Generalized symmetry in noncommutative (complex) geometry
aa r X i v : . [ m a t h . QA ] J u l GENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEXGEOMETRY
SUVRAJIT BHATTACHARJEE, INDRANIL BISWAS, AND DEBASHISH GOSWAMI
Abstract.
We introduce Hopf algebroid covariance on Woronowicz’s differential calculus.Using it, we develop a general framework of noncommutative complex geometry that sub-sumes the one in [ ´B17]. We present transverse complex and K¨ahler structures as examplesand discuss the Connes-Moscovici Hopf algebroid. We prove noncommutative versions of theHodge decomposition and Formality theorems for K¨ahler manifolds. Relation with existingliterature is described. Introduction
Symmetry plays an important, often decisive, role in almost all areas of mathematics;especially in geometry and topology. Classically, symmetry means group action on spaces.However, it necessitates to pass from groups to groupoids to capture local symmetry in anefficient way. For example, it is more natural to consider the (Lie)groupoid of isometries of aRiemannian manifold which is not globally symmetric or homogeneous. The natural domainfor the characteristic classes of certain geometric structures are in fact the cohomology ofthe classifying spaces of (Lie)groupoids. It is possible to go further, saying that groupoidsprovide a concept of generalized symmetry that is essential, as exemplified above and spec-tacularly apparent in the theory of foliations. In the realm of noncommutative geometry,symmetry is captured by action or coaction of Hopf algebras on (co)algebras, which is thenoncommutative version of a space.The concept of Hopf algebroids generalizes that of groupoids, providing a way of con-sidering generalized symmetry in noncommutative geometry. They can be thought of asHopf algebras over noncommutative bases. Initially conceived by algebraic topologists, Hopfalgebroids over commutative bases have been used extensively in geometry and topology.Problems start to appear when we consider generalizing the definition to noncommutativebases. The pre-Hopf algebroid level, i.e., the definition of a bialgebroid is usually acceptedas the correct generalization of a bialgebra over a noncommutative base. The problem iswith the addition of an antipode. A description of the various definitions is given in theintroduction of [KP11]. The definition used in this article also comes from that paper whichwas first given in [BS04].The theory of noncommutative complex geometry was initiated in [ ´B16, ´B17], althoughthere are precursors; see [BPS13, FGR97, KLvS11, PS03]. It attempts to provide a fresh in-sight into various aspects of noncommutative geometry, such as the construction of spectraltriples for quantum groups, by considering “complex structures”. It also promises a fruit-ful interaction between noncommutative geometry and noncommutative projective algebraic geometry. Identifying “differential forms” as the basic objects of study, the framework ofnoncommutative complex geometry is developed in the setting of Woronowicz’s differentialcalculus, see [Wor89]. The classical complex geometry being the obvious example, the setupin [ ´B17] takes as its motivating example the family of quantum flag manifolds. It is pos-sible to proceed, as shown in there, as far as proving a version of the Hard Lefschetz theorem.Singular spaces, such as the leaf space of a foliation, have been studied extensively inclassical geometry as well as noncommutative geometry. These spaces provided the mainimpetus for the development of noncommutative geometry, see [Con82]. Classically, “trans-verse geometry” attempts to study such singular spaces using symmetry, which most of thetime turns out to be a pseudogroup. This was exemplified in the beautiful paper [Hae80].It led to the systematic study of spaces with pseudogroup symmetry. It is natural to askwhether one can do complex geometry over such spaces. That one can, was done in a volumeof works, [CW91, EKA90], to name a few.Now, pseudgroups and groupoids are very much noncommutative in their nature. Thisled to Connes’ construction of the highly noncommutative groupoid C ∗ -algebra of the ho-lonomy groupoid of a foliation, which was successfully applied to the questions in indextheory. However, the fact that groupoids consist of symmetries is not so conspicuous in thisconstruction. To take the symmetry into account, one is naturally led to the language ofHopf algebroids, as shown in [Kal11, Mrˇc99, Mrˇc07].Thus, the study of complex geometry over such singular spaces consists of studying regularspaces with highly noncommutative symmetry, which are also generalized, in that they arenot Hopf algebras.The goal of the present article is to introduce Hopf algebroid symmetry in noncommutativegeometry. We formulate and study a quite general framework of Hopf algebroid covarianceof noncommutative complex and K¨ahler structures. We have been able to accommodate allthe existing examples in our framework. Another notable and novel aspect of our work is anew definition of Hopf algebroid action or covariance on differential calculus which seems towork in a very general context. We present the Connes-Moscovici Hopf algebroid as one ofthe most interesting examples of our setup.Let us briefly describe the plan of the paper. In Section 2, we recall the definition of aHopf algebroid. Foliations and ´etale groupoids in general, are also discussed in some detailand is shown to provide examples. The last subsection is new. It introduces ∗ -structures onHopf algebroids which is essential in order to view them as symmetry objects in noncom-mutative geometry. Sections 3 and 4 describes the whole setup. Hopf algebroid covarianceis introduced. The necessary modifications of the framework in [ ´B17] are described andalong the way, examples coming from foliations are provided. Section 5 proves a version ofHodge decomposition theorem. There are many versions of this theorem in noncommuta-tive geometry. But in order to capture the classical case as well as the cases for foliations([EKA90]) and orbifolds ([BBF + ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 3 not appearing in noncommutative geometry in any essential way. We included it because thecorresponding classical results are proved rather recently, [BBF + Hopf algebroids
We recall the definition of Hopf algebroids from [KP11]. See also [B¨oh09, BS04].2.1.
Bialgebroids.
We begin by defining a generalization of bialgebras.
Definition 2.1.
Let A be a C -algebra. An ( s, t ) -ring over A is a C -algebra H with homo-morphisms s : A → H and t : A op → H whose images commute in H . The functions s and t are referred to as the source and target maps respectively. An( s, t )-ring structure is equivalent to the structure of an A e -algebra on H . Definition 2.2.
