aa r X i v : . [ m a t h . QA ] F e b Braided Cartan Calculi and Submanifold Algebras
Thomas Weber ∗ Universit`a di Napoli “FEDERICO II” and I.N.F.N. Sezione di Napoli,Complesso MSA, Via Cintia, 80126 Napoli, ItalyJanuary 17, 2020
Abstract
We construct a noncommutative Cartan calculus on any braided commutative algebra andstudy its applications in noncommutative geometry. The braided Lie derivative, insertionand de Rham differential are introduced and related via graded braided commutators, alsoincorporating the braided Schouten-Nijenhuis bracket. The resulting braided Cartan calculusgeneralizes the Cartan calculus on smooth manifolds and the twisted Cartan calculus. While itis a necessity of derivation based Cartan calculi on noncommutative algebras to employ centralbimodules our approach allows to consider bimodules over the full underlying algebra. Fur-thermore, equivariant covariant derivatives and metrics on braided commutative algebras arediscussed. In particular, we prove the existence and uniqueness of an equivariant Levi-Civitacovariant derivative for any fixed non-degenerate equivariant metric. Operating in a symmet-ric braided monoidal category we argue that Drinfel’d twist deformation corresponds to gaugeequivalences of braided Cartan calculi. The notions of equivariant covariant derivative andmetric are compatible with the Drinfel’d functor as well. Moreover, we project braided Cartancalculi to submanifold algebras and prove that this process commutes with twist deformation.
Contents
In [42] Stanis law Lech Woronowicz generalized the notion of Cartan calculus to quantumgroups. The crucial ingredient is given by the de Rham differential, which is understood asa linear map d : H → Γ from a Hopf algebra H to a bicovariant H -bimodule Γ, generated by A and d, such that the Leibniz rule d( ab ) = (d a ) b + a d b holds for all a, b ∈ H . It is proventhat such a first order calculus admits an extension to the exterior algebra. Noncommutativecalculi based on derivations rather than generalizations of differential forms are discussed byMichel Dubois-Violette, Peter Michor and Peter Schupp in [18, 19, 36, 37], though differentialforms are included as dual objects to derivations. The latter approaches are suitable for general ∗ [email protected] oncommutative algebras in the setting of noncommutative geometry [13]. However, bimoduleshave to be considered over the center of the algebra. In these notes we are proposing anintermediate procedure, sticking to derivation based calculi while incorporating a Hopf algebrasymmetry to avoid central bimodules. It is motivated by twisted Cartan calculi, a particularclass of noncommutative Cartan calculi in the overlap of deformation quantization [8, 40] and quantum groups [21, 31]. Drinfel’d twists [16] are tools to deform Hopf algebras as well asthe representation theory of the Hopf algebra in a compatible way. They experienced a lot ofattention in the field of deformation quantization since a Drinfel’d twist induces a star productif the corresponding symmetry acts on a smooth manifold by derivations (c.f. [3]). Explicitexamples of star products are quite rare, so this connection was very desirable. However,this should be taken with a grain of salt since there are several situations [11, 15] in whichdeformation quantization can not be obtained via a twisting procedure. More generally, itwas pointed out in [5] that a Drinfel’d twist leads to a noncommutative calculus, the so-called twisted Cartan calculus . The mentioned article even provides twisted covariant derivativesand metrics, generalizing classical Riemannian geometry. The additional braided symmetriesappearing in this work were the main motivation for the author to consider noncommutativeCartan calculi only depending on a triangular structure rather than on the Drinfel’d twistitself. The appropriate categorical framework for this generalization is provided in [6, 7]: thecategory of equivariant braided symmetric bimodules with respect to a triangular Hopf algebraand a braided commutative algebra is symmetric braided and monoidal with respect to thetensor product over the algebra. Generalizing the algebraic construction of the Cartan calculusto this category we obtain the braided Cartan calculus . Vector fields are represented by thebraided Lie algebra of braided derivations, multivector fields become a braided Gerstenhaberalgebra, while differential forms constitute a braided Graßmann algebra. On the categoricallevel a Drinfel’d twist corresponds to a functor and its action can be understood as a gaugeequivalence on the symmetric braided monoidal category (see [4, 28]). We prove that this Drinfel’d functor respects the braided Cartan calculus in the sense that it intertwines thebraided Lie derivative, insertion, de Rham differential and Schouten-Nijenhuis bracket. Notethat both, the classical Cartan calculus and the twisted Cartan calculus, can be regarded asbraided Cartan calculi. The first one with respect to any cocommutative Hopf algebra withtrivial triangular structure and the latter with respect to the twisted Hopf algebra, triangularstructure and algebra. In the same spirit we generalize covariant derivatives and metrics to thebraided symmetric setting. Note however that for simplicity we regard them to be equivariantin addition, a requirement which excludes some interesting examples already in the twistedcase. However, this assumption assures compatibility with the Drinfel’d functor. As yetanother application of the braided Cartan calculus we study the braided Cartan calculus onsubmanifold algebras and prove that they are projected from the ambient algebra in accordanceto gauge equivalences. It would be interesting to generalize the braided Cartan calculus to thesetting of [6], to Lie-Rinehart algebras [27] and furthermore to connect the braided Cartancalculus to Hochschild cohomology and the Cartan calculus introduced by Boris Tsygan (seee.g. [38, 39]).The paper is organized as follows: in Section 2 we recall basic properties of triangular Hopfalgebras and study the symmetric braided monoidal category of equivariant braided symmetricbimodules of a braided commutative algebra. The Drinfel’d functor leads to a braided monoidalequivalence of this category and the one corresponding to the twisted algebra and triangularHopf algebra. Our main result is developed in Section 3: we generalize the construction of theCartan calculus of a commutative algebra to braided commutative algebras by incorporatinga braided symmetry. Starting from the braided Lie algebra of braided derivations we build thebraided Gerstenhaber algebra of braided multivector fields. The braided Schouten-Nijenhuisbracket is obtained by extending the braided commutator. The dual braided exterior algebraconstitutes the braided differential forms. Then, the braided Lie derivative, insertion andde Rham differential are defined, resulting in the braided Cartan relations. In the specialcase of a commutative algebra we regain the commutation relations of the classical Cartancalculus. Connecting to Section 2 we introduce a twist deformation of the braided Cartancalculus and prove that it is isomorphic to the braided Cartan calculus on the twisted algebracorresponding to the twisted triangular structure. This shows that our construction respectsgauge equivalence classes. As an application, we introduce equivariant covariant derivativesand metrics, give several constructions like extending them to braided multivector fields anddifferential forms and proving the existence and uniqueness of an equivariant Levi-Civita co-variant derivative for every non-degenerate equivariant metric. The Drinfel’d functor respectsthe constructions. Finally in Section 4 we study braided Cartan calculi on submanifold alge-bras. We show how to project the algebraic structure and that this procedure commutes withtwist deformation. An explicit example, given by twist quantization of quadric surfaces of R , s elaborated in [25].Throughout these notes every module is considered over a commutative ring k . The cate-gory k M of k -modules is monoidal with respect to the tensor product ⊗ . If not stated otherwiseevery algebra is assumed to be unital and associative. A map Φ : V • → W • between gradedmodules V • = L k ∈ Z V k and W • = L k ∈ Z W k is said to be homogeneous of degree k ∈ Z ifΦ( V ℓ ) ⊆ W k + ℓ . We often write Φ : V • → W • + k in this case. The graded commutator of twohomogeneous maps Φ , Ψ : V • → V • of degree k and ℓ is defined by [Φ , Ψ] = Φ ◦ Ψ − ( − kℓ Ψ ◦ Φ. In this introductory section we recall the notion of triangular Hopf algebra together with itsbraided monoidal category of representations. Afterwards we show how to twist the algebraicstructure by a 2-cocycle and in which sense this induces an equivalence on the categorical level.In the last subsection we discuss equivariant algebra bimodules and their twist deformation.The previous braided monoidal equivalence can be refined to the bimodules which inherit abraided symmetry in addition if the algebra is braided commutative. For more details on(triangular) Hopf algebras we refer to the textbooks [12, 28, 31, 34]. The more experiencedreaders are recommended to [4, 6, 7, 26] for a prompt discussion of what is covered in thissection.
In a shortcut we introduce the category of algebras over a commutative ring k along withtheir representations. Dualizing the definition we obtain coalgebras, combining to the notionof bialgebra if the algebra and coalgebra structures respect each other. From the categoricalperspective bialgebras are those algebras whose category of representations is monoidal withrespect to the usual associativity and unit constraints. Integrating a braiding in this categoryinduces universal R -matrices on the bialgebra, while an additional antipode corresponds toa rigid (braided) monoidal category and accordingly to a (triangular) Hopf algebra on thealgebraic level.A k -algebra is a k -module A endowed with k -linear maps µ : A ⊗ A → A and η : k → A ,called product and unit of A , such that the identities µ ◦ ( µ ⊗ id) = µ ◦ (id ⊗ µ ) : A ⊗ → A (2.1)and µ ◦ ( η ⊗ id) = id = µ ◦ (id ⊗ η ) : A → A (2.2)hold, where we used the k -module isomorphisms k ⊗ A ∼ = A ∼ = A ⊗ k in eq.(2.2). These arethe well-known associativity and unit properties. A k -algebra A is said to be commutative if µ = µ , where µ : A⊗A ∋ ( a ⊗ b ) µ ( b ⊗ a ) ∈ A . In the following we often drop the symbol µ and simply write a · b or ab for the product of two elements a, b ∈ A . The k -algebras form acategory k A with morphisms being k -algebra homomorphisms , i.e. k -linear maps φ : A → A ′ between k -algebras ( A , µ, η ) and ( A ′ , µ ′ , η ′ ) such that φ ◦ µ = µ ′ ◦ ( φ ⊗ φ ) : A ⊗ A → A ′ and φ ◦ η = η ′ : k → A ′ . (2.3)Dualizing this concept we define a k -coalgebra to be a k -module C together with k -linear maps∆ : C → C ⊗ C and ǫ : C → k satisfying(∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆ : C → C ⊗ (2.4)and ( ǫ ⊗ id) ◦ ∆ = id = (id ⊗ ǫ ) ◦ ∆ : C → C . (2.5)The maps ∆ and ǫ are said to be the coproduct and counit of C with the properties of beingcoassociative and counital, respectively. We frequently use Sweedler’s sigma notation c (1) ⊗ c (2) to denote the coproduct ∆( c ) of an element c ∈ C , omitting a possibly finite sum of factorizingelements. By the coassociativity of ∆ we further define c (1) ⊗ c (2) ⊗ c (3) := c (1)(1) ⊗ c (1)(2) ⊗ c (2) = c (1) ⊗ c (2)(1) ⊗ c (2)(2) (2.6)and similarly for higher coproducts. A k -coalgebra C is said to be cocommutative if ∆ = ∆,where ∆ ( c ) = c (2) ⊗ c (1) for all c ∈ C . A k -coalgebra homomorphism is a k -linear map ψ : C → C ′ between k -coalgebras ( C , ∆ , ǫ ) and ( C ′ , ∆ ′ , ǫ ′ ) obeying the relations∆ ′ ◦ ψ = ( ψ ⊗ ψ ) ◦ ∆ : C → C ′ ⊗ C ′ and ǫ ′ ◦ ψ = ǫ : C → k . (2.7)The category of k -comodules is denoted by k C . xample 2.1. We give some elementary examples and constructions of (co)algebras, focusingon the ones we need in the rest of these notes.i.) The tensor product
