1 + i (cid:126) E ) − i (cid:126) H + (cid:126) H E − (cid:126) HE + (cid:126) HE + (cid:126) S ( H )(1 + i (cid:126) E ) HE = S ( E (cid:48) )(1 + i (cid:126) E ) − i (cid:126) H + (cid:126) H E − (cid:126) HE + (cid:126) HE − (cid:126) H (1 + i (cid:126) E ) E + i (cid:126) HE = S ( E (cid:48) )(1 + i (cid:126) E ) − i (cid:126) H + (cid:126) H E − (cid:126) HE + (cid:126) HE − (cid:126) H E − i (cid:126) H E + i (cid:126) HE = S ( E (cid:48) )(1 + i (cid:126) E ) − i (cid:126) H + (cid:126) (cid:18) H − (cid:19) HE + i (cid:126) (cid:18) − H (cid:19) HE . Since N is given in terms of relations of coordinate functions x , x , x , it is worthto study their twist deformation in detail. Note that H (cid:66) x i = λ i x i , where we define λ = 2 = − λ and λ = 0. Then( F − (cid:66) x i ) ⊗ ( F − (cid:66) x j ) = ∞ (cid:88) n =0 ( − n n ! (cid:18)(cid:18) H (cid:19) n (cid:66) x i (cid:19) ⊗ (log(1 + i (cid:126) E ) n (cid:66) x j )= ∞ (cid:88) n =0 ( − n n ! (cid:18) λ i (cid:19) n x i ⊗ (log(1 + i (cid:126) E ) n (cid:66) x j )= x i ⊗ ((1 + i (cid:126) E ) − λi (cid:66) x j )and ( x i ) ∗ F = S ( β ) (cid:66) ( x i ) ∗ = ( F S ( F )) (cid:66) x i = ∞ (cid:88) n =0 ( − n n ! (cid:18) log(1 + i (cid:126) E ) n (cid:18) H (cid:19) n (cid:19) (cid:66) x i =(1 + i (cid:126) E ) − λi (cid:66) x i follow. Since E (cid:66) x i = δ i √ a x − δ i √ ax , E (cid:66) x i = δ i x and E n (cid:66) x i = 0 for n > Lemma 5.4.4. The twisted star products of coordinate functions on N are x (cid:63) F x =( x ) ,x (cid:63) F x = x x − i (cid:126) √ a ( x ) ,x (cid:63) F x = x x + 2i (cid:126) √ ax x − (cid:126) ( x ) ,x (cid:63) F x i = x x i , for all ≤ i ≤ x (cid:63) F x = x x ,x (cid:63) F x = x x + i (cid:126) √ a x x ,x (cid:63) F x =( x ) − (cid:126) √ ax x . Furthermore, the twisted ∗ -involution on coordinate functions is given by ( x ) ∗ F = x , ( x ) ∗ F = x and ( x ) ∗ F = x − (cid:126) √ ax , The relation defining N becomes x (cid:63) F x + a x (cid:63) F x + c = 0 . (5.15)Similarly one calculates the twisted wedge product of coordinate vector fieldsand differential 1-forms and determines the submanifold condition in terms of thedeformed generators (see [57]). Instead we further examine the twisted insertion andLie derivative. Note that the de Rham differential is undeformed since it commuteswith the Hopf ∗ -algebra action. Lemma 5.4.5. One coordinate vector fields the twisted insertion and Lie derivativeread i F ∂ i ω =i ∂ i ((1 + i (cid:126) E ) λi (cid:66) ω ) , L F ∂ i ω = L ∂ i ((1 + i (cid:126) E ) λi (cid:66) ω ) , for all ω ∈ Ω • ( N ) . Using the left C ∞ ( N ) -linearity of i F and L F in the first argu-ment as well as i F X ∧ F Y = i F X i F Y and L F X ∧ F Y = i F X L F Y + ( − (cid:96) L F X i F Y for all X ∈ X • ( N ) and Y ∈ X (cid:96) ( N ) , these formulas can be used to determine the action of higher mul-tivector fields. As a last observation we consider the Minkowski metric g = 12 (d x ⊗ d x + d x ⊗ d x ) + d x ⊗ d x (5.16)in the case a = b = 1, i.e. for the circular -sheet elliptic hyperboloid . Note that(5.16) is in fact the Minkowski metric, however not in Cartesian coordinates butrather in the coordinates (5.9). Proposition 5.4.6. The Minkowski metric (5.16) on the circular -sheet elliptichyperboloid N is U g -equivariant and admits a twist quantization g F = 12 (d x ⊗ F d x + d x ⊗ F d x ) + d x ⊗ F d x + 2i (cid:126) √ a d x ⊗ F d x + (cid:126) d x ⊗ F d x (5.17)ubmanifold Algebras 135 The corresponding twisted Levi-Civita covariant derivative reads ∇ F E H = ∇ E H + 2i ν ∇ E E, ∇ F E (cid:48) H = ∇ E (cid:48) H − ν ∇ E (cid:48) E, ∇ F E E (cid:48) = ∇ E E (cid:48) + i ν ∇ E H − ν ∇ E E, ∇ F E (cid:48) E (cid:48) = ∇ E (cid:48) E (cid:48) − i ν ∇ E (cid:48) H (5.18) on generators of g , while we are not listing the combinations that keep undeformed.Using the left C ∞ ( N ) -linearity in the first argument and the braided Leibniz rulein the second argument these formulas determine the twisted Levi-Civita covariantderivative on higher order vector fields.Proof. Let X ∈ X t ( R ). Then( F − (cid:66) E ) ⊗ ( F − (cid:66) X ) = ∞ (cid:88) n =0 ( − n n ! (cid:18)(cid:18) H (cid:19) n (cid:66) E (cid:19) ⊗ (log(1 + i (cid:126) E ) n (cid:66) X )= ∞ (cid:88) n =0 ( − n n ! E ⊗ (log(1 + i (cid:126) E ) n (cid:66) X )= E ⊗ ((1 + i (cid:126) E ) − (cid:66) X )and similarly ( F − (cid:66) E (cid:48) ) ⊗ ( F − (cid:66) X ) = E (cid:48) ⊗ ((1 + i (cid:126) E ) (cid:66) X ) , ( F − (cid:66) H ) ⊗ ( F − (cid:66) X ) = H ⊗ X follow, implying (5.18). ppendix AMonoidal Categories This first appendix recalls some well-known concepts of category theory which areused without reference throughout the thesis. It is included for convenience of thereader and should be seen as a reference section rather than a self-contained chapter.After recalling the definitions of category, functor and natural transformation wecontinue by discussing their (braided) monoidal versions. We close this appendix byexplaining rigidity and some fundamental properties of rigid monoidal categories.As sources we refer to [68] Part Three, [77] Chap. 9 and [35] Chap. 5. For a generalintroduction to category theory not only focusing on quantum groups see [8].A category C consists of a class ob( C ) of objects and a class hom( C ) of morphisms .For every morphisms f of C there is a source object A and a target object B and wesay that f is a morphism from A to B , writing f : A → B . The class of morphismsfrom A to B is denoted by hom( A, B ). For three objects A , B and C of C werequire the morphisms hom( A, B ) and hom( B, C ) to be composable , i.e. for twomorphisms f : A → B and g : B → C there is a morphisms in hom( A, C ), denotedby g ◦ f . The composition is defined to be associative , which means that for threecomposable morphisms f, g, h one postulates ( h ◦ g ) ◦ f = h ◦ ( g ◦ f ). The lastaxiom says that for each object A of C there is an identity morphism id A : A → A ,satisfying f ◦ id A = f and id A ◦ g = g for any morphism f : A → B and g : B → A . One calls a morphism f : A → B an isomorphism if there exists a morphism f − : B → A such that f − ◦ f = id A and f ◦ f − = id B . The category C is said to be small if ob( C ) and hom( C ) are sets. If for a category C the morphisms hom( A, B )form a set for every pair of objects A, B it is said to be locally small . Prominentexamples are the categories Set with sets as objects and functions between sets asmorphisms, K Vec with objects being vector spaces over a fixed ground field K ofcharacteristic zero and linear maps as morphisms and k A with k -algebras as objectsand algebra homomorphisms as morphisms. The appropriate notion of morphismbetween categories C and D is given by a (covariant) functor F : C → D . It assignsto every object A of C an object F ( A ) in D and to any morphism f : A → B in C a morphism F ( f ) : F ( A ) → F ( B ) such that F ( g ◦ f ) = F ( g ) ◦ F ( f ) and F (id A ) = id F ( A ) for all morphisms f : A → B and g : B → C in C . A contravariantfunctor F : C → D reverses the order in the sense that F ( f ) : F ( B ) → F ( A ) for f : A → B . In this case