Let H be an ( s, t ) -ring over A . The Takeuchi product is the subspace H × A H := { X i h i ⊗ A h ′ i ∈ H ⊗ A H | X i h i t ( a ) ⊗ h ′ i = X i h i ⊗ h ′ i s ( a ) ∀ a ∈ A } of H ⊗ A H , where the tensor product ⊗ A is defined with respect to the following ( A, A ) -bimodule structure on H : a · h · a := s ( a ) t ( a ) h, a , a ∈ A, h ∈ H. (2.1)This Takeuchi product becomes a unital algebra with factorwise multiplication as well asan ( s, t )-ring. Definition 2.3.
Let A l be a C -algebra. A left bialgebroid over A l is an ( s l , t l ) -ring H l equipped with the structure of an A l -coalgebra (∆ l , ε l ) with respect to the ( A l , A l ) -bimodulestructure (2.1) , subject to the following conditions:i) the (left) coproduct ∆ l : H l → H l ⊗ A l H l maps into the subset H l × A l H l and defines amorphism ∆ l : H l → H l × A l H l of unital C -algebras;ii) the (left) counit has the property: ε l ( hh ′ ) = ε l ( hs l ( ε l h ′ )) = ε l ( ht l ( ε l h ′ )) h, h ′ ∈ H l . (2.2)We denote the above left bialgebroid by ( H l , A l , s l , t l , ∆ l , ε l ) or simply by H l . Remark 2.4.
From (2.2) above and the fact that ε l is an ( A l , A l ) -bimodule morphism, itfollows that ε l ( s l ( a ) h ) = aε l ( h ) , ε l ( t l ( a ) h ) = ε l ( h ) a , and it also follows that ε l (1 H l ) = 1 A l .So we have that ε l s l = ε l t l = id A l . Lemma 2.5.
In a left bialgebroid, the left counit is unique.Proof.
Indeed, if both ε l and ε l make ( H l , A l , s l , t l , ∆ l , ε l ) and ( H l , A l , s l , t l , ∆ l , ε l ) left bial-gebroids, then we have: ε l ( h ) = ε l ( s l ε l ( h ) h ) = ε l ( h ) ε l ( h ) = ε l ( t l ε l ( h ) h ) = ε l ( h ) . (cid:3) SUVRAJIT BHATTACHARJEE, INDRANIL BISWAS, AND DEBASHISH GOSWAMI
Given an ( s, t )-ring H , there is another ( A, A )-bimodule structure on H : a · h · a = ht ( a ) s ( a ) , a , a ∈ A h ∈ H. (2.3)With respect to this bimodule structure, the tensor product ⊗ A is defined. Inside H ⊗ A H ,there is the Takeuchi product: H × A H := { X i h i ⊗ A h ′ i ∈ H ⊗ A H | X i s ( a ) h i ⊗ h ′ i = X i h i ⊗ t ( a ) h ′ i ∀ a ∈ A } . This again becomes a unital algebra with factorwise multiplication and also is an ( s, t )-ring.
Definition 2.6.
Let A r be a C -algebra. A right bialgebroid over A r is an ( s r , t r ) -ring H r equipped with the structure of an A r -coalgebra (∆ r , ε r ) with respect to the ( A r , A r ) -bimodulestructure (2.3) , subject to the following conditions:i) the (right) coproduct ∆ r : H r → H r ⊗ A r H r maps into H r × A r H r and defines a morphism ∆ r : H r → H r × A r H r of unital C -algebras;ii) the (right) counit has the property: ε r ( hh ′ ) = ε r ( s r ( ε r h ) h ′ ) = ε r ( t r ( ε r h ) h ′ ) h, h ′ ∈ H r . (2.4)We denote a right bialgebroid by ( H r , A r , s r , t r , ∆ r , ε r ) or simply by H r . Note that if( H l , A l , s l , t l , ∆ l , ε l ) is a left bialgebroid, then ( H opl , A l , t l , s l , ∆ l , ε l ) is a right bialgebroid. Remark 2.7.
As in Remark 2.4, we have ε r s r = ε r t r = id A r . Also as above, the right counitis unique. Sweedler notation.
We shall use Sweedler notation with subscripts ∆ l ( h ) = h ⊗ h forleft coproducts while the right coproducts are indicated by superscripts: ∆ r ( h ) = h ⊗ h . Hopf algebroids.
We now define a Hopf algebroid as an algebra endowed with a leftand a right bialgebroid structure together with an antipode mapping from the left bialgebroidto the right bialgebroid. More precisely:
Definition 2.8.