A ⊗ A ′ of two k -algebras ( A , µ, η ) and ( A ′ , µ ′ , η ′ ) becomes a k -algebrawith product µ A⊗A ′ = ( µ ⊗ µ ′ ) ◦ (id ⊗ τ A ′ , A ⊗ id) : ( A ⊗ A ′ ) ⊗ ( A ⊗ A ′ ) → A ⊗ A ′ and unit η A⊗A ′ = η ⊗ η ′ , where we use the k -module isomorphism k ⊗ k ∼ = k in the latterdefinition and τ A ′ , A : A ′ ⊗ A → A ⊗ A ′ denotes the tensor flip isomorphism. Dually, thetensor product C ⊗ C ′ of two k -coalgebras ( C , ∆ , ǫ ) and ( C ′ , ∆ ′ , ǫ ′ ) can be structured as a k -coalgebra with coproduct ∆ C⊗C ′ = (id ⊗ τ C , C ′ ⊗ id) ◦ (∆ ⊗ ∆ ′ ) : C ⊗ C ′ → ( C ⊗ C ′ ) ⊗ ( C ⊗ C ′ ) and counit ǫ C⊗C ′ = ǫ ⊗ ǫ ′ .ii.) Any commutative ring k is a k -(co)algebra with product and unit given by its ring multi-plication and unit element, while the coproduct and counit are defined by ∆( λ ) = λ (1 ⊗ and ǫ ( λ ) = λ for all λ ∈ k . A k -algebra ( A , µ, η ) which is also a k -coalgebra with coproduct ∆ and counit ǫ is saidto be a k -bialgebra if ∆ and ǫ are k -algebra homomorphisms and µ and η are k -coalgebrahomomorphisms. It is clear by the symmetry in the definition of algebra and coalgebra that a k -algebra and k -coalgebra is a k -bialgebra if and only if its algebra structures are k -coalgebrahomomorphisms if and only if its coalgebra structures are k -algebra homomorphisms. A k -bialgebra homomorphism is a k -algebra homomorphism between k -bialgebras which is also a k -coalgebra homomorphism. Definition 2.2. A k -bialgebra ( H, µ, η, ∆ , ǫ ) is said to be triangular if there is an invertibleelement R ∈ H ⊗ H , called universal R -matrix or triangular structure, with inverse given by R = τ H,H ( R ) , such that ∆ ( ξ ) = R ∆( ξ ) R − for all ξ ∈ H, (2.8) and the hexagon relations (∆ ⊗ id)( R ) = R R and (id ⊗ ∆)( R ) = R R (2.9) are satisfied, where R = R ⊗ , R = 1 ⊗ R , R = (id ⊗ τ H,H )( R ) ∈ H ⊗ . Property(2.8) states that H is quasi-cocommutative. The k -bialgebra H is said to be a k -Hopf algebraif there is a bijective k -linear map S : H → H , called antipode, such that µ ◦ ( S ⊗ id) ◦ ∆ = η ◦ ǫ = µ ◦ (id ⊗ S ) ◦ ∆ : H → H (2.10) holds. A k -bialgebra homomorphism between k -Hopf algebras is said to be a k -Hopf algebrahomomorphism if it intertwines the antipodes. We denote the category of k -Hopf algebras by k H . A k -Hopf algebra ( H, µ, η, ∆ , ǫ, S ) is called triangular if its underlying bialgebra structureis. In the following we often drop the reference to the commutative ring k and simply referto Hopf algebras etc. Remark that there are slightly weaker definitions of Hopf algebra, notassuming the antipode to have an inverse (see [28, 31, 34]). We follow the convention of [12],arguing that in all examples which are relevant for us the antipode is invertible and we do notwant to state this as an additional condition throughout. One can show that the antipode S ofa bialgebra ( H, µ, η, ∆ , ǫ ) is unique if it exists and that it is an anti-bialgebra homomorphismin the sense that S ( ξχ ) = S ( χ ) S ( ξ ) , S (1) = 1 , S ( ξ ) (1) ⊗ S ( ξ ) (2) = S ( ξ (2) ) ⊗ S ( ξ (1) ) and ǫ ◦ S = ǫ (2.11)for all ξ, χ ∈ H . If H is commutative or cocommutative it follows that S = id. Moreover,any cocommutative Hopf algebra is triangular with universal R -matrix given by R = 1 ⊗ R -matrix R satisfies the quantum Yang-Baxter equation R R R = R R R . Fix a triangular k -bialgebra ( H, µ, η, ∆ , ǫ, R ) for the moment. We motivate its definition byelaborating that the representation theory of H has interesting categorical properties. Recallthat a representation of H is nothing but a left H -module , i.e. a k -module M together with a k -linear map λ : H ⊗ M → M , called left H -module action or left H -module structure, suchthat λ ◦ (id H ⊗ λ ) = λ ◦ ( µ ⊗ id M ) : H ⊗ H ⊗ M → M (2.12) nd λ ◦ ( η ⊗ id M ) = id M hold. A left H -module homomorphism is a k -linear map Φ : M → M ′ between left H -modules ( M , λ ) and ( M ′ , λ ′ ) such thatΦ ◦ λ = λ ′ ◦ (id H ⊗ Φ) : H ⊗ M → M ′ . (2.13)We sometimes refer to left H -module homomorphisms as H -equivariant maps . This forms thecategory H M of left H -modules. In the following we often write ξ · m instead of λ ( ξ ⊗ m ) fora left H -module ( M , λ ), where ξ ∈ H and m ∈ M . Note that until now we only used thealgebra structure of H in the definition of H M . In other words, we can consider the categoryof representations for any algebra. However, since ∆ and ǫ are algebra homomorphisms wecan define a left H -module action on the tensor product of two left H -modules ( M , λ ) and( M ′ , λ ′ ) by λ M⊗M ′ = ( λ ⊗ λ ′ ) ◦ (id H ⊗ τ H, M ⊗ id M ′ ) ◦ (∆ ⊗ id M⊗M ′ ) : H ⊗ ( M ⊗ M ′ ) → M ⊗ M ′ and a left H -module action on k by λ k = ( ǫ ⊗ id k ) : H ⊗ k → k ⊗ k ∼ = k . Those actions respect the usual associativity and unit constraints of the tensor product of k -modules because ∆ is coassociative and ǫ satisfies the counit axiom. In other words, ( H M , ⊗ )is a monoidal category. The universal R -matrix R induces a symmetric braiding on thiscategory by defining c RM , M ′ ( m ⊗ m ′ ) = R − · ( m ′ ⊗ m ) ∈ M ′ ⊗ M for all m ∈ M , m ′ ∈ M ′ . (2.14)In fact, the hexagon relations of R correspond to the hexagon relations of c R and c RM , M ′ ◦ c RM ′ , M = id M ′ ⊗M since R is the inverse of R . Conversely, any symmetric braiding c on( H M , ⊗ ) determines a triangular structure R = τ H,H ( c H,H (1 ⊗ ∈ H ⊗ H , where H actson itself by left multiplication. Proposition 2.3 ([28] Proposition XIII.1.4.) . The representation theory H M of a k -bialgebrais a monoidal category. It is braided symmetric if and only if H is triangular. In the case of a Hopf algebra (
H, µ, η, ∆ , ǫ, S ) we receive an additional rigidity propertyof its monoidal category in the sense that every left H -module admits a left and right dualmodule. However, for this we have to restrict our consideration to finitely generated projective k -modules k M f . The antipode of H can be used to transfer the rigidity property from k M f to H M f . Denote the usual dual pairing of a finitely generated projective k -module M andits dual module M ∗ by h· , ·i : M ∗ ⊗ M → k . Proposition 2.4 ([12] Example 5.1.4) . Let H be a k -Hopf algebra and consider the monoidalcategory H M of left H -modules. The monoidal subcategory H M f of finitely generated pro-jective left H -modules is rigid, where the left and right dual M ∗ and ∗ M of an object M in H M f are defined as the finitely generated projective k -module M ∗ with left H -module actiongiven by h ξ · α, m i = h α, S ( ξ ) · m i and h ξ · α, m i = h α, S − ( ξ ) · m i for all ξ ∈ H , m ∈ M and α ∈ M ∗ , respectively. The forgetful functor F : H M f → k M f (2.15) is monoidal. In this subsection we introduce Drinfel’d twists as an instrument to deform (triangular) Hopfalgebra structures. It turns out that the representation theory of the deformed (triangular)Hopf algebra is (braided) monoidally equivalent the representation theory of the undeformed(triangular) Hopf algebra. The definition of Drinfel’d twist originates from [16], while themonoidal equivalence was proven in [17]. We further refer to [4, 26] for a discussion of thistopic. Fix a Hopf algebra (
H, µ, η, ∆ , ǫ, S ) in the following. Definition 2.5.
A (Drinfel’d) twist on H is an invertible element F ∈ H ⊗ H satisfying the -cocycle condition ( F ⊗ ⊗ id)( F ) = (1 ⊗ F )(id ⊗ ∆)( F ) (2.16) and the normalization condition ( ǫ ⊗ id)( F ) = 1 = (id ⊗ ǫ )( F ) . here are several examples and constructions of Drinfel’d twists, showing that this is arich concept. We refer the interested reader to [20, 33]. It follows that the inverse F − of atwist F on H is normalized , i.e. ( ǫ ⊗ id)( F − ) = 1 = (id ⊗ ǫ )( F − ) and satisfies the so called inverse -cocycle condition (∆ ⊗ id)( F − )( F − ⊗
1) = (id ⊗ ∆)( F − )(1 ⊗ F − ) . (2.17)Any element F ∈ H ⊗ H can be written as a finite sum of factorizing elements F i ⊗ F i , F i , F i ∈ H . In the following we usually omit this finite sum and simply write F = F ⊗ F ,which is called leg notation . Using this convention, the 2-cocycle (2.16) condition reads F F ′ ⊗ F F ′ ⊗ F ′ = F ′ ⊗ F F ′ ⊗ F F ′ , (2.18)where we marked the second copy of F by F = F ′ ⊗ F ′ to distinguish the summations. Thefollowing proposition (c.f. [31] Theorem 2.3.4) reveals the utility of Drinfel’d twists as theyprovide a construction of (triangular) Hopf algebras from given ones. Proposition 2.6.
Consider a twist F on H . Then H F = ( H, µ, η, ∆ F , ǫ, S F ) is a Hopf algebrawith coproduct and antipode given by ∆ F ( ξ ) = F ∆( ξ ) F − and S F ( ξ ) = βS ( ξ ) β − , (2.19) respectively, for all ξ ∈ H , where β = F S ( F ) ∈ H . If H is triangular with universal R -matrix R , so is H F with universal R -matrix R F = F RF − . Let F be a twist on H and consider the corresponding monoidal category ( H F M , ⊗ F ) ofrepresentations of H F . Since H and H F coincide as algebras every left H -module is automati-cally a left H F -module and vice versa. However, the actions on the tensor product of modulesdiffer in general. For this reason we denote the monoidal structure of H F M by ⊗ F . Namely,for two left H -modules (or equivalently two left H F -modules) M and M ′ the tensor product M ⊗ F M ′ coincides with M ⊗ M ′ as a k -module but M ⊗ F M ′ is a left H F -module via ξ · ( m ⊗ F m ′ ) = ( ξ c (1) · m ) ⊗ F ( ξ c (2) · m ′ ) , (2.20)where ∆ F ( ξ ) = ξ c (1) ⊗ ξ c (2) , while M ⊗ M ′ is a left H F -module via ξ · ( m ⊗ m ′ ) = ( ξ (1) · m ) ⊗ ( ξ (2) · m ′ )for all ξ ∈ H F , m ∈ M and m ′ ∈ M ′ . We are able to compare those pictures via a left H F -module isomorphism ϕ M , M ′ : M ⊗ F M ′ ∋ ( m ⊗ F m ′ ) ( F − · m ) ⊗ ( F − · m ′ ) ∈ M ⊗ M ′ . (2.21)In fact, ϕ M , M ′ intertwines the left H F -module actions, since ϕ M , M ′ ( ξ · ( m ⊗ F m ′ )) = (( ξ (1) F − ) · m ) ⊗ (( ξ (2) F − ) · m ) = ξ · ϕ M , M ′ ( m ⊗ F m ′ )for all ξ ∈ H , m ∈ M and m ′ ∈ M ′ and admits an inverse left H F -module homomorphism ϕ − M , M ′ : M ⊗ M ′ ∋ ( m ⊗ m ′ ) ( F · m ) ⊗ F ( F · m ′ ) ∈ M ⊗ F M ′ . The map ϕ gives rise to a monoidal equivalence. We formulate this in the following theorem(c.f. [28] Lemma XV.3.7.). Theorem 2.7.