F ( g ◦ f ) = F ( f ) ◦ F ( g ) has to be adapted in the definition.Functors can be composed in an obvious way and for every category there exists theidentity functor. This leads to the category of categories with categories as objectsand functors as morphisms. Two categories C and D are said to be isomorphic natural transformation Θ : F → G of two(contravariant) functors F, G : C → D to be a collection of morphisms in D of theform Θ A : F ( A ) → G ( A ) for every object A of C , such that for every morphism f : A → B in C one has Θ B ◦ F ( f ) = G ( f ) ◦ Θ A . If all Θ A are isomorphisms thefunctors F and G are said to be naturally equivalent . Furthermore, we say that twocategories C and D are naturally equivalent if there are two functors F : C → D and G : D → C such that G ◦ F and F ◦ G are naturally equivalent to the identityfunctors. To digest this bunch of definitions and absorb some of its nutrition weinvite the reader to consider Proposition 2.2.1 in Section 2.2.The notion of categories is extremely useful to organize concepts. However, manyof the examples we want to encounter inherit more structure. For this reason wereview the definition of a monoidal category . In a nutshell one mimics the prop-erties of the tensor product of vector spaces on a categorical level. Starting witha general category C we assume the existence of a functor ⊗ : C × C → C obeyingan associativity constraint . Namely we postulate the existence of an isomorphism α U,V,W : ( U ⊗ V ) ⊗ W → U ⊗ ( V ⊗ W ) for any triple U, V, W of objects in C , suchthat ( U ⊗ V ) ⊗ W U ⊗ ( V ⊗ W )( U (cid:48) ⊗ V (cid:48) ) ⊗ W (cid:48) U (cid:48) ⊗ ( V (cid:48) ⊗ W (cid:48) ) α U,V,W ( f ⊗ g ) ⊗ h f ⊗ ( g ⊗ h ) α U (cid:48) ,V (cid:48) ,W (cid:48) commutes for all morphisms f : U → U (cid:48) , g : V → V (cid:48) and h : W → W (cid:48) in C . In otherwords, there is a natural equivalence α : ⊗ ◦ ( ⊗ × id) → ⊗ ◦ (id × ⊗ ) of functors C × → C . For a K -vector space V we know that K ⊗ V ∼ = V ∼ = V ⊗ K . This resultis generalized by the left and right unit constraint of an object I in C , which arenatural equivalences (cid:96) : ⊗ ◦ ( I × id) → id and r : ⊗ ◦ (id × I ) → id of functors C → C ,respectively. Definition A.1 (Monoidal Category) . A category C together with a functor ⊗ : C ×C → C satisfying an associativity constraint with respect to α and an object I of C satisfying a left and right unit constraint with respect to (cid:96) and r is said to be amonoidal category, if in addition the pentagon relation (( U ⊗ V ) ⊗ W ) ⊗ X ( U ⊗ V ) ⊗ ( W ⊗ X ) U ⊗ ( V ⊗ ( W ⊗ X ))( U ⊗ ( V ⊗ W )) ⊗ X U ⊗ (( V ⊗ W ) ⊗ X ) α U ⊗ V,W,X α U,V,W ⊗ id X α U,V,W ⊗ X α U,V ⊗ W,X id U ⊗ α V,W,X and the triangle relation ( V ⊗ I ) ⊗ W V ⊗ ( I ⊗ W ) V ⊗ W α V,I,W r V ⊗ id W id V ⊗ (cid:96) W hold, for all objects U, V, W, X in C . If in addition α, (cid:96) and r are identities thecategory C is called strict monoidal. 38 Appendix AIn this thesis we assume all monoidal categories to be strict monoidal, which canbe done without loss of generality according to [68] Sec. XI.5. As already mentionedas a motivating example, the category K Vec is monoidal with functor given by theusual tensor product of vector spaces and unit object K . Also the categories Setand k A of sets and associative unital algebras over a commutative unital ring k are monoidal with functor given by the Cartesian product and the tensor product ofalgebras, respectively. The latter coincides with the tensor product of k -modules butendows the tensor product of two algebras with the tensor