A Hopf algebroid is given by a triple ( H l , H r , S ) , where H l = ( H l , A l , s l , t l , ∆ l , ε l ) is a left A l -bialgebroid and H r = ( H r , A r , s r , t r , ∆ r , ε r ) is a right A r -bialgebroid on the same C -algebra H , and S : H → H is invertible C -linear. These structures are subject to thefollowing four conditions:i) the images of s l and t r as well as those of t l and s r , coincide: s l ε l t r = t r , t l ε l s r = s r , s r ε r t l = t l , t r ε r s l = s l ; (2.5) ii) twisted coassociativity holds: (∆ l ⊗ id H )∆ r = ( id H ⊗ ∆ r )∆ l , (∆ r ⊗ id H )∆ l = ( id H ⊗ ∆ l )∆ r ; (2.6) iii) for all a ∈ A l , a ∈ A r and h ∈ H , we have S ( t l ( a ) ht r ( a )) = s r ( a ) S ( h ) s l ( a ); (2.7) iv) the antipode axioms hold: µ H ( S ⊗ id H )∆ l = s r ε r , µ H ( id H ⊗ S )∆ r = s l ε l . (2.8) ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 5
We apply ε r to the first two and ε l to the second pair of identities in (2.5) and get that A l and A r are anti-isomorphic as C -algebras: φ := ε r s l : A opl → A r , φ − := ε l t r : A r → A opl ,θ := ε r t l : A l → A opr , θ − := ε l s r : A opr → A l . (2.9)The antipode is anti-algebra and anti-coalgebra morphism (between different coalgebras)and satisfies the equationsflip ◦ ( S ⊗ S )∆ l = ∆ r S, flip ◦ ( S ⊗ S )∆ r = ∆ l S, (2.10)where flip : H ⊗ C H → H ⊗ C H is the flip permuting two factors of the tensor product (thisbecomes an ( A l , A l )-respectively ( A r , A r )-bimodule). Similar formulas hold for the inverse S − . The following identities will be used: s r ε r s l = Ss l , s l ε l s r = Ss r , s r ε r t l = S − s l , s l ε l t r = S − s r ,t r ε r s l = St l , t l ε l s r = St r , t r ε r t l = S − t l , t l ε l t r = S − t r ,ε r s l ε l = ε r S, ε l s r ε r = ε l S, ε r t l ε l = ε r S − , ε l t r ε r = ε l S − , (2.11)and µ H ( S ⊗ s l ε l )∆ l = S, µ H ( s r ε r ⊗ S )∆ r = S,µ H op ( id H ⊗ S − )∆ l = t r ε r , µ H op ( S − ⊗ id H )∆ r = t l ε l ,µ H op ( t l ε l ⊗ S − )∆ l = S − , µ H op ( S − ⊗ t r ε r )∆ r = S − . (2.12) Lemma 2.9.
In a Hopf algebroid, the antipode is unique.Proof.
Indeed, if both S and S make ( H l , H r , S ) and ( H l , H r , S ) Hopf algebroids then wehave S ( a ) = s r ε r ( a ) S ( a ) = S ( a ) a S ( a ) = S ( a ) a S ( a ) = S ( a ) s l ε l ( a ) = S ( a ) . (cid:3) Finally, note that if ( H l , H r , S ) is a Hopf algebroid, then ( H opr , H opl , S − ) is also a Hopfalgebroid.2.3. ´Etale groupoids. We now introduce our main example besides Hopf algebras. A Hopfalgebra is a Hopf algebroid with A l = A r = C . We follow [MM03]. See also [Con94, Har15,Kal11]. Definition 2.10.
A groupoid G is a small category in which each arrow is invertible. Moreexplicitly, a groupoid consists of a space of objects G , a space of arrows G (often denotedby G itself ) and five structure maps relating the two:i) source and target maps s, t : G → G , assigning to each arrow g its source s ( g ) andtarget t ( g ) ; one says that g is from s ( g ) to t ( g ) ;ii) a partially defined composition of arrows, that is, only for those arrows g, h for whichsource and target match that is s ( g ) = t ( h ) ; in other words, a map m : G := G s × tG G → G , ( g, h ) gh that is associative whenever defined, producing the compositearrow going from s ( gh ) = s ( h ) to t ( gh ) = t ( g ) ;iii) a unit map G → G , x x , that has the property t ( g ) g = g s ( g ) = g ;iv) an inversion inv : G → G , g g − that produces the inverse arrow going from s ( g − ) = t ( g ) to t ( g − ) = s ( g ) , fulfilling g − g = 1 s ( g ) , gg − = 1 t ( g ) . SUVRAJIT BHATTACHARJEE, INDRANIL BISWAS, AND DEBASHISH GOSWAMI
These maps can be assembled into a diagram G G G G G m inv st (2.13)An arrow may be denoted by x g −→ y to indicate that y = s ( g ) and x = t ( g ).A topological groupoid is a groupoid in which both G and G are topological spaces andall the structure maps are continuous. Similarly one defines smooth groupoids, where in addi-tion s and t are required to be surjective submersions in order to ensure that G = G s × tG G remains a manifold. A topological (or smooth) groupoid is called ´etale if the source map isa local homeomorphism (or local diffeomorphism); this condition implies that all structuresmaps are local homeomorphisms (or local diffeomorphisms, respectively). In the smoothcase, this equivalently amounts to saying that dim G = dim G . In particular, an ´etalegroupoid has zero-dimensional source and target fibers, and hence they are discrete. Weshall only be dealing with smooth ´etale groupoids.We give some examples of ´etale groupoids below. Example 2.11. i) The unit groupoid has a single manifold M as both its object and arrow space. All themaps are identity functions.ii) A (discrete) group is a one-object groupoid (called the point groupoid).iii) The translation groupoid Γ ⋉ M of a smooth left action of a discrete group has as objectspace M and arrow space Γ × M . The source is ( g, m ) m , the target is ( g, m ) gm and the multiplication is ( g, m )( g ′ , m ′ ) = ( gg ′ , m ′ ) .iv) Orbifold groupoids or proper ´etale groupoids. We refer to [MM03, Har15] for moredetails.v) Let ( M, F ) be a foliated manifold. Then the (reduced) holonomy groupoid is ´etale. As the last example is one of our main motivating examples, we shall describe it in aslightly greater details. See [CW91, MM03, CLN85, CM01]. A foliation F on M is given bya cocycle U = { U i , f i , g ij } modeled on a manifold N ( R n or C n ), i.e.,i) { U i } is an open covering of M ;ii) f i : U i → N are submersions with connected fibers defining F ;iii) g ij are local diffeomorphisms of N and g ij f j = f i on U i ∩ U j .The manifold N = ⊔ f i ( U i ) is called the transverse manifold of F associated to the cocycle U , and the pseudogroup P generated by g ij is called the holonomy pseudogroup on thetransverse manifold. To any pseudogroup P on some manifold X we can associate an ´etale(effective) groupoid Γ( P ) over X as follows: for any x, y ∈ X letΓ( P )( x, y ) = { germ x g | g ∈ P, x ∈ dom ( g ) , g ( x ) = y } . (2.14)The multiplication in Γ( P ) is given by the composition of transitions. Equipped with classicalsheaf topology Γ( P ) becomes a smooth manifold and Γ( P ) becomes an ´etale groupoid. Inour case, Γ( P ) is called the reduced holonomy groupoid of ( M, F ) and is denoted Hol N ( M, F )(but also we write Γ( P ) sometimes). ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 7
We now show one gets Hopf algebroids naturally from ´etale groupoids following [KP11,Mrˇc07]. Before that we introduce the following.