For any twist F on H there is a monoidal equivalence of the monoidal cate-gories ( H M , ⊗ ) and ( H F M , ⊗ F ) . If H is triangular we obtain a braided monoidal equivalencebetween braided monoidal categories ( H M , ⊗ , c R ) and ( H F M , ⊗ F , c R F ) . In the light of this theorem Drinfel’d twists are sometimes referred to as gauge transfor-mations or gauge equivalences (see e.g. [28] Section XV.3). This nomenclature is affirmed bythe observation that 1 ⊗ ∈ H ⊗ H is a Drinfel’d twist on any Hopf algebra H and if F isa twist on H and F ′ a twist on H F , the product F ′ F is a Drinfel’d twist on H such that H F ′ F = ( H F ) F ′ . .3 Equivariant Hopf Algebra Module Algebra Representations For some applications the monoidal equivalence H M ∼ = H F M of Theorem 2.7 is too arbitrary.Motivated from differential geometry we want to study equivariant module algebra bimodulesinstead, which generalize equivariant vector bundles. However, the restriction of the monoidalequivalence to those bimodules fails to be braided in general. To fix this we have to restrictourselves to braided commutative algebras and equivariant braided symmetric algebra bimod-ules. Nonetheless, this setting is rich enough to allow for several interesting examples, e.g. thebraided multivector fields and differential forms of a braided commutative algebra, as we seein Sections 3.1.Fix a Hopf algebra ( H, µ, η, ∆ , ǫ, S ) and consider a left H -module ( A , λ ) which is an algebrawith product µ A and unit η A in addition. It is said to be a left H -module algebra if the moduleaction respects the algebra structure, i.e. if λ ◦ (id H ⊗ µ A ) = µ A ◦ ( λ ⊗ λ ) ◦ (id H ⊗ τ H, A ⊗ id A ) ◦ (∆ ⊗ id A⊗A ) : H ⊗ A ⊗ A → A and λ ◦ (id H ⊗ η A ) = η A ◦ ǫ : H → A hold. In the following we often write µ A ( a ⊗ b ) = a · b for a, b ∈ A and ξ ⊲ a for the module action of ξ ∈ H on a ∈ A . The units of A and H aresometimes denoted by 1 A and 1 H , respectively or simply by 1. In this notation the modulealgebra axioms read ξ ⊲ ( a · b ) = ( ξ (1) ⊲ a ) · ( ξ (2) ⊲ b ) and ξ ⊲ A = ǫ ( ξ )1 A (2.22)for all ξ ∈ H and a, b ∈ A . A left H -module algebra homomorphism is a left H -modulehomomorphism between left H -module algebras which is also an algebra homomorphism. Thecategory of left H -module algebras is denoted by H A . Lemma 2.8 ([4] Theorem 3.4) . Let F be a twist on H and consider a left H -module algebra ( A , · , A ) . Then A F = ( A , · F , A ) is a left H F -module algebra with respect to the same left H -module action, where a · F b = ( F − ⊲ a ) · ( F − ⊲ b ) (2.23) for all a, b ∈ A . Fix a left H -module algebra A in the following and consider the category A M of left A -modules. In order to compare it to the representation theory of the deformed algebra A F wehave to incorporate an additional action of the Hopf algebra H on the modules. To obtaininteresting results this action has to respect the A -module structure. Accordingly we considerthe subcategory H A M of H -equivariant left A -modules . The objects of H A M are left H -modules M , which are left A -modules in addition such that ξ ⊲ ( a · m ) = ( ξ (1) ⊲ a ) · ( ξ (2) ⊲ m ) (2.24)for all ξ ∈ H , a ∈ A and m ∈ M . Morphisms are left H -module homomorphisms between H -equivariant left A -modules which are also left A -module homomorphisms. Lemma 2.9.
Let F be a twist on H and A a left H -module algebra. Then there is a functor Drin F : H A M → H F A F M , (2.25) called Drinfel’d functor, which is the identity on morphisms and assigns to every H -equivariantleft A -module M the same left H -module but with left A F -module structure given by a · F m = ( F − ⊲ a ) · ( F − ⊲ m ) (2.26) for all a ∈ A and m ∈ M .Proof. In fact, the obtained k -module M F is an object in H F A F M , since( a · F b ) · F m = a · F ( b · F m ) and ξ ⊲ ( a · F m ) = ( ξ c (1) ⊲ a ) · F ( ξ c (2) ⊲ m )follow for all ξ ∈ H , a, b ∈ A and m ∈ M in complete analogy to Lemma 2.8. Furthermore, anymorphisms φ : M → M ′ in H A M is automatically a morphism in H F A F M , where left H F -linearityis trivially given and left A F -linearity follows since φ ( a · F m ) = φ (( F − ⊲ a ) · ( F − ⊲ m )) = ( F − ⊲ a ) · φ ( F − ⊲ m )=( F − ⊲ a ) · ( F − ⊲ φ ( m )) = a · F φ ( b )for all a ∈ A and m ∈ M . ne might ask if the monoidal equivalence of Theorem 2.7 restricts to H A M . However, H A M is not monoidal with respect to the usual tensor product of k -modules, since there is nocoproduct on A in general to distribute the left A -module action to the tensor factors. Toobtain a monoidal category we need two specifications: first we consider the subcategory of H -equivariant A -bimodules H A M A , i.e. there are commuting left and right A -actions whichare equivariant with respect to the left H -action. Secondly, we consider the tensor product ⊗ A over A , which is defined for two objects M and M ′ by the quotient M ⊗ M ′ /N M , M ′ , where N M , M = im( ρ M ⊗ id M ′ − id M ⊗ λ M ′ ) and λ M ′ and ρ M denote the left and right A -actions on M ′ and M , respectively. As a consequence one has( m · a ) ⊗ A m ′ = m ⊗ A ( a · m ′ ) (2.27)for all a ∈ A , m ∈ M and m ′ ∈ M ′ . Then M ⊗ A M ′ is an H -equivariant A -bimodule, withinduced left H -action and left and right A -actions given by a · ( m ⊗ A m ′ ) = ( a · m ) ⊗ A m ′ and ( m ⊗ A m ′ ) · a = m ⊗ A ( m ′ · a ) (2.28)for all a ∈ A , m ∈ M and m ′ ∈ M ′ . On morphisms φ : M → N and ψ : M ′ → N ′ of H A M A one defines ( φ ⊗ A ψ )( m ⊗ A m ′ ) = φ ( m ) ⊗ A ψ ( m ′ ) for all m ∈ M and m ′ ∈ M ′ . Proposition 2.10.
The tuple ( H A M A , ⊗ A ) is a monoidal category and for a twist F on H the monoidal equivalence of Theorem 2.7 descends to a monoidal equivalence of ( H A M A , ⊗ A ) and ( H F A F M A F , ⊗ A F ) . We refer to [6] Theorem 3.13 for a proof and more information. In contrast to Theorem 2.7we do not obtain a symmetric braided monoidal structure on H A M A if H is triangular ingeneral. The H -equivariant A -bimodules are still too arbitrary. One has to demand moresymmetry before. We do so by considering a braided commutative left H -module algebra A for a triangular Hopf algebra ( H, R ) instead of a general left A -module algebra. This meansthat b · a = ( R − ⊲ a ) · ( R − ⊲ b ) holds for all a, b ∈ A . On the level of A -bimoduleswe want to keep this symmetry: an H -equivariant braided symmetric A -bimodule M for abraided commutative left H -module algebra A is an H -equivariant A -bimodule such that m · a = ( R − ⊲ a ) · ( R − ⊲ m ) for all a ∈ A and m ∈ M . In other words, the left and right A -actions are related via the universal R -matrix, mirroring the braided commutativity of A .These bimodules form a category H A M RA with morphisms being the usual left H -linear and leftand right A -linear maps. A proof of the following statement can be found in [6] Theorem 5.21. Theorem 2.11. If H is triangular and A is braided commutative we obtain a braided monoidalequivalence ( H A M RA , ⊗ A , c R ) ∼ = ( H F A F M R F A F , ⊗ A F , c R F ) (2.29) between braided monoidal categories. We enter the main section of these notes with the aim to construct a noncommutative Cartancalculus for any braided commutative algebra. Since its development is entirely parallel to theclassical Cartan calculus on a commutative algebra, with basically no choices on the way, itfeels justified to call it the braided Cartan calculus on a fixed braided commutative algebra.Before proving this result we recall the notion of multivector fields and differential forms ona commutative algebra, also to indicate the naturality of the generalization. The correspond-ing Graßmann and Gerstenhaber structures are equivariant with respect to a cocommutativeHopf algebra if the commutative algebra is a Hopf algebra module algebra in addition. Morein general we give the definitions of braided Graßmann and Gerstenhaber algebra and providebraided multivector fields and differential forms on a braided commutative algebra as exam-ples. In the second subsection we introduce a differential on braided differential forms via abraided version of the Chevalley-Eilenberg formula. Remark that the differential is a gradedbraided derivation with respect to the braided wedge product, however, since it is equivari-ant, it resembles a graded (non-braided) derivation. Using graded braided commutators therelations between the braided Lie derivative, insertion and differential are generalizing andentirely mirror the commutation relations of the classical Cartan calculus. We end the secondsubsection by applying the gauge equivalence given by the Drinfel’d functor to the braidedCartan calculus and proving that the result is isomorphic to the braided Cartan calculus onthe twisted algebra with respect to the twisted triangular structure. Some ramifications of his gauge equivalence, in particular for the interpretation of the twisted Cartan calculus ona commutative algebra, are discussed. As an application of the braided Cartan calculus thethird and last subsection deals with equivariant covariant derivatives and metrics. The mainresults are the extension of an equivariant covariant derivative to braided multivector fieldsand differential forms and the existence of a unique equivariant Levi-Civita covariant deriva-tive for a fixed non-degenerate equivariant metric. We prove that the Drinfel’d functor iscompatible with all constructions. For the Cartan calculus on a commutative algebra A the two most important A -bimodulesare the multivector fields X • ( A ) and differential forms Ω • ( A ). They are graded and possessa Graßmann structure. If A is a left H -module algebra for a cocommutative Hopf algebra H , X • ( A ) and Ω • ( A ) are H -equivariant symmetric A -bimodules and the module actions respectthe grading. Let us briefly recall the construction of those modules and then generalize themto the category H A M RA for a triangular Hopf algebra ( H, R ) and a braided commutative left H -module algebra A .Fix a cocommutative Hopf algebra H and a commutative left H -module algebra A for themoment. The derivations Der( A ) of A are an H -equivariant symmetric A -bimodule with left H -action given by the adjoint action ( ξ ⊲ X )( a ) = ξ (1) ⊲ ( X ( S ( ξ (2) ) ⊲ a )) (3.1)and left and right A -module actions ( a · X )( b ) = a · X ( b ) = ( X · a )( b ), for all ξ ∈ H , X ∈ Der( A )and a ∈ A . In particular, the tensor algebraT • Der( A ) = A ⊕