product multiplication,i.e. ( a ⊗ x ) · ( b ⊗ y ) = ( ab ) ⊗ ( xy ) for all a, b ∈ A , x, y ∈ X defines an associative unitalproduct on the tensor product of two algebras A and X . However, the category A M of left A -modules for an algebra A is not monoidal in general. In fact, it is monoidalwith respect to the usual associativity and unit constraints of k -modules if and onlyif A is a bialgebra (see Proposition 2.2.2). Since categories are always consideredtogether with their morphisms, the next definition is relevant in this context. Definition A.2 (Monoidal Functor) . A functor F : C → D between two monoidalcategories ( C , ⊗ C , I C , α C , (cid:96) C , r C ) and ( D , ⊗ D , I D , α D , (cid:96) D , r D ) is said to be a monoidalfunctor if there is a natural transformation Ξ A,B : F ( A ) ⊗ D F ( B ) → F ( A ⊗ C B ) for any pair ( A, B ) of objects in C and a morphism φ : I D → F ( I C ) such that ( F ( A ) ⊗ D F ( B )) ⊗ D F ( C ) F ( A ) ⊗ D ( F ( B ) ⊗ D F ( C )) F ( A ⊗ C B ) ⊗ D F ( C ) F ( A ) ⊗ D F ( B ⊗ C C ) F (( A ⊗ C B ) ⊗ C C ) F ( A ⊗ C ( B ⊗ C C )) α D F ( A ) ,F ( B ) ,F ( C ) Ξ A,B ⊗ D id F ( C ) id F ( A ) ⊗ D Ξ B,C Ξ A ⊗C B,C Ξ A,B ⊗C C F ( α C A,B,C ) ,F ( A ) ⊗ D I D F ( A ) ⊗ D F ( I C ) F ( A ) F ( A ⊗ C I C ) id ⊗ D φr D Ξ A,I C F ( r C ) and I D ⊗ D F ( A ) F ( I C ) ⊗ D F ( A ) F ( A ) F ( I C ⊗ C A ) φ ⊗ D id (cid:96) D Ξ I C ,A F ( (cid:96) C ) commute as diagrams in D . If Ξ and φ are isomorphisms in D , the monoidal functor F is said to be strong monoidal. If Ξ and φ are even identities in D , F is said tobe strict monoidal. It is clear that natural transformations of monoidal functors should respect theunderlying monoidal structure in addition. Definition A.3 (Monoidal Equivalence) . A natural transformation Θ : F → F (cid:48) between monoidal functors ( F, Ξ , φ ) and ( F (cid:48) , Ξ (cid:48) , φ (cid:48) ) between monoidal categories C and D is said to be a monoidal natural transformation if I D F ( I C ) F (cid:48) ( I C ) φ φ (cid:48) Θ I C and F ( A ) ⊗ D F ( B ) F ( A ⊗ C B ) F (cid:48) ( A ) ⊗ D F (cid:48) ( B ) F (cid:48) ( A ⊗ C B ) Ξ A,B Θ A ⊗ D Θ B Θ A ⊗C B Ξ (cid:48) A,B onoidal Categories 139 commute for all pairs ( A, B ) of objects in C as diagrams in D . If Θ is a naturalisomorphism in addition, it is said to be a monoidal natural isomorphism. Finally,two monoidal categories C and D are called monoidally equivalent if there existtwo monoidal functors F : C → D and F (cid:48) : D → C together with monoidal naturalisomorphisms F (cid:48) ◦ F → id C and F ◦ F (cid:48) → id D .Commutativity constraints of a monoidal category ( C , ⊗ , I, α, (cid:96), r ) are naturalisomorphisms β : ⊗ → ⊗ ◦ τ of functors C × C → C , where τ : C × C → C × C denotesthe flip functor τ ( A, B ) = ( B, A ) for any pair ( A, B ) of objects in C . Demandingcompatibility with the associativity constraint we end up with the definition of abraiding. Definition A.4 (Braided Monoidal Category) . A monoidal category ( C , ⊗ , I, α, (cid:96), r ) is said to be braided monoidal if there is a commutativity constraint β satisfying thehexagon relations ( A ⊗ B ) ⊗ C A ⊗ ( B ⊗ C ) ( B ⊗ C ) ⊗ A ( B ⊗ A ) ⊗ C B ⊗ ( A ⊗ C ) B ⊗ ( C ⊗ A ) α A,B,C β A,B ⊗ id C β A,B ⊗ C α B,C,A α B,A,C id B ⊗ β A,C and A ⊗ ( B ⊗ C ) ( A ⊗ B ) ⊗ C C ⊗ ( A ⊗ B ) A ⊗ ( C ⊗ B ) ( A ⊗ C ) ⊗ B ( C ⊗ A ) ⊗ B α − A,B,C id A ⊗ β B,C β A ⊗ B,C α − C,A,B α − A,C,B β A,C ⊗ id B , which are commutative diagrams in C . If β B,A ◦ β A,B = id A ⊗ B the braided monoidalcategory is said to be symmetric. A monoidal functor F : C → D between braidedmonoidal categories is called braided monoidal functor if F ( A ) ⊗ D F ( B ) F ( A ⊗ C B ) F ( B ) ⊗ D F ( A ) F ( B ⊗ C A ) Ξ A,B β D F ( A ) ,F ( B ) F ( β C A,B )Ξ B,A commutes in D for any pair ( A, B ) of objects in C . The commutativity constraint β of a braided monoidal category is also called braiding . It follows that the braiding respects the unit object, i.e. that A ⊗ I I ⊗ AA β A,I r A (cid:96) A commutes for every object A in the braided monoidal category. Another interest-ing class of monoidal categories is given by rigid ones. Their key feature is thatthey admit dual objects in a way which generalizes the following example: a finite-dimensional K -vector spaces V possesses a basis e , . . . , e n ∈ V and a corresponding dual basis e , . . . , e n ∈ V ∗ = Hom K ( V, K ), such that e i ( e j ) = δ ij and every element v ∈ V can be represented as v = (cid:80) ni =1 e i ( v ) e i .40 Appendix A Definition A.5 (Duality) . For a strict monoidal category ( C , ⊗ , I ) we say that anobject A ∗ of C is an left dual object of an object A of C if there are morphisms ev A : A ∗ ⊗ A → I and π A : I → A ⊗ A ∗ in C such that (id A ⊗ ev A ) ◦ ( π A ⊗ id A ) = id A and (ev A ⊗ id A ∗ ) ◦ (id A ∗ ⊗ π A ) = id A ∗ hold. If there are left duals for two objects A and B we can define the left transpose f ∗ : B ∗ → A ∗ of a morphism f : A → B by f ∗ = (ev B ⊗ id A ∗ ) ◦ (id B ∗ ⊗ f ⊗ id A ∗ ) ◦ (id B ∗ ⊗ π A ) . An object ∗ A is said to be a right dual of an object A if there are morphisms ev (cid:48) A : A ⊗ ∗ A → I and π (cid:48) A : I → ∗ A ⊗ A such that (ev (cid:48) A ⊗ id A ) ◦ (id A ⊗ π (cid:48) A ) = id A and (id ∗ A ⊗ ev (cid:48) A ) ◦ ( π (cid:48) A ⊗ id ∗ A ) = id ∗ A hold. If there are right duals ∗ A and ∗ B of two objects A and B , the right transpose ∗ f : ∗ B → ∗ A of a morphism f : A → B is defined by ∗ f = (id ∗ B ⊗ ev (cid:48) B ) ◦ (id ∗ B ⊗ f ⊗ id ∗ A ) ◦ ( π (cid:48) A ⊗ id ∗ A ) . If there is a left and a right dual for every object in C we call the strict monoidalcategory C rigid. In fact, rigid strict monoidal categories behave very much like the category K Vec f of finite-dimensional vector spaces. This is underlined in the next proposition, takenfrom [68] Prop. XIV.2.2. Proposition A.6. Let ( C , ⊗ , I ) be a rigid strict monoidal category.i.) For all morphisms f : A → B and g : B → C in C one has ( g ◦ f ) ∗ = f ∗ ◦ g ∗ and id ∗ A = id A ∗ . ii.) There are natural bijections Hom( A ⊗ B, C ) ∼ = Hom( A, C ⊗ B ∗ ) and Hom( A ∗ ⊗ B, C ) ∼ = Hom( B, A ⊗ C ) , where A, B, C are objects in C .iii.) There is an isomorphism ( A ⊗ B ) ∗ ∼ = B ∗ ⊗ A ∗ for any two objects A, B in C .Similar statements hold for the right transpose. Moreover, there are isomorphisms ∗ ( A ∗ ) ∼ = A ∼ = ( ∗ A ) ∗ . Note that in general ( A ∗ ) ∗ (cid:29) A and ∗ ( ∗ A ) (cid:29) A . ppendix BBraided Graßmann andGerstenhaber Algebras Let k be a commutative ring with unit 1. A graded module over k is a direct sum V • = (cid:76) k ∈ Z V k of k -modules. Note that V • itself is a k -module with respect tothe component-wise action. Remark that the notion of graded modules is usuallyutilized in the context of graded rings (see [26] Chap. II Sec. 11). For our purpose itis sufficient to restrict our consideration to usual rings which can be seen as gradedrings concentrated in degree zero. A map Φ : V • → W • between graded modules V • = (cid:76) k ∈ Z V k and W • = (cid:76) k ∈ Z W k is said to be homogeneous of degree k ∈ Z ifΦ( V (cid:96) ) ⊆ W k + (cid:96) . We often write Φ : V • → W • + k in this