Fiber sum notation.
Let E and F are vector bundles over two manifolds X and Y ,respectively. Suppose φ : X → Y is an ´etale map (i.e., a local homeomorphism) and α : E ∼ = φ ∗ F an isomorphism of vector bundles. Then the push-forward (or fiber sum) of φ ,denoted by φ ∗ : Γ c ( X, E ) → Γ c ( Y, F ), is defined by( φ ∗ s )( y ) = X φ ( x )= y α ( s ( x )) , (2.15)where x ∈ X, y ∈ Y and s ∈ Γ c ( X, E ). Here we identify the fiber φ ∗ F z with F φ ( z ) using thedefinition of pullback.If G is an ´etale groupoid over a compact Hausdorff G , the space C ∞ c ( G ) of smoothfunctions on G = G with compact support carries a Hopf algebroid structure. Although G = G often happens to be non-Hausdorff in examples, we assume this condition in thispaper since the reduced holonomy groupoid of a Riemannian foliation is always Hausdorff.We have two C ∞ ( G )-actions on C ∞ c ( G ) by left and right multiplication with respect to whichwe define the four tensor products denoted by ⊗ llC ∞ ( G ) , ⊗ rrC ∞ ( G ) , ⊗ rlC ∞ ( G ) and ⊗ lrC ∞ ( G ) . Weneed the following isomorphismsΩ s,t : C ∞ c ( G ) ⊗ rlC ∞ ( G ) C ∞ c ( G ) → C ∞ c ( G s × tG G ) = C ∞ c ( G )Ω t,t : C ∞ c ( G ) ⊗ llC ∞ ( G ) C ∞ c ( G ) → C ∞ c ( G t × tG G ) = C ∞ c ( G )Ω s,s : C ∞ c ( G ) ⊗ rrC ∞ ( G ) C ∞ c ( G ) → C ∞ c ( G s × sG G ) = C ∞ c ( G )Ω t,s : C ∞ c ( G ) ⊗ lrC ∞ ( G ) C ∞ c ( G ) → C ∞ c ( G t × sG G ) = C ∞ c ( G ) (2.16)all given by the formulas Ω − . − ( u ⊗ −− C ∞ ( G ) u ′ )( g, g ′ ) = u ( g ) u ( g ′ ) , (2.17)for u, u ′ ∈ C ∞ c ( G ) and ( g, g ′ ) in the respective pullback G − × − G G . The maps are isomor-phism, as it was shown in [Mrˇc07]. We now give the Hopf algebroid structure maps for C ∞ c ( G ) over C ∞ ( G ): Ring structure.
On the base algebra C ∞ ( G ) one has the commutative pointwise product,whereas the total algebra C ∞ c ( G ) is equipped with a convolution product, defined as thecomposition ∗ : C ∞ c ( G ) ⊗ rlC ∞ ( G ) C ∞ c ( G ) Ω s,t −−→ C ∞ c ( G ) m ∗ −→ C ∞ c ( G ) . (2.18)Explicitly, ( u ∗ v )( g ) := ∗ ( u ⊗ v ) = ( m ∗ Ω s,t ( u ⊗ v ))( g ) = X g = g g u ( g ) u ( g ) , (2.19)which can be used in showing associativity of the product. SUVRAJIT BHATTACHARJEE, INDRANIL BISWAS, AND DEBASHISH GOSWAMI
Source and target maps.
For f ∈ C ∞ ( G ) and u ∈ C ∞ c ( G ),( f ∗ u )( g ) = f ( t ( g )) u ( g ) and ( u ∗ f )( g ) = u ( g ) f ( s ( g )) . (2.20)It can be shown that C ∞ ( G ), identified with those functions in C ∞ c ( G ) having support on1 G ⊂ G , is a commutative subalgebra of C ∞ c ( G ). We put for the (left and right bialgebroid)source and target maps s l ≡ s r ≡ t l ≡ t r ≡ ∗ : C ∞ ( G ) → C ∞ c ( G ) , (2.21)i.e., the injection as subalgebra given by the fiber sum of the unit map 1 : G → G . Moreexplicitly, s l : f f , where f ( g ) = ( f ( x ) if g = 1 x for some x ∈ G Left and right coproducts.
Using the isomorphism Ω − , − , the left and right coproducts aregiven as follows: ∆ l : C ∞ c ( G ) → C ∞ c ( G t × tG G ) ∼ = C ∞ c ( G ) ⊗ ll C ∞ c ( G ) , (∆ l u )( g, g ′ ) = ( u ( g ) if g = g ′ , , (2.23a)∆ r : C ∞ c ( G ) → C ∞ c ( G s × sG G ) ∼ = C ∞ c ( G ) ⊗ rr C ∞ c ( G ) , (∆ r u )( g, g ′ ) = ( u ( g ) if g = g ′ , l = d l ∗ and ∆ r = d r ∗ for the diagonal maps d l : G → G t × tG G , g ( g, g )and d r : G → G s × sG G , g ( g, g ). Left and right counits.
Both left and right counits are respectively determined by the fibersum of the target and source maps of the groupoid. For any x ∈ G , ε l : C ∞ c ( G ) → C ∞ ( G ) , ( ε l ( u ))( x ) = P t ( g )= x u ( g ) ε r : C ∞ c ( G ) → C ∞ ( G ) , ( ε r ( u ))( x ) = P s ( g )= x u ( g ) . (2.24) Antipode.
The antipode is given by the groupoid inversion, S : C ∞ c ( G ) → C ∞ c ( G ) , ( S ( u ))( g ) = u ( g − ) = ( inv ∗ u )( g ) . (2.25) Theorem 2.12.