Der( A ) ⊕ (Der( A ) ⊗ A Der( A )) ⊕ · · · of Der( A ) with respect to the tensor product ⊗ A over A is well-defined. It is an H -equivariantsymmetric A -bimodule with module actions defined on factorizing elements X ⊗ A · · ·⊗ A X k ∈ T k Der( A ) by ξ ⊲ ( X ⊗ A · · · ⊗ A X k ) =( ξ (1) ⊲ X ) ⊗ A · · · ⊗ A ( ξ ( k ) ⊲ X k ) ,a · ( X ⊗ A · · · ⊗ A X k ) =( a · X ) ⊗ A · · · ⊗ A X k , ( X ⊗ A · · · ⊗ A X k ) · a = X ⊗ A · · · ⊗ A ( X k · a ) (3.2)for all ξ ∈ H and a ∈ A . Furthermore, there is an ideal I in T • Der( A ) generated by elements X ⊗ A · · · ⊗ A X k ∈ T k Der( A ) such that X i = X j for a pair ( i, j ) such that 1 ≤ i < j ≤ k . Thequotient T • Der( A ) /I is the exterior algebra. It is the Graßmann algebra X • ( A ) of multivectorfields on A and the induced product, the wedge product, is denoted by ∧ . Since H is cocom-mutative and the A -actions symmetric, they respect the ideal I . Consequently, the inducedactions on X • ( A ) are well-defined, structuring the multivector fields as an H -equivariant sym-metric A -bimodule with the additional property that the module actions respect the grading.Moreover, the usual commutator of endomorphisms [ · , · ] is a Lie bracket for the derivationsof A . It extends uniquely to a Gerstenhaber bracket J · , · K on X • ( A ) by defining J a, b K = 0, J X, a K = X ( a ) for all a, b ∈ A , X ∈ Der( A ) and inductively declaring the graded Leibniz rule J X, Y ∧ Z K = J X, Y K ∧ Z + ( − ( k − ℓ Y ∧ J X, Z K (3.3)for all X ∈ X k ( A ), Y ∈ X ℓ ( A ) and Z ∈ X • ( A ). In detail this means that J · , · K : X k ( A ) × X ℓ ( A ) → X k + ℓ − ( A ) is a graded (with respect to the degree shifted by −
1) Lie bracket, i.e. itis graded skew-symmetric J Y, X K = − ( − ( k − ℓ − J X, Y K (3.4)and satisfies the graded Jacobi identity J X, J Y, Z KK = JJ X, Y K , Z K + ( − ( k − ℓ − J Y, J X, Z KK , (3.5)where X ∈ X k ( A ), Y ∈ X ℓ ( A ) and Z ∈ X • ( A ), such that the graded Leibniz rule (3.3) holdsin addition. Using the formula J X ∧ · · · ∧ X k , Y ∧ · · · ∧ Y ℓ K = k X i =1 ℓ X j =1 ( − i + j [ X i , Y j ] ∧ X ∧ · · · ∧ c X i ∧ · · · ∧ X k ∧ Y ∧ · · · ∧ c Y j ∧ · · · ∧ Y ℓ , (3.6) hich holds for all X , . . . , X k , Y , . . . , Y ℓ ∈ X ( A ), it is easy to prove that the Gerstenhaberbracket J · , · K is H -equivariant, i.e. that ξ ⊲ J X, Y K = J ξ (1) ⊲ X, ξ (2) ⊲ Y K (3.7)for all ξ ∈ H and X, Y ∈ X • ( A ). Note that c X i and c Y j means that X i and Y j are left out inthe wedge product of eq.(3.6).Differential forms on A are defined in the following way: consider Hom A (Der( A ) , A ), the k -module of k -linear and A -linear maps Der( A ) → A . It is an H -equivariant symmetric A -bimodule with respect to the adjoint H -action and ( a · ω )( X ) = a · ω ( X ) = ( ω · a )( X ) forall a ∈ A , ω ∈ Hom A (Der( A ) , A ) and X ∈ Der( A ). The corresponding exterior algebra isdenoted by Ω • ( A ). One can define a differential d of ω ∈ Ω k ( A ) via(d ω )( X , . . . , X k +1 ) = k +1 X i =1 ( − i +1 X i ( ω ( X , . . . , c X i , . . . , X k +1 ))+ X i The braided derivations Der R ( A ) are an H -equivariant braided symmetric A -bimodule. Furthermore, the braided commutator [ X, Y ] R = XY − ( R − ⊲ Y )( R − ⊲ X ) , (3.10) where X, Y ∈ Der R ( A ) , structures Der R ( A ) as a braided Lie algebra. The latter means that [ · , · ] R is braided skew-symmetric, i.e. [ Y, X ] R = − [ R − ⊲ X, R − ⊲ Y ] R and satisfies thebraided Jacobi identity, i.e. [ X, [ Y, Z ] R ] R = [[ X, Y ] R , Z ] R + [ R − ⊲ Y, [ R − ⊲ X, Z ] R ] R (3.11) for all X, Y, Z ∈ Der R ( A ) . This is an elementary consequence of the properties of the triangular structure. In a nextstep we want to generalize the construction of multivector fields of a commutative algebra(compare also to [10]). Since Der R ( A ) is an A -bimodule we can build the tensor algebraT • Der R ( A ) with respect to ⊗ A and with module actions on factorizing elements X ⊗ A · · · ⊗ A X k ∈ T k Der R ( A ) defined by ξ ⊲ ( X ⊗ A · · · ⊗ A X k ) =( ξ (1) ⊲ X ) ⊗ A · · · ⊗ A ( ξ ( k ) ⊲ X k ) ,a · ( X ⊗ A · · · ⊗ A X k ) =( a · X ) ⊗ A · · · ⊗ A X k , ( X ⊗ A · · · ⊗ A X k ) · a = X ⊗ A · · · ⊗ A ( X k · a ) (3.12)for all ξ ∈ H and a ∈ A . There is an ideal I in T • Der R ( A ) generated by elements X ⊗ A · · · ⊗ A X k ∈ T k Der R ( A ) which equal X ⊗ A · · · ⊗ A X i − ⊗ A (cid:18) R ′ − ⊲ (cid:18) ( R − ⊲ X j ) ⊗ A ( R − ⊲ ( X i +1 ⊗ A · · · ⊗ A X j − )) (cid:19)(cid:19) ⊗ A ( R ′ − ⊲ X i ) ⊗ A X j +1 ⊗ A · · · ⊗ A X k (3.13) or a pair ( i, j ) such that 1 ≤ i < j ≤ k . One can prove that the module actions (3.12) respect I (see [41]). This induces an H -equivariant graded associative braided commutative product ∧ R on the quotient, declaring the braided multivector fields ( X •R ( A ) , ∧ R ) on A . In general, the as-sociative unital graded algebra and H -equivariant braided symmetric A -bimodule (Λ • M , ∧ R )associated to an H -equivariant braided symmetric A -bimodule M in this way is said to be the braided Graßmann algebra or braided exterior algebra corresponding to M . Coming back tothe example of braided multivector fields we can use the braided commutator of vector fieldsto obtain additional structure on the braided Graßmann algebra. Namely, we are defining a k -bilinear operation J · , · K R : X k R ( A ) × X ℓ R ( A ) → X k + ℓ − R ( A ) in the following way. If a, b ∈ A we set J a, b K R = 0. For a ∈ A and a factorizing element X = X ∧ R · · · ∧ R X k ∈ X k R ( A ) where k > J X, a K R = k X i =1 ( − k − i X ∧ R · · · ∧ R X i − ∧ R ( X i ( R − ⊲ a )) ∧ R (cid:18) R − ⊲ (cid:18) X i +1 ∧ R · · · ∧ R X k (cid:19)(cid:19) (3.14)and J a, X K R = k X i =1 ( − i (cid:18) R − ⊲ (cid:18) X ∧ R · · · ∧ R X i − (cid:19)(cid:19) ∧ R (( R − ⊲ X i )( R − ⊲ a )) ∧ R X i +1 ∧ R · · · ∧ R X k . (3.15)Furthermore, on factorizing elements X = X ∧ R · · ·∧ R X k ∈ X k R ( A ) and Y = Y ∧ R · · ·∧ R Y ℓ ∈ X ℓ R ( A ), where k, ℓ > 0, we define J X, Y K R = k X i =1 ℓ X j =1 ( − i + j [ R − ⊲ X i , R ′ − ⊲ Y j ] R ∧ R (cid:18) R ′ − ⊲ (cid:18)(cid:18) R − ⊲ ( X ∧ R · · · ∧ R X i − ) (cid:19) ∧ R c X i ∧ R X i +1 ∧ R · · · ∧ R X k ∧ R Y ∧ R · · · ∧ R Y j − (cid:19)(cid:19) ∧ R c Y j ∧ R Y j +1 ∧ R · · · ∧ R Y ℓ , (3.16)where [ · , · ] R denotes the braided commutator and c X i and c Y j means that X i and Y j are omittedin above product. The operation J · , · K R is said to be the braided Schouten-Nijenhuis bracket . Proposition 3.2. The braided multivector fields ( X •R ( A ) , ∧ R , J · , · K R ) on A are an associativeunital graded algebra and an H -equivariant braided symmetric A -bimodule equipped with an H -equivariant graded (with degree shifted by − ) braided Lie bracket J · , · K R : X k R ( A ) ⊗ X ℓ R ( A ) → X k + ℓ − R ( A ) , which means that J · , · K R is graded braided skewsymmetric, i.e. J Y, X K R = − ( − ( k − · ( ℓ − J R − ⊲ X, R − ⊲ Y K R , (3.17) and satisfies the graded braided Jacobi identity J X, J Y, Z K R K R = JJ X, Y K R , Z K R + ( − ( k − · ( ℓ − J R − ⊲ Y, J R − ⊲ X, Z K R K R , (3.18) such that the graded braided Leibniz rule J X, Y ∧ R Z K R = J X, Y K R ∧ R Z + ( − ( k − · ℓ ( R − ⊲ Y ) ∧ R J R − ⊲ X, Z K R (3.19) holds in addition, where X ∈ X k R ( A ) , Y ∈ X ℓ R ( A ) and Z ∈ X •R ( A ) . More in general we make the following definition. Definition 3.3 (Braided Gerstenhaber algebra) . An associative unital graded algebra and H -equivariant braided symmetric A -bimodule ( G • , ∧ R ) is said to be a braided Gerstenhaberalgebra if the module actions respect the degree and if there is an H -equivariant graded (withdegree shifted by − ) braided Lie bracket satisfying a graded braided Leibniz rule with respectto ∧ R . Let G • be a braided Gerstenhaber algebra. It follows that G is a braided commutative left H -module algebra and G is a braided Lie algebra. Moreover, G is an H -equivariant braidedsymmetric G -bimodule and G k is an H -equivariant braided symmetric G -bimodule. Thismeans that for any X ∈ G we can define the braided Lie derivative L R X = J X, · K R : G k → G k which is a braided derivation, i.e. L R X ( Y ∧ R Z ) = L R X Y ∧ R Z + ( R − ⊲ Y ) ∧ R ( R − ⊲ L R X ) Z (3.20) or all X ∈ G and Y, Z ∈ G • . It furthermore satisfies L R [ X,Y ] R = L R X L R Y − L RR − ⊲ Y L RR − ⊲ X for all X, Y ∈ G . On the other hand one can start with a braided commutative left H -modulealgebra A and construct the braided Gerstenhaber algebra of its braided multivector fields,as discussed before. Note that the braided Schouten-Nijenhuis bracket J · , · K R is the uniquebraided Gerstenhaber bracket on ( X •R ( A ) , ∧ R ) such that J X, a K R = X ( a ) and J X, Y K R = [ X, Y ] R (3.21)hold for all a ∈ A and X, Y ∈ X R ( A ).Dually we consider k -linear maps ω : Der R ( A ) → A such that ω ( X · a ) = ω ( X ) · a for all X ∈ Der R ( A ) and a ∈ A and denote the corresponding k -module by Ω R ( A ). We structureΩ R ( A ) as a braided symmetric A -bimodule with left and right A -actions defined by( a · ω )( X ) = a · ω ( X ) and ( ω · a )( X ) = ω ( R − ⊲ X ) · ( R − ⊲ a ) , (3.22)respectively, and left H -action ( ξ ⊲ ω )( X ) = ξ (1) ⊲ ( ω ( S ( ξ (2) ) ⊲ X )) , the adjoint action, for all ξ ∈ H , a ∈ A , ω ∈ Ω R ( A ) and X ∈ Der R ( A ). It follows that ω ( a · X ) = ( R − ⊲ a ) · ( R − ⊲ ω )( X )and ξ ⊲ ( ω ( X )) = ( ξ (1) ⊲ ω )( ξ (2) ⊲ X ) for all ξ ∈ H , ω ∈ Ω R ( A ), a ∈ A and X ∈ Der R ( A ).There is an H -equivariant insertion i R : X R ( A ) ⊗ Ω R ( A ) → A , defined for any X ∈ Der R ( A )and ω ∈ Ω R ( A ) by i R X ω = ( R − ⊲ ω )( R − ⊲ X ) . In fact, ξ ⊲ (i R X ω ) = ξ ⊲ (( R − ⊲ ω )( R − ⊲ X )) = (( ξ (1) R − ) ⊲ ω )(( ξ (2) R − ) ⊲ X )=(( R − ξ (2) ) ⊲ ω )(( R − ξ (1) ) ⊲ X ) = i R ξ (1) ⊲ X ( ξ (2) ⊲ ω )for all ξ ∈ H , X ∈ Der R ( A ) and ω ∈ Ω R ( A ). It follows that the braided exterior algebraΩ •R ( A ) of Ω R ( A ) is an H -equivariant braided symmetric A -bimodule. In the following lineswe show that it is also compatible with the braided evaluation. For ω, η ∈ Ω R ( A ) we define a k -bilinear map ω ∧ R η : Der( A ) × Der( A ) → A by( ω ∧ R η )( X, Y ) = ( ω ( R − ⊲ X ))(( R − ⊲ η )( Y )) − ( ω ( R − ⊲ Y ))(( R − ⊲ η )( R − ⊲ X ))for all X, Y ∈ Der R ( A ). One proves that − ( ω ∧ R η )( R − ⊲ Y, R − ⊲ X ) = ( ω ∧ R η )( X, Y ) = − (( R − ⊲ η ) ∧ R ( R − ⊲ ω ))( X, Y )and that ( ω ∧ R η )( X, Y · a ) =(( ω ∧ R η )( X, Y )) · a, ( ω ∧ R η )( a · X, Y ) =( R − ⊲ a ) · (( R − ⊲ ( ω ∧ R η ))( X, Y )) ,ξ ⊲ (( ω ∧ R η )( X, Y )) =(( ξ (1) ⊲ ω ) ∧ R ( ξ (2) ⊲ η ))( ξ (3) ⊲ X, ξ (4) ⊲ Y ) (3.23)hold for all ξ ∈ H , ω, η ∈ Ω R ( A ), a ∈ A and X, Y ∈ Der R ( A ). The evaluations of the H -action and A -module actions read( ξ ⊲ ( ω ∧ R η ))( X, Y ) = ξ (1) ⊲ (( ω ∧ R η )( S ( ξ (3) ) ⊲ X, S ( ξ (2) ) ⊲ Y )) , ( a · ( ω ∧ R η ))( X, Y ) = a · (( ω ∧ R η )( X, Y )) , (( ω ∧ R η ) · a )( X, Y ) =(( ω ∧ R η )( R − ⊲ X, R − ⊲ Y )) · ( R − ⊲ a ) . (3.24)Inductively one defines the evaluation of higher wedge products. Explicitly, the evaluatedmodule actions on factorizing elements ω ∧ R . . . ∧ R ω k ∈ Ω k R ( A ) read( ξ ⊲ ( ω ∧ R . . . ∧ R ω k ))( X , . . . , X k )= ξ (1) ⊲ (( ω ∧ R . . . ∧ R ω k )( S ( ξ ( k +1) ) ⊲ X , . . . , S ( ξ (2) ) ⊲ X k )) , (3.25)( a · ( ω ∧ R . . . ∧ R ω k ))( X , . . . , X k ) = a · (( ω ∧ R . . . ∧ R ω k )( X , . . . , X k )) (3.26)and (( ω ∧ R . . . ∧ R ω k ) · a )( X , . . . , X k )=(( ω ∧ R . . . ∧ R ω k )( R − ⊲ X , . . . , R − k ) ⊲ X k )) · ( R − ⊲ a ) . (3.27)for all X , . . . , X k ∈ Der R ( A ), a ∈ A and ξ ∈ H . It is useful to further define the insertioni R X : Ω •R ( A ) → Ω •− R ( A ) of an element X ∈ Der R ( A ) into the last slot an element ω ∈ Ω k R ( A )by i R X ω = ( − k − ( R − ⊲ ω )( · , . . . , · , R − ⊲ X ) . (3.28)Inductively we set i R X ∧ R Y = i R X i R Y (3.29)for all X, Y ∈ X •R ( A ). emma 3.4. (Ω •R ( A ) , ∧ R ) is a graded braided commutative associative unital algebra and an H -equivariant braided symmetric A -bimodule. The insertion i R : X •R ( A ) ⊗ Ω •R ( A ) → Ω •R ( A ) (3.30) of braided multivector fields is H -equivariant such that i R X is a right A -linear and braided left A -linear homogeneous map of degree − k for all X ∈ X k R ( A ) . Furthermore, i R X is left A -linearand braided right A -linear in X . If k = 1 i R X is a graded braided derivation of degree − .Proof. Fix a, b ∈ A , X ∈ Der R ( A ), ξ ∈ H and ω ∈ Ω R ( A ). First of all, the left and right A and left H -module actions are well-defined on Ω R ( A ), since ( b · ω )( X · a ) = b · ( ω ( X · a )) =(( b · ω )( X )) · a ,( ω · b )( X · a ) = ω (( R − ⊲ X ) · ( R − ⊲ a )) · ( R − ⊲ b )= ω ( R − ⊲ X ) · (( R ′ − R − ) ⊲ b ) · (( R ′ − R − ) ⊲ a )= ω ( R − ⊲ X ) · ( R − ⊲ b ) · a =(( ω · b )( X )) · a and ( ξ ⊲ ω )( X · a ) = ξ (1) ⊲ ( ω (( S ( ξ (2) ) (1) ⊲ X ) · ( S ( ξ (2) ) (2) ⊲ a )))= ξ (1) ⊲ ( ω ( S ( ξ (3) ) ⊲ X ) · ( S ( ξ (2) ) ⊲ a ))=( ξ (1) ⊲ ω ( S ( ξ (4) ) ⊲ X )) · (( ξ (2) S ( ξ (3) )) ⊲ a )=(( ξ (1) ⊲ ω )(( ξ (2) S ( ξ (3) )) ⊲ X )) · a =(( ξ ⊲ ω )( X )) · a hold by the hexagon relations and the bialgebra anti-homomorphism properties of S . The A -bimodule is H -equivariant, since( ξ ⊲ ( a · ω · b ))( X ) = ξ (1) ⊲ (( a · ω · b )( S ( ξ (2) ) ⊲ X ))=( ξ (1) ⊲ a ) · ( ξ (2) ⊲ ( ω (( R − S ( ξ (4) )) ⊲ X ))) · (( ξ (3) R − ) ⊲ b )=( ξ (1) ⊲ a ) · (( ξ (2) ⊲ ω )(( ξ (3) R − S ( ξ (5) )) ⊲ X )) · (( ξ (4) R − ) ⊲ b )=( ξ (1) ⊲ a ) · (( ξ (2) ⊲ ω )( R − ⊲ X )) · (( R − ξ (3) ) ⊲ b )=(( ξ (1) ⊲ a ) · ( ξ (2) ⊲ ω ) · ( ξ (3) ⊲ b ))( X )and it is braided symmetric because(( R − ⊲ ω ) · ( R − ⊲ a ))( X ) =(( R − ⊲ ω )( R ′ − ⊲ X )) · (( R ′ − R − ) ⊲ a )=(( R ′′ − R ′ − R − ) ⊲ a ) · ( R ′′ − ⊲ (( R − ⊲ ω )( R ′ − ⊲ X )))=( a · ω )( X ) . These properties extend to the braided Graßmann algebra Ω •R ( A ), giving an associative gradedbraided commutative product ∧ R . We further prove that i R X is a graded braided derivation ofthe wedge product for X ∈ Der R ( A ). Let ω, η ∈ Ω R ( A ). Theni R X ( ω ∧ R η ) =( − − (( R − ⊲ ω ) ∧ R ( R − ⊲ η ))( · , R − ⊲ X )= − ( R − ⊲ ω )( R − ⊲ η )( R − ⊲ X )+ ( R − ⊲ ω )(( R ′ − R − ) ⊲ X )(( R ′ − R − ) ⊲ η )=i R X ( ω ) ∧ R η + ( − · ( R − ⊲ ω ) ∧ R i RR − ⊲ X η. In particular this implies ξ ⊲ (i R X ( ω ∧ R η )) = i R ξ (1) ⊲ X (( ξ (2) ⊲ ω ) ∧ R ( ξ (3) ⊲ η )) for all ξ ∈ H .Inductively, one showsi R X ( ω ∧ R η ) = (i R X ω ) ∧ R η + ( − k ( R − ⊲ ω ) ∧ R i RR − ⊲ X η and ξ ⊲ (i R X η ) = i R ξ (1) ⊲ X ( ξ (2) ⊲ 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 ⊲ i R X ∧ R X ω = ξ ⊲ (i R X i R X ω ) = i R ξ (1) ⊲ X i R ξ (2) ⊲ X ( ξ (3) ⊲ ω ) = i R ξ (1) ⊲ ( X ∧ R X ) ( ξ (2) ⊲ ω ) or all ξ ∈ H and ω ∈ Ω •R ( A ) and inductively one obtains ξ ⊲ (i R X ω ) = i R ξ (1) ⊲ X ( ξ (2) ⊲ ω ) forany X ∈ Ω •R ( A ). It is easy to verify that i R sarisfies the linearity propertiesi R a · X ω = a · (i R X ω ) , i R X · a ω = (i R X ( R − ⊲ ω )) · ( R − ⊲ a ) , i R X ( ω · a ) =(i R X ω ) · a, i R X ( a · ω ) = ( R − ⊲ a ) · (i RR − ⊲ X ω ) (3.31)for all X ∈ X •R ( A ), a ∈ A and ω ∈ Ω •R ( A ). This concludes the proof of the lemma. In the following pages we construct a noncommutative Cartan calculus for any braided sym-metric algebra. The development is entirely parallel to the Cartan calculus of a commutativealgebra, however in a symmetric braided monoidal category. In particular, we are not con-strained to use the center of the algebra. Afterwards we define a twist deformation of anybraided Cartan calculus and show that it is isomorphic to the braided Cartan calculus of thetwist deformed algebra with respect to the twisted triangular structure.One defines a differential 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 − ⊲ X )(( R − ⊲ ω )( Y )) − ( R − ⊲ Y )( R − ⊲ ( ω ( X ))) − ω ([ X, Y ] R ) (3.32)for all X, Y ∈ Der R ( A ) and extends d to higher wedge powers by demanding it to be a gradedderivation with respect to ∧ R , i.e.d( ω ∧ R ω ) = (d ω ) ∧ R ω + ( − k ω ∧ R (d ω ) (3.33)for ω ∈ Ω k R ( A ) and ω ∈ Ω •R ( A ). Alternatively we can directly define d ω ∈ Ω k +1 R ( A ) for any ω ∈ Ω k R ( A ) by(d ω )( X , . . . , X k ) = k X i =0 ( − i ( R − ⊲ X i ) (cid:18) ( R − ⊲ ω ) (cid:18) R − ⊲ X , . . . , R − i +1) ⊲ X i − , c X i , X i +1 , . . . , X k (cid:19)(cid:19) + X i 2, since d isa graded braided derivation. The computations can be found in [41]. Define now the braideddifferential forms Ω •R ( A ) on A to be the smallest differential graded subalgebra of Ω •R ( A ) suchthat A ⊆ Ω •R ( A ). Every element of Ω k R ( A ) can be written as a finite sum of elements of theform a d a ∧ R . . . ∧ R d a k , where a , . . . , a k ∈ A . Using eq.(3.34) and the fact that the braidedcommutator is H -equivariant it immediately follows that d commutes with the left H -moduleaction. In other words, d is equivariant with respect to the adjoint action, implying( ξ ⊲ d) ω = ξ (1) ⊲ (d( S ( ξ (2) ) ⊲ ω )) = ( ξ (1) S ( ξ (2) )) ⊲ (d ω ) = ǫ ( ξ )d ω (3.35)for all ξ ∈ H and ω ∈ Ω •R ( A ). Recall that the graded braided commutator of two homogeneousmaps Φ , Ψ : G • → G • of degree k and ℓ between braided Graßmann algebras is defined by[Φ , Ψ] R = Φ ◦ Ψ − ( − kℓ ( R − ⊲ Ψ) ◦ ( R − ⊲ Φ) . (3.36)If Φ or Ψ is equivariant, the graded braided commutator coincides with the graded commutator.Furthermore, if Φ , Ψ : X •R ( A ) ⊗ G • → G • are H -equivariant maps such that Φ X , Ψ Y : G • → G • are homogeneous of degree k and ℓ for any X ∈ X k R ( A ) and Y ∈ X ℓ R ( A ), respectively, thegraded braided commutator of Φ X and Ψ Y reads[Φ X , Ψ Y ] R = Φ X ◦ Ψ Y − ( − kℓ Ψ R − ⊲ Y ◦ Φ R − ⊲ X . (3.37)For any X ∈ X •R ( A ) we define the braided Lie derivative L R : X •R ( A ) ⊗ Ω •R ( A ) → Ω •R ( A ) by L R X = [i R X , d] R . It is H -equivariant and if X ∈ X k R ( A ), L R X is a homogeneous map of degree − ( k − k = 1 we obtain a braided derivation L R X of Ω •R ( A ). emma 3.5. One has L R a ω = − (d a ) ∧ R ω and L R X ∧ R Y = i R X L R Y + ( − ℓ L R X i R Y (3.38) for all a ∈ A , ω ∈ Ω •R ( A ) , X ∈ X •R ( A ) and Y ∈ X ℓ R ( A ) . If X, Y ∈ X R ( A )[ L R X , i R Y ] R = i R [ X,Y ] R (3.39) holds.Proof. By the 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 follows. From the graded braided Leibniz rule of the graded braided commutator we obtain 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 + ( − − · ℓ [i R X , d] R i R Y =i R X L R Y + ( − ℓ L R X i R Y . The missing formula trivially holds on braided differential forms of degree 0, while for ω ∈ Ω R ( A ) one obtains[ L R X , i R Y ] R ω = L R X i R Y ω − ( − · i RR − ⊲ Y L RR − ⊲ X ω =(i R X d + di R X )i R Y ω − i RR − ⊲ Y (i RR − ⊲ X d + di RR − ⊲ X ) ω = X (i R Y ω ) + 0 + (d(( R ′′ − R ′ − ) ⊲ ω ))(( R ′′ − R − ) ⊲ Y, ( R ′ − R − ) ⊲ X ) − ( R − ⊲ Y )(i RR − ⊲ 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 all we have toprove.We are prepared to prove the main theorem of this section. It assigns to any braidedcommutative left H -module algebra A a noncommutative Cartan calculus, which we call thebraided Cartan calculus of A in the following. Theorem 3.6 (Braided Cartan calculus) . Let A be a braided commutative left H -modulealgebra and consider the braided differential forms (Ω •R ( A ) , ∧ R , d) and braided multivectorfields ( X •R ( A ) , ∧ R , J · , · K 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 ) , (3.40) 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 J X,Y K R , [ L R X , i R Y ] R =i R J X,Y K 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 , (3.41) for all X, Y ∈ X •R ( A ) .Proof. We are going to prove the above formulas in reversed order. Since d is a differentialit follows that [d , d] R = 2d = 0. Recall that there is no braiding appearing here since dis 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 ℓ R ( A ). Then[i R X , i R Y ] R = i R X i R Y − ( − kℓ i RR − ⊲ Y i RR − ⊲ X = i R X ∧ R Y − ( − kℓ ( R − ⊲ Y ) ∧ R ( R − ⊲ X ) = 0follows from the definition of i R X ∧ R Y = i R X i R Y . Using the graded braided Jacobi identity of thegraded 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 braiding appearing since dis equivariant. Recall that the braided Schouten-Nijenhuis bracket of a homogeneous element Y = Y ∧ R · · · ∧ R Y ℓ ∈ X ℓ R ( A ) with a ∈ A and X ∈ X R ( A ) read J a, Y K R = ℓ X j =1 ( − j +1 ( R − ⊲ Y ) ∧ R · · ·∧ R ( R − j − ⊲ Y j − ) ∧ R J R − ⊲ a, Y j K R ∧ R Y j +1 ∧ R · · ·∧ R Y ℓ nd J X, Y K R = ℓ X j =1 ( R − ⊲ Y ) ∧ R · · · ∧ R ( R − j − ⊲ Y j − ) ∧ R [ R − ⊲ X, Y j ] R ∧ R Y j +1 ∧ R · · · ∧ R Y ℓ , respectively. If ℓ = 1 we obtain[ L R a , i R Y ] R ω =( L R a i R Y − ( − ( − · i RR − ⊲ Y L RR − ⊲ a ) ω = − d a ∧ R i R Y ω − i RR − ⊲ Y (d( R − ⊲ a ) ∧ R ω )= − d a ∧ R i R Y ω − ( R − ⊲ Y )( R − ⊲ a ) · ω + d(( R ′ − R − ) ⊲ a ) ∧ R i R ( R ′− R − ) ⊲ Y ω =i R J a,Y K R ω for all ω ∈ Ω •R ( A ) by Lemma 3.5. Using the graded braided Leibniz rule this extends to any ℓ > 1, namely[ L R a , i R Y ∧ R ···∧ R Y ℓ ] R =[ L R a , i R Y ] R i R Y ∧ R ···∧ R Y ℓ + ( − ( − · i RR − ⊲ Y [ L RR − ⊲ a , i R Y ∧ R ···∧ R Y ℓ ]=i R