case. As an example, considerthe Graßmann algebra (Ω • ( M ) , ∧ ) of differential forms for a smooth manifold M .It is a graded R -module as well as a graded C ∞ ( M )-module and the de Rhamdifferential d : Ω • ( M ) → Ω • +1 ( M ) is a homogeneous map of degree 1. The gradedcommutator of two homogeneous maps Φ , Ψ : V • → V • of degree k and (cid:96) is definedby [Φ , Ψ] = Φ ◦ Ψ − ( − k(cid:96) Ψ ◦ Φ . In Section 4.3 braided differential forms (Ω •R ( A ) , ∧ R ) of a braided commutativealgebra A for a triangular Hopf algebra ( H, R ) are introduced as a generalization todifferential forms on a smooth manifold. In the following lines we give the generalframework for this space to fit in (see also [13]). It is the generalization of Graßmannalgebra in the category of H -equivariant braided symmetric A -bimodules. Remarkthat braided Lie algebras and their quantum analogues can be formulated in a moregeneral categorical setting (see e.g. [61, 78]).Fix a triangular Hopf algebra ( H, R ) and a braided commutative algebra A . Forany H -equivariant braided symmetric A -bimodule we are able to define the tensoralgebra T • M = (cid:77) k ∈ Z M k = A ⊕ M ⊕ ( M ⊗ A M ) ⊕ · · · with respect to the tensor product ⊗ A over A , where by definition M k = { } if k < M = A . The tensor algebra T • M is an associative unital algebra withrespect to the product given by the tensor product ⊗ A and the unit 1 ∈ A . Lemma B.1. The tensor algebra T • M of an H -equivariant braided symmetric A -bimodule is an H -equivariant braided symmetric A -bimodule with respect to the fol- lowing module actions, given on factorizing elements m ⊗ A · · · ⊗ A m k by ξ (cid:66) ( m ⊗ A · · · ⊗ A m k ) =( ξ (1) (cid:66) m ) ⊗ A · · · ⊗ A ( ξ ( k ) (cid:66) m k ) ,a · ( m ⊗ A · · · ⊗ A m k ) =( a · m ) ⊗ A · · · ⊗ A m k , ( m ⊗ A · · · ⊗ A m k ) · a = m ⊗ A · · · ⊗ A ( m k · a ) for all ξ ∈ H and a ∈ A , where k ≥ . It is an easy exercise to verify this lemma. Furthermore, there is an ideal I in(T • M , ⊗ A ), generated by elements m ⊗ A · · · ⊗ A m k ∈ T k M which equal m ⊗ A · · · ⊗ A m i − ⊗ A (cid:18) R (cid:48) − (cid:66) (cid:18) ( R − (cid:66) m j ) ⊗ A ( R − (cid:66) ( m i +1 ⊗ A · · · ⊗ A m j − )) (cid:19)(cid:19) ⊗ A ( R (cid:48) − (cid:66) m i ) ⊗ A m j +1 ⊗ A · · · ⊗ A m k for a pair ( i, j ) such that 1 ≤ i < j ≤ k . Lemma B.2. The left H -action and the left and right A -actions from Lemma B.1respect the ideal I .Proof. Let ξ ∈ H , a ∈ A and m ⊗ A · · · ⊗ A m k ∈ I for a k > 1. Then( ξ (1) (cid:66) m ) ⊗ A · · · ⊗ A ( ξ ( k ) (cid:66) m k ) = ξ (cid:66) ( m ⊗ A · · · ⊗ A m k )= ξ (cid:66) (cid:18) m ⊗ A · · · ⊗ A m i − ⊗ A (cid:18) R (cid:48) − (cid:66) (cid:18) ( R − (cid:66) m j ) ⊗ A ( R − (cid:66) ( m i +1 ⊗ A · · · ⊗ A m j − )) (cid:19)(cid:19) ⊗ A ( R (cid:48) − (cid:66) m i ) ⊗ A m j +1 ⊗ A · · · ⊗ A m k (cid:19) = ξ (1) (cid:66) (cid:18) m ⊗ A · · · ⊗ A m i − (cid:19) ⊗ A (cid:18) ( ξ (2) R (cid:48) − ) (cid:66) (cid:18) ( R − (cid:66) m j ) ⊗ A ( R − (cid:66) ( m i +1 ⊗ A · · · ⊗ A m j − )) (cid:19)(cid:19) ⊗ A (( ξ (3) R (cid:48) − ) (cid:66) m i ) ⊗ A ξ (4) (cid:66) (cid:18) m j +1 ⊗ A · · · ⊗ A m k (cid:19) = ξ (1) (cid:66) (cid:18) m ⊗ A · · · ⊗ A m i − (cid:19) ⊗ A (cid:18) R (cid:48) − (cid:66) (cid:18) (( ξ (3) R − ) (cid:66) m j ) ⊗ A (( ξ (4) R − ) (cid:66) ( m i +1 ⊗ A · · · ⊗ A m j − )) (cid:19)(cid:19) ⊗ A (( R (cid:48) − ξ (2) ) (cid:66) m i ) ⊗ A ξ (5) (cid:66) (cid:18) m j +1 ⊗ A · · · ⊗ A m k (cid:19) =( ξ (1) (cid:66) m ) ⊗ A · · · ⊗ A ( ξ ( i − (cid:66) m i − ) ⊗ A (cid:18) R (cid:48) − (cid:66) (cid:18) (( R − ξ ( j ) ) (cid:66) m j ) ⊗ A ( R − (cid:66) (( ξ ( i +1) (cid:66) m i +1 ) ⊗ A · · · ⊗ A ( ξ ( j − (cid:66) m j − ))) (cid:19)(cid:19) ⊗ A (( R (cid:48) − ξ ( i ) ) (cid:66) m i ) ⊗ A ( ξ ( j +1) (cid:66) m j +1 ) ⊗ A · · · ⊗ A ( ξ ( k ) (cid:66) m k )raided Graßmann and Gerstenhaber Algebras 143for a pair ( i, j ) such that 1 ≤ i < j ≤ k , which implies ξ (cid:66) I ⊆ I . The left A -moduleaction is only affected if i = 1. In this case( a · m ) ⊗ A m ⊗ A · · · ⊗ A m k = a · ( m ⊗ A · · · ⊗ A m k )= a · (cid:18)(cid:18) R (cid:48) − (cid:66) (cid:18) ( R − (cid:66) m j ) ⊗ A ( R − (cid:66) ( m ⊗ A · · · ⊗ A m j − )) (cid:19)(cid:19) ⊗ A ( R (cid:48) − (cid:66) m ) ⊗ A m j +1 ⊗ A · · · ⊗ A m k (cid:19) = (cid:18) a · (( R (cid:48) − R − ) (cid:66) m j ) ⊗ A (( R (cid:48) − R − ) (cid:66) ( m ⊗ A · · · ⊗ A m j − )) (cid:19) ⊗ A ( R (cid:48) − (cid:66) m ) ⊗ A m j +1 ⊗ A · · · ⊗ A m k = (cid:18) (( R (cid:48)(cid:48) − R (cid:48) − R − ) (cid:66) m j ) ⊗ A (( R (cid:48)(cid:48) − R (cid:48) − R − ) (cid:66) ( m ⊗ A · · · ⊗ A m j − )) (cid:19) ⊗ A (( R (cid:48)(cid:48) − (cid:66) a ) · ( R (cid:48) − (cid:66) m )) ⊗ A m j +1 ⊗ A · · · ⊗ A m k = (cid:18) (( R (cid:48) − R − ) (cid:66) m j ) ⊗ A (( R (cid:48) − R − ) (cid:66) ( m ⊗ A · · · ⊗ A m j − )) (cid:19) ⊗ A (( R (cid:48) − (cid:66) a ) · ( R (cid:48) − (cid:66) m )) ⊗ A m j +1 ⊗ A · · · ⊗ A m k = (cid:18) R (cid:48) − (cid:66) (cid:18) ( R − (cid:66) m j ) ⊗ A ( R − (cid:66) ( m ⊗ A · · · ⊗ A m j − )) (cid:19)(cid:19) ⊗ A ( R (cid:48) − (cid:66) ( a · m )) ⊗ A m j +1 ⊗ A · · · ⊗ A m k . On the other hand, if 1 ≤ i < j = km ⊗ A · · · ⊗ A m k − ⊗ A ( m k · a ) = ( m ⊗ A · · · ⊗ A m k ) · a = m ⊗ A · · · ⊗ A m i − ⊗ A (cid:18) R (cid:48) − (cid:66) (cid:18) ( R − (cid:66) m k ) ⊗ A ( R − (cid:66) ( m i +1 ⊗ A · · · ⊗ A m k − )) (cid:19)(cid:19) ⊗ A (( R (cid:48) − (cid:66) m i ) · a )= m ⊗ A · · · ⊗ A m i − ⊗ A (cid:18) (( R (cid:48) − R − ) (cid:66) m k ) · ( R (cid:48)(cid:48) − (cid:66) a ) ⊗ A (( R (cid:48)(cid:48) − R (cid:48) − R − ) (cid:66) ( m i +1 ⊗ A · · · ⊗ A m k − )) (cid:19) ⊗ A (( R (cid:48)(cid:48) − R (cid:48) − ) (cid:66) m i )= m ⊗ A · · · ⊗ A m i − ⊗ A (cid:18) (( R (cid:48) − R − ) (cid:66) m k ) · ( R (cid:48)(cid:48) − (cid:66) a ) ⊗ A (( R (cid:48)(cid:48) − R (cid:48)(cid:48)(cid:48) − R − ) (cid:66) ( m i +1 ⊗ A · · · ⊗ A m k − )) (cid:19) ⊗ A (( R (cid:48)(cid:48) − R (cid:48)(cid:48)(cid:48) − R (cid:48) − ) (cid:66) m i )= m ⊗ A · · · ⊗ A m i − ⊗ A (cid:18) ( R − (cid:66) m k ) · ( R (cid:48)(cid:48) − (cid:66) a ) ⊗ A (( R (cid:48)(cid:48)(cid:48) − R (cid:48)(cid:48) − R − ) (cid:66) ( m i +1 ⊗ A · · · ⊗ A m k − )) (cid:19) ⊗ A (( R (cid:48)(cid:48)(cid:48) − R (cid:48)(cid:48) − R − ) (cid:66) m i )= m ⊗ A · · · ⊗ A m i − ⊗ A (cid:18) ( R − (cid:66) ( m k · a ))44 Appendix B ⊗ A (( R (cid:48)(cid:48)(cid:48) − R − ) (cid:66) ( m i +1 ⊗ A · · · ⊗ A m k − )) (cid:19) ⊗ A (( R (cid:48)(cid:48)(cid:48) − R − ) (cid:66) m i )= m ⊗ A · · · ⊗ A m i − ⊗ A (cid:18) (( R (cid:48) − R − ) (cid:66) ( m k · a )) ⊗ A (( R (cid:48) − R (cid:48)(cid:48)(cid:48) − ) (cid:66) ( m i +1 ⊗ A · · · ⊗ A m k − )) (cid:19) ⊗ A (( R − R (cid:48)(cid:48)(cid:48) − ) (cid:66) m i )= m ⊗ A · · · ⊗ A m i − ⊗ A (cid:18) (( R − R (cid:48) − ) (cid:66) ( m k · a )) ⊗ A (( R − R (cid:48) − ) (cid:66) ( m i +1 ⊗ A · · · ⊗ A m k − )) (cid:19) ⊗ A ( R − (cid:66) m i )= m ⊗ A · · · ⊗ A m i − ⊗ A (cid:18) R (cid:48) − (cid:66) (cid:18) ( R − (cid:66) ( m k · a )) ⊗ A ( R − (cid:66) ( m i +1 ⊗ A · · · ⊗ A m k − )) (cid:19)(cid:19) ⊗ A ( R (cid:48) − (cid:66) m