With the above structure maps, C ∞ c ( G ) becomes a Hopf algebroid over C ∞ ( G ) . The proof is in [KP11]. See also [Con82, Con85, Kor08, Kor09].
ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 9
Modules.
Let H = ( H l , H r , S ) be a Hopf algebroid. A left module over H is simply aleft module over the underlying C -algebra H . We denote the structure map by ( h, m ) h · m .The left bialgebroid structure H l induces an ( A l , A l )-bimodule structure on each module anda monoidal structure on the category of modules. More explicitly, let M be an H -module.Then the ( A l , A l )-bimodule structure is given by a · m · a = s l ( a ) · t l ( a ) · m, (2.26)for all a , a ∈ A l and m ∈ M . The left coproduct defines the monoidal structure ( M, N ) M ⊗ A N , where M ⊗ A N is equipped with the H -module structure h · ( m ⊗ n ) := h · m ⊗ h · n, h ∈ H, m ∈ M, n ∈ N. (2.27)The monoidal unit is given by A l with left H -action h · a = ε l ( hs l ( a )) . Note that ε l ( ht l ( a )) = ε l ( hs l ( ε l ( t l ( a )))) = ε l ( hs l ( a )) . Also A l being the monoidal unit it is an algebra in the categoryof H -modules, i.e., it is an H -module algebra. This structure will be important for us in theexamples we consider. Remark 2.13.
We state the definition of an H -module algebra explicitly. It is a C -algebraand left H -module B such that the multiplication in B is A l -balanced andi) h · B = s l ε l ( h ) · B ;ii) h · ( bb ′ ) = ( h · b )( h · b ′ ) .for b, b ′ ∈ B and h ∈ H . Note that B has a canonical A l -ring structure. Its unit is the map A l → B , a s l ( a ) · B = t l ( a ) · B . Similarly, one can consider right H -modules as modules over the C -algebra H . Suchmodules get the structure of an ( A r , A r )-bimodule and the category becomes monoidal usingthe right coproduct. The monoidal unit is A r . We now see some examples coming from thegeometry of groupoids. We follow [Kal11]. Definition 2.14.
A smooth left action of a Lie groupoid G on a smooth manifold P along asmooth map π : P → G is a smooth map µ : G s × πG P → P , ( g, p ) g · p , which satisfiesthe conditions φ ( g · p ) = t ( g ) , π ( p ) · p = p and g ′ · ( g · p ) = ( g ′ g ) · p for all g ′ , g ∈ G and p ∈ P with s ( g ′ ) = t ( g ) and s ( g ) = π ( p ) . We define right actions of ´etale groupoids on smooth manifolds in a similar way.
Definition 2.15.
Let G be an ´etale groupoid, and let E be a smooth complex vector bundleover G . A representation of the groupoid G on E is a smooth left action ρ : G s × pG E → E ,denoted by ρ ( g, v ) = g · v , of G on E along the bundle projection p : E → G such that forany arrow x g −→ y the induced map g ∗ : E x → E y , v g · v , is a linear isomorphism. Asection u : G → E is called G -invariant if for any arrow x g −→ y , it holds that g · u ( x ) = u ( y ) . Let us see what representations mean in the examples above.
Example 2.16. i) Representations of the unit groupoid associated to a smooth manifold correspond preciselyto complex vector bundles.ii) Representations of the point groupoid associated to a (discrete) group Γ correspond torepresentations of the group on finite dimensional complex vector spaces. iii) Representations of the translation groupoid Γ ⋉ M corresponds to Γ -equivariant complexvector bundles over M .iv) Representations of the orbifold groupoid are the orbibundles.v) Representations of the holonomy groupoid are the transversal vector bundles.vi) For an ´etale groupoid G the complexified tangent bundle of G becomes a representationof G . The cotangent bundle, exterior bundle all inherit this natural representation, soit makes sense to speak of vector fields, differential forms or Riemannian metrics etc.on ´etale groupoids (vector fields, differential forms or Riemannian metrics etc. on G ,respectively, invariant under the action). Also note that the exterior derivative d isinvariant under the G -action. This follows from naturality of d and a local argument. Proposition 2.17.
Let E be representation of the ´etale groupoid G . The space of smoothsections Γ ∞ ( E ) over G becomes a module over C ∞ c ( G ) by the formulas ( a · u )( x ) = X t ( g )= x a ( g )( g · u ( s ( g ))) , (2.28) for a ∈ C ∞ c ( G ) and u ∈ Γ ∞ ( E ) . The proof is in [Kal11]. Moreover, each module of finite type and constant rank appearsin this way, giving a version of Serre-Swan theorem. See [Con85] for an example comingfrom Sobolev spaces.2.5. ∗ -structures. We introduce ∗ -structures on Hopf algebroids which will be needed inorder to view them as symmetry objects. This is one of the main results of the presentpaper. We view the ensuing structures as the first step in defining a “compact”-type Hopfalgebroid in analogy with CQG -algebras [DK94], though we do not go in that direction here.Let ( H l , H r , S ) be a Hopf algebroid such that H , A l and A r are ∗ -algebras, s l and s r are ∗ -preserving (the involutions for H , A r and A l are denoted by the same symbol ∗ ). Assumethat ε l t r ( a ∗ ) = ( ε l s r ( a )) ∗ , ε r t l ( a ∗ ) = ( ε r s l ( a )) ∗ (2.29)hold for all a ∈ A r , a ∈ A l . Lemma 2.18.
We have h ∗ t l ( a ) ∗ ⊗ A r h ′∗ = h ∗ ⊗ A r h ′∗ s l ( a ) ∗ . (2.30) Proof.