J a,Y K R ∧ R Y ∧ R ···∧ R Y ℓ − i RR − ⊲ Y [ L RR − ⊲ a , i R Y ∧ R ···∧ R Y ℓ ]= · · · = i R J a,Y K R . Again by Lemma 3.5 we know that [ L R X , i R Y ] R = i R [ X,Y ] R holds for ℓ = 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 J X,Z K R holds for all X ∈ X k R ( A ) and Z ∈ X •R ( A ) for a 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 − ⊲ Z ] R L RR − ⊲ Y − L R X [i R Y , i R Z ] R − ( − m [ L R X , i RR − ⊲ Z ] R i RR − ⊲ Y =i R X [ L R Y , i R Z ] R − ( − m [ L R X , i RR − ⊲ Z ] R i RR − ⊲ Y =i R X i R J Y,Z K R − ( − m i R J X, R − ⊲ Z K R i R ( R − ⊲ Y ) =i R X ∧ R J Y,Z K R + ( − m − i R J X, R − ⊲ Z K R ∧ R ( R − ⊲ Y ) =i R J X ∧ R Y,Z K R for all X ∈ X k R ( A ), Y ∈ X R ( A ) and Z ∈ X m R ( A ) using Lemma 3.5. By induction [ L R X , i R Y ] R =i R J X,Y K R for all X, Y ∈ X •R ( A ). The remaining formula is verified via[ 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 − ℓ [i RR − ⊲ Y , [ L RR − ⊲ X , d] R ] R =[i R J X,Y K R , d] R + 0= L R J X,Y K R for all X ∈ X k R ( A ) and Y ∈ X ℓ R ( A ). This concludes the proof of the theorem.In particular, the Cartan calculus on a commutative algebra is a braided Cartan calculuswith respect to the trivial triangular structure and a (possibly trivial) action of a cocommu-tative Hopf algebra. We discuss a further class of examples which is to some extent alreadypresent in the literature, see [5] for R = 1 ⊗ H, R ), a braided commutative left H -module algebra A and a Drinfel’dtwist F on H in the following. Recall from Theorem 2.11 that the Drinfel’d functorDrin F : ( H A M RA , ⊗ A , c R ) → ( H F A F M R F A F , ⊗ A F , c R F ) (3.42)is a braided monoidal equivalence of braided monoidal categories with braided monoidal nat-ural transformation given on objects M and M ′ of H A M RA by ϕ M , M ′ : M F ⊗ A F M ′F ∋ ( m ⊗ A F m ′ ) ( F − ⊲ m ) ⊗ A ( F − ⊲ m ′ ) ∈ ( M ⊗ A M ′ ) F . or X ∈ Der R ( A ) F we define a k -linear map X F : A → A by X F ( a ) = ( F − ⊲ X )( F − ⊲ a ) for all a ∈ A . (3.43)This declares an isomorphism ( X R ( A )) F ∋ X X F ∈ X R F ( A F ) of H F -equivariant braidedsymmetric A F -modules. In particular, ξ ⊲ X F = ( ξ ⊲ X ) F , a · R F X F = ( a · F X ) F , X F · R F a = ( X · F a ) F (3.44)for all ξ ∈ H , a ∈ A and X ∈ X R ( A ) F , where we denoted the A F -module actions on X R F ( A F )by · R F . We define the twisted wedge product ∧ F = Drin F ( ∧ R ) ◦ ϕ X R ( A ) , X R ( A ) : X R ( A ) F ⊗ A F X R ( A ) F → X R ( A ) F (3.45)and extend the isomorphism (3.43) to higher wedge powers as a homomorphism of the twistedwedge product, i.e. ( X ∧ F Y ) F = X F ∧ R F Y F (3.46)for all X, Y ∈ X •R ( A ) F , where ∧ F = Drin F ( ∧ R ) ◦ ϕ X •R ( A ) , X •R ( A ) . Inductively this leads toan 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 J · , · K F : Drin F ( J · , · K R ) ◦ ϕ X •R ( A ) , X •R ( A ) : X •R ( A ) F ⊗ A F X •R ( A ) F → X •R ( A ) F (3.47)can be defined. On elements X, Y ∈ X •R ( A ) F the twisted operations read X ∧ F Y = ( F − ⊲ X ) ∧ R ( F − ⊲ Y ) and J X, Y K F = J F − ⊲ X, F − ⊲ Y K R , (3.48)respectively. Similarly we define an isomorphism F : Ω •R ( A ) F → Ω •R F ( A F ) of H F -equivariantbraided symmetric A F -bimodules, the twisted Lie derivative and twisted insertion 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 , (3.49)while the de Rham differential becomes d : Ω •R ( A ) F → Ω • +1 R ( A ) F after utilizing the Drinfel’dfunctor (see [41] for more details). On elements X ∈ X •R ( A ) F and ω ∈ Ω •R ( A ) F we obtain L F X ω = L RF − ⊲ X ( F − ⊲ ω ) and i F X ω = i RF − ⊲ X ( F − ⊲ ω ) , (3.50)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 , J · , · K F ) (3.51)as the twisted Cartan calculus with respect to F and R . Proposition 3.7. This assignment F : ( X •R ( A ) F , ∧ F , J · , · K F ) → ( X •R F ( A F ) , ∧ R F , J · , · K R F ) , (3.52) defined by eq.(3.43) and eq.(3.46), is an isomorphism of braided Gerstenhaber algebras and thetwisted Cartan calculus with respect to R and F is isomorphic to the braided Cartan calculuson A F with respect to R F via the isomorphism F . In particular ( J X, Y K F ) F = J X F , Y F K R F , ( 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 (3.53) for all X, Y ∈ X •R ( A ) F and ω ∈ Ω •R ( A ) F .Proof. By the inverse 2-cocycle property the twisted concatenation of X, Y ∈ Der( A ) F equals( X · F Y ) F ( a ) = (( F − F ′ − ) ⊲ X )(( F − F ′ − ) ⊲ Y )( F − ⊲ a ) = ( X F · R F Y F )( a )for all a ∈ A , where · R F denotes the concatenation of endomorphisms of A F . Then([ X, Y ] F ) F =([ F − ⊲ X, F − ⊲ Y ] R ) F =(( F − ⊲ X ) · R ( F − ⊲ Y )) F − ((( R − F − ) ⊲ Y ) · R (( R − F − ) ⊲ X )) F =( X · F Y ) F − (( R − F ⊲ Y ) · F ( R − F ⊲ X )) F = X F · R F Y F − ( R − F ⊲ Y F ) · R F ( R − F ⊲ X F )=[ X F , Y F ] R F where we also employed (3.44). Using formula (3.21) our previous computations together witheq.(3.46) imply ( J X, Y K F ) F = J X F , Y F K R F for all X, Y ∈ X •R ( A ) F . The other equationsfollow similarly. We refer to [41] for a full proof. n other words, the above proposition shows that the twisted Cartan calculus is gaugeequivalent to the untwisted Cartan calculus. Since the construction of the braided Cartan cal-culus is determined by the triangular structure and the twisted Cartan calculus is braided withrespect to the twisted triangular structure our construction respects the gauge equivalence.In this light twist deformations seem trivial. On the other hand, there are situations whereit is worth to distinguish the braided Cartan calculus and its twist deformations. Imagine forexample a commutative left H -module algebra A for a cocommutative Hopf algebra H . For anontrivial twist F on H the twisted Cartan calculus is noncommutative while the untwistedone is commutative. In this sense one might consider the twisted Cartan calculus as a quan-tization of the untwisted one even if both are gauge equivalent. This might be interpreted asa quantization which is in 1-1-correspondence to its classical counterpart. Having the braided Cartan calculus at hand we wonder if other concepts of differential geom-etry generalize to this setting. Focusing on the algebraic properties of covariant derivatives,namely function linearity in the first argument and a Leibniz rule in the second argument,we introduce equivariant covariant derivatives on equivariant braided symmetric bimodules.Note that there are several notions of covariant derivatives on noncommutative algebras (seee.g. [1, 2, 4, 5, 6, 9, 18, 25, 29, 35]). In particular one has to distinguish between left and rightcovariant derivatives. In the spirit of these notes we demand the covariant derivative to beequivariant in addition, for which the definitions of left and right covariant derivatives coin-cide. Curvature and Torsion of equivariant covariant derivatives are discussed and we extendan equivariant covariant derivative on the algebra to braided multivector fields and differentialforms. We furthermore give a generalization of metrics to the braided commutative settingand prove that there exists a unique equivariant Levi-Civita covariant derivative for everynon-degenerate equivariant metric. Fix in the following a triangular Hopf algebra ( H, R ) anda braided commutative left H -module algebra A . Definition 3.8 (Equivariant covariant derivative) . Consider an H -equivariant braided sym-metric A -bimodule M . An H -equivariant map ∇ R : X R ( A ) ⊗ M → M is said to be anequivariant 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 ) (3.54) and ∇ R X ( a · s ) = ( L R X a ) · s + ( R − ⊲ a ) · ( ∇ RR − ⊲ X s ) . (3.55)Note that H -equivariance of a k -linear map ∇ R : X R ( A ) ⊗ M → M reads ξ ⊲ ( ∇ R X s ) = ∇ R ξ (1) ⊲ X ( ξ (2) ⊲ s ) for all ξ ∈ H , X ∈ X R ( A ) and s ∈ M . The curvature of an equivariantcovariant derivative ∇ R on M is defined by R ∇ R ( X, Y ) = ∇ R X ∇ R Y − ∇ RR − ⊲ Y ∇ RR − ⊲ X − ∇ R [ X,Y ] R (3.56)for X, Y ∈ X R ( A ). If M = X R ( A ) we can further define the torsion of ∇ R byTor ∇ R ( X, Y ) = ∇ R X Y − ∇ RR − ⊲ Y ( R − ⊲ X ) − [ X, Y ] R , (3.57)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. While eq.(3.54) and eq.(3.55) usually only refer to a left covariant derivativewe prove in the following lemma (c.f. [41]) that in the equivariant setup the notions of leftand right covariant derivatives are equivalent. Lemma 3.9. Let ∇ R be a covariant derivative on an H -equivariant braided symmetric A -bimodule M . Then for all a ∈ A , X ∈ X R ( A ) and s ∈ M , ∇ R X · a s = ( ∇ R X ( R − ⊲ s )) · ( R − ⊲ a ) (3.58) and ∇ R X ( s · a ) = ( ∇ R X s ) · a + ( R − ⊲ s ) · ( L RR − ⊲ X a ) (3.59) hold. On the other hand, every H -equivariant map ∇ R : X R ( A ) ⊗M → M satisfying eq.(3.58)and eq.(3.59) is an equivariant covariant derivative on M . There are natural extensions of an equivariant covariant derivative ∇ R : X R ( A ) ⊗ X R ( A ) → X R ( A ) to braided multivector fields and differential forms in analogy to differential geometry.We define the braided dual pairing h· , ·i R : Ω R ( A ) ⊗ X R ( R ) → A by h ω, X i R = ω ( X ) for all ω ∈ Ω R ( A ) and X ∈ X R ( A ). It is H -equivariant, left A -linear in the first and right A -linearin the second argument. roposition 3.10. An equivariant covariant derivative ∇ R on X R ( A ) induces an equivariantcovariant derivative ˜ ∇ R on Ω R ( A ) via h ˜ ∇ R X ω, Y i R = L R X h ω, Y i R − hR − ⊲ ω, ∇ RR − ⊲ X Y i R (3.60) for all X, Y ∈ X R ( A ) and ω ∈ Ω R ( A ) . Moreover, ∇ R and ˜ ∇ R can be extended as braidedderivations to equivariant covariant derivatives on X •R ( A ) and Ω •R ( A ) , respectively.Proof. Let X, Y ∈ X R ( A ), ω ∈ Ω R ( A ) and a ∈ A . Then ˜ ∇ R X ω ∈ Ω R ( A ) is well-defined, since h ˜ ∇ R X ω, Y · a i R = L R X h ω, Y · a i R − hR − ⊲ ω, ∇ RR − ⊲ X ( Y · a ) i R =( L R X h ω, Y i R ) · a + ( R − ⊲ h ω, Y i R ) · L RR − ⊲ X a − hR − ⊲ ω, ( ∇ RR − ⊲ X Y ) · a + ( R ′ − ⊲ Y ) · L RR ′− R − ⊲ X a i R = h ˜ ∇ R X ω, Y i R · a. Similarly one proves that ˜ ∇ R is left A -linear in the first argument and satisfies the braidedLeibniz rule in the second argument. For another η ∈ Ω R ( A ) one verifies that˜ ∇ R X ( ω ∧ R η ) = ˜ ∇ R X ω ∧ R η + ( R − ⊲ ω ) ∧ R ˜ ∇ RR − ⊲ X η (3.61)defines an equivariant covariant derivative on Ω R ( A ) and inductively ˜ ∇ R extends as a braidedderivation of ∧ R to Ω •R ( A ). The extension of ∇ R to braided multivector fields is entirelysimilar.Let ∇ R : X R ( A ) ⊗ M → M be an equivariant covariant derivative with respect to R on an H -equivariant braided symmetric A -bimodule M . For any twist F on H we define the twistedequivariant covariant derivative ∇ F = Drin F ( ∇ R ) ◦ ϕ X R ( A ) , M : X R ( A ) F ⊗ F M F → M F , (3.62)which reads ∇ F X s = ∇ RF − ⊲ X ( F − ⊲ s ) . (3.63)on elements X ∈ X R ( A ) F and s ∈ M F . Proposition 3.11. The twisted equivariant covariant derivative is an equivariant covariantderivative with respect to the twisted triangular structure, where we identify X R ( A ) F with X R F ( A F ) according to Proposition 3.7.Proof. Let ξ ∈ H , a ∈ A , X ∈ X R ( A ) F and s ∈ M F . Then ξ ⊲ ( ∇ F X s ) = ∇ R ( ξ (1) F − ) ⊲ X (( ξ (2) F − ) ⊲ s ) = ∇ F ξ d (1) ⊲ X ( ξ c (2) ⊲ s )shows that ∇ F is H F -equivariant, while ∇ F a · F X s =(( F − F ′ − ) ⊲ a ) · ( ∇ R ( F − F ′− ) ⊲ X ( F − ⊲ s ))=( F − ⊲ a ) · ( ∇ R ( F − F ′− ) ⊲ X (( F − F ′ − ) ⊲ s ))=( F − ⊲ a ) · ( F − ⊲ ( ∇ RF ′− ⊲ X ( F ′ − ⊲ s )))= a · F ( ∇ F X s )and ∇ F X ( a · F s ) = ∇ RF − ⊲ X ((( F − F ′ − ) ⊲ a ) · (( F − F ′ − ) ⊲ a ))=( L RF − ⊲ X (( F − F ′ − ) ⊲ a )) · (( F − F ′ − ) ⊲ s )+ (( R − F − F ′ − ) ⊲ a ) · ( ∇ R ( R − F − ) ⊲ X (( F − F ′ − ) ⊲ s ))=( F − ⊲ ( L RF ′− ⊲ X ( F ′ − ⊲ a ))) · ( F − ⊲ s )+ (( F − R − F ′ − ) ⊲ a ) · ( ∇ R ( F − R − F ′− ) ⊲ X ( F − ⊲ s ))=( L F X a ) · F s + (( F − F − R − F ) ⊲ a ) · ( ∇ R ( F − F − R − F ) ⊲ X ( F − ⊲ s ))=( L F X a ) · F s + ( R − F ⊲ a ) · F ( ∇ FR F ⊲ X s )are the correct linearity properties, proving that ∇ F is an equivariant covariant derivativewith respect to R F . n Riemannian geometry covariant derivatives are always considered together with a Rie-mannian metric. We want to generalize them to braided commutative algebras: a k -linear map g : X R ( A ) ⊗ A X R ( A ) → A , which is left A -linear in the first argument and H -equivariant, issaid to be an equivariant metric if it is braided symmetric , i.e. if g ( Y, X ) = g ( R − ⊲ X, R − ⊲ Y )for all X, Y ∈ X R ( A ). It follows that g is braided right A -linear in the first argument as wellas right A -linear and braided left A -linear in the second argument. An equivariant metric issaid to be non-degenerate if g ( X, Y ) = 0 for all Y ∈ X R ( A ) implies X = 0, it is said to be strongly non-degenerate if g ( X, X ) = 0 for all X = 0 and it is said to be Riemannian if itis strongly non-degenerate and there is a partial order ≥ on A such that g ( X, X ) ≥ X = 0 in addition. Note that strongly non-degeneracy implies non-degeneracy. An equivariantcovariant derivative ∇ R : X R ( A ) ⊗ X R ( A ) → X R ( A ) on A is said to be a metric equivariantcovariant derivative with respect to an equivariant metric g , if L R X ( g ( Y, Z )) = g ( ∇ R X Y, Z ) + g ( R − ⊲ Y, ∇ RR − ⊲ X Z ) (3.64)holds for all X, Y, Z ∈ X R ( A ). Note that equivariance ξ ⊲ g ( X, Y ) = g ( ξ (1) ⊲ X, ξ (2) ⊲ Y ) forall ξ ∈ H and X, Y ∈ X R ( A ), of a metric is a quite strong requirement. Similar approacheswhich omit this condition are e.g. [2, 5, 25]. Lemma 3.12. Let g be a non-degenerate equivariant metric on A . Then there is a uniquetorsion-free metric equivariant covariant derivative on A .Proof. Fix an equivariant metric g on A . Any equivariant covariant derivative ∇ R on A ,which is torsion free and metric with respect to g , satisfies2 g ( ∇ R X Y, Z ) = X ( g ( Y, Z )) + ( R − ⊲ Y )( g ( R − ⊲ Z, R − ⊲ X )) − ( R − ⊲ Z )( g ( R − ⊲ X, R − ⊲ Y )) − g ( X, [ Y, Z ] R ) + g ( R − ⊲ Y, [ R − ⊲ Z, R − ⊲ X ] R )+ g ( R − ⊲ Z, [ R − ⊲ X, R − ⊲ Y ] R ) (3.65)for all X, Y, Z ∈ X R ( A ). In particular, this shows the uniqueness of a torsion-free equivariantcovariant derivative which is metric with respect to g , if g is non-degenerate. It remains toprove that a k -bilinear map ∇ R determined by the above formula is a metric torsion-freeequivariant covariant derivative. This follows by the (braided) linearity properties of g andthe braided Leibniz rule. A full proof can be found in [41].The unique torsion-free metric equivariant covariant derivative on ( A , g ) is said to be the equivariant Levi-Civita covariant derivative . We want to remark that Lemma 3.12 admitsa generalization in the sense that for any value of the torsion there exists a unique metricequivariant covariant derivative. As a last observation of this section we prove that the twistdeformation of an equivariant metric is an equivariant metric on the twisted algebra and theassignment LC: g 7→ ∇ LC , attributing to a non-degenerate equivariant metric its equivariantLevi-Civita covariant derivative, respects the Drinfel’d functor. Corollary 3.13. Let g be an equivariant metric on A . Then, the twisted equivariant metric g F , which is defined by g F ( X, Y ) = g ( F − ⊲ X, F − ⊲ Y ) (3.66) for all X, Y ∈ X R ( A ) , is an equivariant metric with respect to R F on A F . Moreover, assumingthat g and g F are non-degenerate, twisting the equivariant Levi-Civita covariant derivative withrespect to g leads to the equivariant Levi-Civita covariant derivative with respect to g F .Proof. One immediately verifies that g F is an H F -equivariant left A F -linear map X R ( A ) F ⊗ A F X R ( A ) F → A F which is braided symmetric with respect to R F . Via the identification X R ( A ) F ∼ = X R F ( A F ) of Proposition 3.7 g F becomes an equivariant metric on A F . Let X, Y, Z ∈ X R ( A ) and denote the equivariant Levi-Civita covariant derivative of g by ∇ R .From eq.(3.64) it follows that the twisted equivariant covariant derivative ∇ F satisfies L F X ( g F ( Y, Z )) = g F ( ∇ F X Y, Z ) + g F ( R − F ⊲ Y, ∇ FR − F ⊲ X Z ) . (3.67)Since Tor ∇ R = 0 we obtain Tor ∇ F = 0, proving that ∇ F is the unique equivariant Levi-Civitacovariant derivative corresponding to g F if the latter is non-degenerate. Submanifolds in Braided Commutative Geometry In this section we show that the braided Cartan calculus is compatible with the concept ofsubmanifold algebras if the triangular Hopf algebra respects the corresponding submanifoldideal. This can be understood as a construction of new examples of braided Cartan calculi fromknown ones. The projection to submanifold algebras respects Drinfel’d twist gauge equivalenceclasses, which is an interesting supplement to Proposition 3.7. The second subsection isdevoted to the study of equivariant covariant derivatives on submanifold algebras. Dependingon the choice of a strongly non-degenerate equivariant metric one is able to project equivariantcovariant derivatives as well as curvature and torsion if the submanifold algebra obeys two mildaxioms. Furthermore, the notion of twisted equivariant covariant derivative and metric arecompatible with the projections. While the main Section 3 stands out with quite an amount ofdetails, we are relatively short-spoken in the present section. The interested reader is relegatedto [41] for a more circumstantial discussion. A different approach to Riemannian geometry onnoncommutative submanifolds, based on the choice of a finite-dimensional Lie subalgebra g ofDer( A ) and a vector space homomorphism g → M into a right A -module M , is consideredin [1]. Yet another approach to noncommutative (fuzzy) submanifolds S of R n , based on theimposition of an energy cutoff on a quantum particle in R n , subject to a confining potentialwith a very sharp minimum on S , has been recently proposed and applied to spheres in [24]. In noncommutative geometry there is a well-known notion of submanifold ideal (c.f. [32])generalizing the concept of closed embedded smooth submanifolds. We refer to [14] for a recentdiscussion of submanifold algebras. In braided commutative geometry the submanifold idealshave to be respected by the Hopf algebra action in order to inherit a braided symmetry on thequotient algebra. We continue by describing the braided Cartan calculus on the submanifoldalgebra in this situation. It is given by the projection of the braided Cartan calculus ofthe ambient algebra. Moreover, the Drinfel’d functor intertwines the submanifold algebraprojections. The following discussion is also motivated by [22, 23].Fix a triangular Hopf algebra ( H, R ) and a braided commutative left H -module algebra A .For any algebra ideal C ⊆ A the coset space A / C becomes an algebra with unit and productinduced from A . The elements of A / C are equivalence classes of elements in A , where a, b ∈ A are identified if and only if there exists an element c ∈ C such that a = b + c . The correspondingsurjective projection is denoted by pr : A ∋ a a + C ∈ A / C . If the left H -module actionrespects C , i.e. if H ⊲ C ⊆ C , the quotient A / C is a braided commutative left H -module algebrawith respect to R and the left H -action defined by ξ ⊲ pr( a ) = pr( ξ ⊲ a ) for all a ∈ A . Braidedvector fields on the braided commutative algebra A / C can be obtained as projections from acertain class of braided vector fields on A . A braided derivation X ∈ Der R ( A ) is said to be tangent to C if X ( C ) ⊆ C . The k -module of all braided derivations of A which are tangent to C is denoted by X t ( A ). It is a braided Lie subalgebra and an H -equivariant braided symmetric A -sub-bimodule of X R ( A ). Consider the k -linear mappr: X t ( A ) → Der R ( A / C ) , (4.1)defined for any X ∈ X t ( A ) by pr( X )(pr( a )) = pr( X ( a )) for all a ∈ A . Note that there are leftand right A / C -actions and an H -action on the image of (4.1) defined by ξ ⊲ pr( X ) = pr( ξ ⊲ X ) , pr( a ) · pr( X ) = pr( a · X ) , pr( X ) · pr( a ) = pr( X · a ) (4.2)for all ξ ∈ H , a ∈ A and X ∈ X t ( A ). Those structure the image pr( X t ( A )) ⊆ X R ( A ) asan H -equivariant braided symmetric A / C -sub-bimodule and braided Lie subalgebra. On theother hand pr( X t ( A )) can be viewed as an H -equivariant braided symmetric A -bimodule with A -actions a · pr( X ) = pr( a ) · pr( X ) and pr( X ) · a = pr( X ) · pr( a ) for all a ∈ A and X ∈ X t ( A ).With respect to the latter structures (4.1) becomes a homomorphism of H -equivariant braidedsymmetric A -bimodules and braided Lie algebras. It follows that the kernel X ( A ) of (4.1) isan H -equivariant braided symmetric A -sub-bimodule and a braided Lie ideal of X t ( A ). Definition 4.1. An algebra ideal C ⊆ A is said to be a submanifold ideal and the quotient A / C is said to be a submanifold algebra if there is a short exact sequence → X ( A ) → X t ( A ) pr −→ Der R ( A / C ) → of H -equivariant braided symmetric A -bimodules and braided Lie algebras. ix a submanifold ideal C of A in the following. The short exact sequence (4.3) extends toa short exact sequence 0 → X • ( A ) → X • t ( A ) pr −→ X •R ( A / C ) → X ∧ R Y ) = (pr( X )) ∧ R (pr( Y )) forall X, Y ∈ X • t ( A ), where X • ( A ) and X • t ( A ) denote the braided exterior algebras of X ( A ) and X t ( A ), respectively. In particular pr( J X, Y K R ) = J pr( X ) , pr( Y ) K R holds for all X, Y ∈ X • t ( A ).