i )shows that the right A -action also respects the ideal. This concludes the proof ofthe lemma.By Lemma B.1 and Lemma B.2 we conclude the following statement. Proposition B.3. The quotient T • M /I is a well-defined associative unital gradedalgebra and an H -equivariant braided symmetric A -bimodule. We denote the quotient by Λ • M and the induced product by ∧ R . On factorizingelements m ∧ R · · · ∧ R m k ∈ Λ k M the induced module actions read ξ (cid:66) ( m ∧ R · · · ∧ R m k ) =( ξ (1) (cid:66) m ) ∧ R · · · ∧ R ( ξ ( k ) (cid:66) m k ) ,a · ( m ∧ R · · · ∧ R m k ) =( a · m ) ∧ R · · · ∧ R m k , ( m ∧ R · · · ∧ R m k ) · a = m ∧ R · · · ∧ R ( m k · a ) , where ξ ∈ H and a ∈ A . Note that the module actions respect the degree bydefinition. Definition B.4. The associative unital graded algebra and H -equivariant braidedsymmetric A -bimodule (Λ • M , ∧ R ) is said to be the braided Graßmann algebra orbraided exterior algebra of the H -equivariant braided symmetric A -bimodule M . We prove that the product of a braided Graßmann algebra inherits the braidedsymmetry from M . Lemma B.5. Let M be an H -equivariant braided symmetric A -bimodule. Thebraided wedge product ∧ R is graded braided commutative, i.e. Y ∧ R X = ( − k(cid:96) ( R − (cid:66) X ) ∧ R ( R − (cid:66) Y ) for all X ∈ Λ k M and Y ∈ Λ (cid:96) M .Proof. Consider two factorizing elements X = X ∧ R · · · ∧ R X k ∈ Λ k M and Y = Y ∧ R · · · ∧ R Y (cid:96) ∈ Λ (cid:96) M . First remark that X = − X ∧ R · · · ∧ R X i − ∧ R ( R − (cid:66) X i +1 ) ∧ R ( R − (cid:66) X i ) ∧ R X i +2 ∧ R · · · ∧ R X k , raided Graßmann and Gerstenhaber Algebras 145for any 1 ≤ i < k since X ⊗ A · · ·⊗ A X k + X ⊗ A · · ·⊗ A X i − ⊗ A ( R − (cid:66) X i +1 ) ⊗ A ( R − (cid:66) X i ) ⊗ A X i +2 ⊗ A · · ·⊗ A X k is an element of I . Then X ∧ R Y = X ∧ R · · · ∧ R X k ∧ R Y ∧ R · · · ∧ R Y (cid:96) =( − (cid:96) X ∧ R · · · ∧ R X k − ∧ R ( R − (cid:66) Y ) ∧ R · · · ∧ R ( R − (cid:96) ) (cid:66) Y (cid:96) ) ∧ R ( R − (cid:66) X k )=( − · (cid:96) X ∧ R · · · ∧ R X k − ∧ R (( R (cid:48) − R − ) (1) (cid:66) Y ) ∧ R · · ·∧ R (( R (cid:48) − R − ) ( (cid:96) ) (cid:66) Y (cid:96) ) ∧ R ( R (cid:48) − (cid:66) X k − ) ∧ R ( R − (cid:66) X k )=( − · (cid:96) X ∧ R · · · ∧ R X k − ∧ R ( R − (cid:66) Y ) ∧ R · · ·∧ R ( R − (cid:96) ) (cid:66) Y (cid:96) ) ∧ R ( R − (cid:66) ( X k − ∧ R X k ))= · · · =( − k · (cid:96) ( R − (cid:66) Y ) ∧ R ( R − (cid:66) X )follows.It remains to generalize the concept of Gerstenhaber algebra to our setting. Definition B.6. An associative unital graded algebra and H -equivariant braidedsymmetric A -bimodule ( G • , ∧ R ) is said to be a braided Gerstenhaber algebra if themodule actions respect the degree and if there is an H -equivariant graded (with degreeshifted by − ) braided Lie bracket (cid:74) · , · (cid:75) R : G k × G (cid:96) → G k + (cid:96) − , i.e. (cid:74) X, Y (cid:75) R = − ( − ( k − (cid:96) − (cid:74) ( R − (cid:66) Y ) , ( R − (cid:66) X ) (cid:75) R and (cid:74) X, (cid:74) Y, Z (cid:75) R (cid:75) R = (cid:74)(cid:74) X, Y (cid:75) R , Z (cid:75) R + ( − ( k − (cid:96) − (cid:74) ( R − (cid:66) Y ) , (cid:74) ( R − (cid:66) X ) , Z (cid:75) R (cid:75) R satisfying a graded braided Leibniz rule (cid:74) X, Y ∧ R Z (cid:75) R = (cid:74) X, Y (cid:75) R ∧ R Z + ( − ( k − (cid:96) ( R − (cid:66) Y ) ∧ R (cid:74) ( R − (cid:66) X ) , Z (cid:75) R with respect to ∧ R in addition. Above X ∈ G k , Y ∈ G (cid:96) and Z ∈ G • . Let G • be a braided Gerstenhaber algebra. 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