We compute h ∗ t l ( a ) ∗ ⊗ A r h ′∗ = h ∗ s r ( ε r ( t l ( a ))) ∗ ⊗ A r h ′∗ = h ∗ s r (( ε r t l ( a )) ∗ ) ⊗ A r h ′∗ = h ∗ s r ε r s l ( a ∗ ) ⊗ A r h ′∗ = h ∗ · ε r s l ( a ∗ ) ⊗ A r h ′∗ = h ∗ ⊗ A r ε r s l ( a ∗ ) · h ′∗ = h ∗ ⊗ A r h ′∗ t r ε r s l ( a ∗ )= h ∗ ⊗ A r h ′∗ s l ( a ∗ )= h ∗ ⊗ A r h ′∗ s l ( a ) ∗ . (2.31) (cid:3) ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 11
Lemma 2.18 says that the map ( − ) ∗ ⊗ ( − ) ∗ : H l ⊗ C H l → H r ⊗ A r H r descends to anisomorphism ( − ) ∗ ⊗ ( − ) ∗ : H l ⊗ A l H l → H r ⊗ A r H r . So we can make sense of∆ r ( − ) ∗ = ( − ) ∗ ⊗ ( − ) ∗ ∆ l . (2.32)In Sweedler notation, ( h ∗ ) ⊗ ( h ∗ ) = ( h ) ∗ ⊗ ( h ) ∗ . (2.33) Definition 2.19.
Let ( H l , H r , S ) be a Hopf algebroid such that H , A l and A r are ∗ -algebraswhile s l and s r are ∗ -preserving. Then ( H l , H r , S ) is said to be a Hopf ∗ -algebroid if (2.29) and (2.32) hold. Some immediate corollaries of Definition 2.19 are:i) ( − ) ∗ ⊗ ( − ) ∗ : H l ⊗ A l H l → H r ⊗ A r H r induces an isomorphism H l × A l H l → H r × A r H r .ii) From (2.5), t l ( − ) ∗ = s r ε r t l ( − ) ∗ = s r ( − ) ∗ ε r s l = ( − ) ∗ s r ε r s l = ( − ) ∗ Ss l , with the lastequality following from (2.11).iii) Similarly, t r ( − ) ∗ = ( − ) ∗ Ss r . Proposition 2.20.
Let ( H l , H r , S ) be a Hopf ∗ -algebroid. Then the counits and the antipodesatisfy ε r S − ( − ) ∗ = ( − ) ∗ ε r , ε l S − ( − ) ∗ = ( − ) ∗ ε l , S ( − ) ∗ S ( − ) ∗ = id H (2.34) and A l becomes an H -module ∗ -algebra, i.e., the H -action satisfies ( h · a ) ∗ = S ( h ) ∗ · a ∗ h ∈ H, a ∈ A l . (2.35) Proof.
We have h ∗ = s l ε l (( h ∗ ) )( h ∗ ) = s l ε l (( h ) ∗ )( h ) ∗ , so h = h ( s l ε l (( h ) ∗ )) ∗ . Similarly, h = h ( t l ε l (( h ) ∗ )) ∗ . Now, ( − ) ∗ s l ε l ( − ) ∗ = s l ( − ) ∗ ε l ( − ) ∗ = t r ε r s l ( − ) ∗ e l ( − ) ∗ = t r ( − ) ∗ ε r t l ε l ( − ) ∗ . Similarly, ( − ) ∗ t l ε l ( − ) ∗ = s r ( − ) ∗ ε r t l ε l ( − ) ∗ . So we conclude that ( − ) ∗ ε r t l ε l ( − ) ∗ satisfies the right counit axioms. Hence ε r = ( − ) ∗ ε r t l ε l ( − ) ∗ =( − ) ∗ ε r S − ( − ) ∗ . Similarly, ε l = ( − ) ∗ ε l S − ( − ) ∗ . From this we observe that s l ε l ( − ) ∗ =( − ) ∗ t r ε r and s r ε r ( − ) ∗ = ( − ) ∗ t l ε l . Using the above observation and proceeding exactlyas before, it follows that ( − ) ∗ S − ( − ) ∗ satisfies the antipode axioms. By uniqueness, we have S = ( − ) ∗ S − ( − ) ∗ , which implies S ( − ) ∗ S ( − ) ∗ = id H . Finally, S ( h ) ∗ · a ∗ = ε l ( S ( h ) ∗ s l ( a ∗ ))= ε l ( S ( h ) ∗ s l ( a ) ∗ )= ε l ∗ ( s l ( a ) S ( h ))= ε l ∗ ( St l ( a ) S ( h ))= ε l ∗ S ( ht l ( a ))= ∗ ε l ( ht l ( a ))= ( ε l ( hs l ( a ))) ∗ = ( h · a ) ∗ . (cid:3) Besides Hopf ∗ -algebras, the Hopf algebroid in Theorem 2.12 becomes a central exampleof Hopf ∗ -algebroids: Proposition 2.21.
The space C ∞ c ( G ) becomes a Hopf ∗ -algebroid over C ∞ ( G ) with ∗ -structure given by u ∗ ( g ) = u ( g − ) for u ∈ C ∞ c ( G ) and f ∗ ( x ) = f ( x ) for f ∈ C ∞ ( G ) . (2.36) Proof.
This follows from direct computations. (cid:3)
Another class of examples, which we have not mentioned above, comes from weak Hopfalgebras studied in [BNS99]. Our ∗ -structure is the same as C ∗ -structure mentioned in[BNS99]. Following this and the standard theory of CQG -algebras, leads to opening upa new direction of study, namely, (co)representation theory of Hopf ∗ -algebroids and theinterplay of the ∗ -structure and (co)integrals.2.6. Conjugate modules.