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 )) , (4.5)leading to a short exact sequence of differential graded algebras0 → ker(pr) → Ω •R ( A ) pr −→ Ω •R ( A / C ) → , (4.6)where ker(pr) = L k ≥ ker(pr) k is defined recursively by ker(pr) = C andker(pr) k +1 = { ω ∈ Ω k +1 R ( A ) | i R X ω ∈ ker(pr) k for all X ∈ X t ( A ) } (4.7)for k ≥ 0. As in the case of X R ( A / C ), the projected actions, defined by intertwining theprojections, structure Ω •R ( A / C ) as an object in H A / C M RA / C . Theorem 4.2. The braided Cartan calculus on A / C is the projection of the braided Cartancalculus 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 ) and d(pr( ω )) = pr(d ω ) (4.8) for all X ∈ X • t ( A ) and ω ∈ Ω •R ( A ) .Proof. Equations (4.8) are easily verified on braided differential forms of order 0 and 1. Sincepr is a homomorphism of the braided wedge product the claim follows.As a special case we recover that the Cartan calculus on a closed embedded submanifold ι : N → M of a smooth manifold M is obtained by the pullback ι ∗ : Ω • ( M ) → Ω • ( N ) ofdifferential forms and restriction ι ∗ : X • t ( M ) → X • ( N ) of tangent multivector fields to N . Thelatter is defined for any X ∈ X t ( M ) as the unique vector field X | N ∈ X ( N ), which is ι -relatedto X , i.e. T q ι ( X | N ) q = X ι ( q ) for all q ∈ N , where T q ι : T q N → T ι ( q ) M denotes the tangentmap (c.f. [30] Lemma 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 ).In the next proposition we prove that the gauge equivalence given by the Drinfel’d functoris compatible with the notion of submanifold ideal, i.e. the projection to submanifold algebrasand twisting commute. In the particular case of a cocommutative Hopf algebra with trivialtriangular structure this means that twist quantization and projection to the submanifoldalgebra commute (see also [25]). Proposition 4.3. For any twist F on H , the submanifold algebra projection of the twistdeformation ( X • t ( A ) F , ∧ F , J · , · K F ) of the braided Gerstenhaber algebra of tangent multivec-tor fields on A coincides with the twist deformation ( X •R ( A / C ) F , ∧ F , J · , · K F ) of the braidedGerstenhaber algebra of braided multivector fields 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 ) (4.9) for all X ∈ X • t ( A ) F and ω ∈ Ω •R ( A ) F .Proof. Note that the twist deformation of X • t ( A ) is a braided Gerstenhaber algebra since thebraided multivector fields which are tangent to C are an H -submodule and a braided symmetric A -sub-bimodule of X •R ( A ). We already noticed that pr: X • t ( A ) → X •R ( A / C ) is surjective. Let X, Y ∈ X • t ( A ) F and a ∈ A . Thenpr( X ) ∧ F pr( Y ) = ( F − ⊲ pr( X )) ∧ R ( F − ⊲ pr( Y )) = pr( X ∧ F Y ) , and similarly J pr( X ) , pr( Y ) K F = pr( J X, Y K F ) and pr( a ) · F pr( X ) = pr( a · F X ) follow. Moreover, L F pr( X ) pr( ω ) = L RF − ⊲ pr( X ) ( F − ⊲ pr( ω )) = pr( L F X ω )and i F pr( X ) pr( ω ) = i RF − ⊲ pr( X ) ( F − ⊲ pr( ω )) = pr(i F X ω )for all X ∈ X • t ( A ) and ω ∈ Ω •R ( A ) by Theorem 4.2. This concludes the proof of the proposition. .2 Equivariant Covariant Derivatives on Submanifolds In this section we discuss equivariant covariant derivatives on submanifold algebras and studyunder which conditions equivariant covariant derivatives and metrics allow for projections.Accepting two mild axioms the latter is possible for a given strongly non-degenerate equivariantmetric. Furthermore, the projection of the equivariant covariant derivative is compatible withthe notion of curvature, torsion and twist deformation.Fix a submanifold ideal C of A and a strongly non-degenerate equivariant metric g on A .Then there is a direct sum decomposition X R ( A ) = X t ( A ) ⊕ X n ( A ) , (4.10)where X n ( A ) are the so-called braided normal vector fields with respect to C and g , definedto be the subspace orthogonal to X t ( A ) with respect to g . Then, pr g : X R ( A ) → X R ( A / C )is the k -linear map which first projects to the first addend in the above decomposition andapplies pr: X t ( A ) → X R ( A / C ) afterwards. In particular pr g ( X ) = pr( X ) for all X ∈ X t ( A ).In a next step we define a k -linear map g A / C : X R ( A / C ) ⊗ A / C X R ( A / C ) → A / C by g A / C (pr g ( X ) , pr g ( Y )) = pr g ( g ( X, Y )) (4.11)for all X, Y ∈ X R ( A ). It is well-defined if X ( A ) = ker pr has the following property. Axiom 1: for every X ∈ X ( A ) there are finitely many c i ∈ C and X i ∈ X t ( A )such that X = X i c i X i .This is for example the case if X ( A ) is finitely generated as a C -bimodule. If g is non-degenerate the projection g A / C is not non-degenerate in general. However, if we assume thefollowing property of g , the projection g A / C is strongly non-degenerate if g is. Axiom 2: if X ∈ X t ( A ), then g ( X, X ) ∈ C implies X ∈ X ( A ).Note that in the case of closed embedded smooth manifolds both axiom 1 and 2 are satisfied. Proposition 4.4. For any strongly non-degenerate equivariant metric g on A such that theaxioms 1 and 2 are satisfied, g A / C is a well-defined strongly non-degenerate equivariant metricon A / C . The projection ∇ A / C pr( X ) pr( Y ) = pr g ( ∇ R X Y ) , (4.12) of an equivariant covariant derivative ∇ R : X R ( A ) ⊗ X R ( A ) → X R ( A ) on A , where X, Y ∈ X t ( A ) , is an equivariant covariant derivative with respect to R on A / C . If furthermore, ∇ R is the equivariant Levi-Civita covariant derivative with respect to g , ∇ A / C is the equivariantLevi-Civita covariant derivative on A / C with respect to g A / C .Proof. 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) X i c i · X i (cid:19) , pr g ( Y ) (cid:19) = pr (cid:18) g (cid:18) X i c i · X i , Y (cid:19)(cid:19) =pr (cid:18) X i c i · g ( X i , Y ) | {z } ∈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 ). 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 implies X = 0, which is equivalent to the statement that X = 0 implies g A / C ( X, X ) = 0, i.e. strongnon-degeneracy of g A / C . From Axiom 1 it follows that ∇ A / C is well-defined. In fact, for X = P 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) X i c i · ∇ R X i Y | {z } ∈ X ( A ) (cid:19) = 0 nd ∇ A / C pr( Y ) pr( X ) =pr g (cid:18) X i ∇ R Y ( c i · X i ) (cid:19) = X i pr g ( ∈C z }| { ( L R Y c i ) · X i | {z } ∈ X ( A ) + ∈C z }| { ( R − ⊲ c i ) ·∇ RR − ⊲ X i Y | {z } ∈ X ( A ) ) = 0 , since ∇ R is left A -linear in the first argument and satisfies a braided Leibniz rule in the secondargument. The remaining results are proven in [41].We would like to stress that the assumptions of Proposition 4.4 are sufficient to projectstrongly non-degenerate equivariant metrics and equivariant covariant derivatives to subman-ifold algebras. Whether those conditions are also necessary is part of further investigation.Fix an equivariant covariant derivative ∇ R on A and a strongly non-degenerate equivariantmetric g such that axiom 1 and 2 hold. The curvature and torsion of a projected equivariantcovariant derivative coincide with the projection of the curvature and torsion of ∇ R . Corollary 4.5. The curvature R ∇ A / C and the torsion Tor ∇ A / C of the projected equivariantcovariant derivative ∇ A / C are given by R ∇ A / C (pr( X ) , pr( Y ))(pr( Z )) = pr g ( R ∇ R ( X, Y ) Z ) (4.13) and Tor ∇ A / C (pr( X ) , pr( Y )) = pr g (Tor ∇ R ( X, Y )) (4.14) for all X, Y, Z ∈ X t ( A ) . One extends the projection pr g : X •R ( A ) → X •R ( A / R ) to braided multivector fields bydefining it to coincide with pr on A and to be a homomorphism of the braided wedge producton higher wedge powers. On braided differential forms we set pr g = pr. Corollary 4.6. The equivariant covariant 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 equivariant covariant derivative ∇ A / C on A / C according to Proposi-tion 3.10, 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 g ( ˜ ∇ R X ω ) (4.15) for all X ∈ X t ( A ) , Y ∈ X • t ( A ) and ω ∈ Ω •R ( A ) . Furthermore, twisted equivariant covariant derivatives behave well under projection. Proposition 4.7. For any twist F on H , the projection of the twisted equivariant covariantderivative coincides with the twist deformation of the projected equivariant covariant derivative,i.e. ( ∇ A / C ) F pr( X ) pr( Y ) = pr g ( ∇ F X Y ) for all X, Y ∈ X t ( A ) F . Its curvature and torsion aregiven by R ( ∇ A / C ) F (pr( X ) , pr( Y ))(pr( Z ))= R ∇ A / C (cid:18) ( F − F ′ − ) ⊲ pr( X ) , ( F − F ′ − ) ⊲ pr( Y ) (cid:19) ( F − ⊲ pr( Z ))=pr (cid:18) R ∇ R (cid:18) ( F − F ′ − ) ⊲ X, ( F − F ′ − ) ⊲ Y (cid:19) ( F − ⊲ Z ) (cid:19) (4.16) and Tor ( ∇ A / C ) F (pr( X ) , pr( Y )) =Tor ∇ A / C ( F − ⊲ pr( X ) , F − ⊲ pr( Y ))=pr (cid:18) Tor ∇ R ( F − ⊲ X, F − ⊲ Y ) (cid:19) (4.17) for all X, Y, Z ∈ X t ( A ) F , respectively. Similar statements hold for the induced (twisted)equivariant covariant derivatives on braided differential forms and braided multivector fields.Proof. For all X, Y ∈ X t ( A ) F one obtainspr g ( ∇ F X Y ) =pr g ( ∇ RF − ⊲ X ( F − ⊲ Y )) = ∇ A / C pr( F − ⊲ X ) (pr( F − ⊲ Y ))= ∇ A / CF − ⊲ pr( X ) ( F − ⊲ pr( Y )) = ( ∇ A / C ) F pr( X ) pr( Y )and similarly one proves the statements about the induced equivariant covariant derivatives. cknowledgments The author is grateful to Francesco D’Andrea and Gaetano Fiore for their constant support.In particular, he wants to thank the latter for introducing him to the concept of twistedCartan calculus and proposing to prove compatibility with projections to submanifold algebras.Special thanks go to Paolo Aschieri and the referee for their valuable comments on the firstversion of this paper. Furthermore, the author wants to thank Stefan Waldmann for posinga question at the DQ seminar about the existence of a braided Cartan calculus for everytriangular structure. References [1] Arnlind, J. and Norkvist, A. T.: Noncommutative minimal embeddings and morphismsof pseudo-Riemannian calculi . 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