We shall systematically use the language of conjugate modulesin order to keep track of various aspects. See [BM09, BPS13].Let ( H l , H r , S ) be a Hopf ∗ -algebroid and M an H -module. We define the conjugatemodule M by declaring thati) M = M as abelian group;ii) we write m for an element m ∈ M when we consider it as an element of M ;iii) the module operation for M is h · m = S ( h ) ∗ · m .Again, let B be a ∗ -algebra and let E be a ( B, B ) bimodule. The conjugate bimodule E is defined by the following three conditions:i) E = E as abelian group;ii) We write e for an element e ∈ E when we consider it as an element of E ;iii) The bimodule operations for E are b · e = e · b ∗ and e · b = b ∗ · e. If θ : E → F is any morphism, then we define θ : E → F by θ ( e ) = θ ( e ).We make B an associative algebra by defining the multiplication bb ′ := b ′ b . As an R -algebra, B is isomorphic to B op via the map b b . We make B a C -algebra through the ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 13 algebra homomorphism C → b , λ λ ∗ . We now define B → B , b b ∗ . Then C algebras. If θ : B → B ′ is a morphism then we say that θ is ∗ -preservingif θ = θ A l is an H -module ∗ algebra, is nothing butthe assertion that A l → A l is an H -module morphism. We also see that for an H -module M , the induced ( A l , A l )-bimodule structure matches with the prescription above. Thus our ∗ -structure naturally produces examples of “Bar categories” in the sense of [BM09]. Lemma 2.22.
Let B be an H -module ∗ -algebra, and let the invariant subalgebra B H bedefined as B H = { b ∈ B | h · b = s l ε l ( h ) · b } . Then B → B induces an isomorphism B H → B H .Proof. This follows from the fact that H -module morphism. (cid:3) In fact, we can say more:
Proposition 2.23.
Let B be an H -module ∗ -algebra. Then B H is also a ∗ -algebra. So that,by Lemma 2.22 we can identify ( B ) H = ( B H ) as algebras.Proof. Let b ∈ B H . We compute ( h · b ∗ ) ∗ = S ( h ) ∗ · b = s l ε l ( S ( h ) ∗ ) · b = s l ( ε l ( h ) ∗ ) · b = ( s l ε l ( h )) ∗ · b so that h · b ∗ = (( s l ε l ( h )) ∗ · b ) ∗ = ( S ( s l ε l ( h ) ∗ )) ∗ · b ∗ = S − s l ε l ( h ) · b ∗ = t l ε l ( h ) · b ∗ . Next observe that taking h = s l ( a ) for a ∈ A l in the last equality gives s l ( a ) · b ∗ = t l ( a ) · b ∗ .So that for all h ∈ H we get s l ε l ( h ) · b ∗ = t l ε l ( h ) · b ∗ which in turn implies that b ∗ ∈ B H . (cid:3) Noncommutative complex structures
Differential calculi.
Let H = ( H l , H r , S ) be a Hopf ∗ -algebroid. We start by defininga differential calculus. We follow the setup in [ ´B17]. Definition 3.1. An N -graded H -module is an N -graded C -vector space which is also an H -module such that the H -action preserves the N -grading. Definition 3.2. An N -graded H -module algebra is an N -graded algebra which is also an H -module algebra such that the H -action preserves the N -grading. Definition 3.3.
A pair ( B, d ) is called an H -covariant complex if B is an N -graded H -module algebra, and d is homogeneous of degree one satisfying d = 0 , such that A l and H generate H as algebra, (3.1) where H := { h ∈ H | [ h − s l ε l ( h ) , d ] = [ h − t l ε l ( h ) , d ] = 0 } . (3.2) Definition 3.4.
A triple ( B, ∂, ∂ ) is called an H -covariant double complex if B is an N -graded H -module algebra, ∂ is homogeneous of degree (1 , , and ∂ is homogeneous of degree (0 , , such that ∂ = 0 , ∂ = 0 , ∂∂ + ∂∂ = 0 and they satisfy (3.1) . For any H -covariant complex ( B, d ), we call an element d -closed if it is contained in ker ( d ) and d -exact if it is contained in im ( d ). For an H -covariant double complex ( B, ∂, ∂ ),we define ∂ -closed, ∂ -closed, ∂ -exact and ∂ -exact elements analogously. Definition 3.5. An H -covariant complex ( B, d ) is called an H -covariant differential gradedalgebra if d satisfies the graded Leibniz rule d ( bb ′ ) = d ( b ) b ′ + ( − k bd ( b ′ ) b ∈ B k , b ′ ∈ B. (3.3) Definition 3.6. An H -covariant differential calculus over an H -module algebra B (with unitmap ι B ) is an H -covariant differential graded algebra (Ω , d ) (with unit map ι Ω ) such that Ω = B , the two H -action on B coming from B itself and Ω coincide, and Ω k = span C { b db ∧ · · · ∧ db k | b , · · · , b k ∈ B } . (3.4) Notation.
We use ∧ to denote the multiplication between elements of a differential calcu-lus when both are of order greater that 0. We call an element of a differential calculus a form.Observe that the coincidence of the two H -actions on B implies that the two unit mapsalso coincide. Observe also that the induced ( A l , A l )-bimodule structure on Ω coincide withthe one coming from the unit map. Definition 3.7. An H -covariant differential calculus (Ω , d ) over an H -module ∗ -algebra B is a ∗ -differential calculus if the involution of B extends to a degree zero involutive conjugatelinear map on Ω , for which ( dω ) ∗ = d ( ω ∗ ) for all ω ∈ Ω , and ( ω ∧ η ) ∗ = ( − kl η ∗ ∧ ω ∗ , ω ∈ Ω k , η ∈ Ω l making Ω an H -module ∗ -algebra. We say that a form is real if ω ∗ = ω . Lemma 3.8.
For an H -covariant ∗ -differential calculus (Ω , d ) , we havei) [ h − s l ε l ( h ) , d ] = 0 = ⇒ [ S − ( h ∗ ) − t l ε l ( S − ( h ∗ ))] = 0 ;ii) [ h − t l ε l ( h ) , d ] = 0 = ⇒ [ S − ( h ∗ ) − s l ε l ( S − ( h ∗ ))] = 0 ;for h ∈ H . Thus combining the two, we get that h ∈ H if and only if S − ( h ∗ ) ∈ H . ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 15
Proof.
For ω ∈ Ω, we compute0 = { [ h − s l ε l ( h ) , d ]( ω ∗ ) } ∗ = { ( h − s l ε l ( h )) · d ( ω ∗ ) − d (( h − s l ε l ( h )) · ω ∗ ) } ∗ = { ( h − s l ε l ( h )) · ( dω ) ∗ } ∗ − d ( { ( h − s l ε l ( h )) · ω ∗ } ∗ )= ( S ( h − s l ε l ( h ))) ∗ · dω − d ( S ( h − s l ε l ( h )) ∗ · ω )= [( S ( h − s l ε l ( h ))) ∗ , d ]( ω ) . And similarly, 0 = { [ h − t l ε l ( h ) , d ]( ω ∗ ) } ∗ = [( S ( h − t l ε l ( h ))) ∗ , d ]( ω ). Now { S ( h − s l ε l ( h )) } ∗ = S ( h ) ∗ − S ( s l ε l ( h )) ∗ = S − ( h ∗ ) − S − ( s l ( ε l ( h ) ∗ ))= S − ( h ∗ ) − t l ε l ( S − ( h ∗ ))and { S ( h − t l ε l ( h )) } ∗ = S ( h ) ∗ − S ( t l ε l ( h )) ∗ = S − ( h ∗ ) − s l ( ε l ( h ) ∗ )= S − ( h ∗ ) − s l ε l ( S − ( h ∗ )) . Thus the lemma follows. (cid:3)
Lemma 3.9. On Ω , defining the product as ω ∧ η = ( − kl η ∧ ω for ω ∈ Ω k , η ∈ Ω l makes (Ω , d ) an H -covariant differential graded algebra. Then an H -covariant ∗ -differentialcalculus is an H -covariant differential calculus such that , d ) → (Ω , d ) is H -linear anda differential graded algebra homomorphism.Proof. The second part follows from the discussion prior to Lemma 2.22. For the first part,we observe that given ω ∈ Ω and h ∈ H ,[ h − s l ε l ( h ) , d ]( ω ) = ( h − s l ε l ( h )) · d ( ω ) − d (( h − s l ε l ( h )) · ω )= ( h − s l ε l ( h )) · dω − d (( S ( h − s l ε l ( h ))) ∗ · ω )= S ( h − s l ε l ( h )) ∗ · dω − d ( S ( h − s l ε l ( h )) ∗ · ω )= [ S ( h − s l ε l ( h )) ∗ , d ]( ω )and similarly, [ h − t l ε l ( h ) , d ]( ω ) = [ S ( h − t l ε l ( h )) ∗ , d ]( ω ). Now the lemma follows fromLemma 3.8. (cid:3) Definition 3.10.
We define the space of invariant forms Ω of Ω as Ω = { ω ∈ Ω | h · ω = s l ε l ( h ) · ω = t l ε l ( h ) · ω for all h ∈ H } . Observe that we recover the usual definition of invariant subalgebra as in Lemma 2.22 ifthe differential d is identically 0. Proposition 3.11.
For the space of invariant forms we have,i) (Ω , d | Ω ) is a differential graded algebra;ii) Ω is a ∗ -algebra;iii) d | Ω satisfies d | Ω ( ω ∗ ) = ( d | Ω ω ) ∗ for all ω ∈ Ω ;iv) , d | Ω ) → (Ω , d | Ω ) is a differential graded algebra homomorphism.Proof. i) That Ω is an algebra follows from the same proof as in d identically 0 case. More-over, that d preserves Ω follows from the definition of H . ii) Observe that for h ∈ H and ω ∈ Ω ( h · ω ∗ ) ∗ = S ( h ) ∗ · ω = S − ( h ∗ ) · ω = t l ε l ( S − ( h ∗ )) · ω = t l ( ε l ( h ) ∗ ) · ω = ( Ss l ε l ( h )) ∗ · ω so that h · ω ∗ = (( Ss l ε l ( h )) ∗ · ω ) ∗ = s l ε l ( h ) · ω ∗ . Again ( h · ω ∗ ) ∗ = S ( h ) ∗ · ω = S − ( h ∗ ) · ω = s l ε l ( S − ( h ∗ )) · ω = s l ( ε l ( h ) ∗ ) · ω = ( s l ε l ( h )) ∗ · ω so that h · ω ∗ = (( s l ε l ( h )) ∗ · ω ) ∗ = S − s l ε l ( h ) · ω ∗ = t l ε l ( h ) · ω ∗ . iii) holds because d satisfies the property.iv) Follows from ii). (cid:3) We shall denote the differential on Ω only by d , assuming that it really means d isrestricted to Ω . Now we come to our example. According to Haefliger [Kor08]: Definition 3.12.
A transverse structure on a foliated manifold ( M, F ) is a structure on thetransversal manifold N , invariant under the action of the holonomy pseudogroup P . Since the groupoid Γ( P ) is constructed out of P , it follows that P invariant structuresare Γ( P ) invariant. The normal bundle N ( M, F ) of the foliation F is isomorphic to thetangent bundle T N of N . Thus, basic forms on the foliated manifold ( M, F ) are in bijectivecorrespondence with Γ( P )-invariant forms on the transverse manifold N (see [Kor08]). Tosee what does Γ( P ) invariant forms correspond to, we introduce the following. Definition 3.13.
A local bisection of a Lie groupoid G is a local section σ : U → G of s : G → G defined on an open subset U ⊂ G such that tσ is an open embedding. If G is ´etale, any arrow g induces a germ of a homeomorphism σ g : ( U, s ( g )) → ( V, t ( g ))from a neighborhood U of s ( g ) to a neighborhood V of t ( g ) as follows: choosing U smallenough such that a bisection σ exists and t | σU is a homeomorphism into V := t ( σU ), we set σ g := tσ . We do not distinguish between σ g and the actual germ of this map at the point s ( g ). Lemma 3.14.
Let G be an ´etale groupoid, and let E be a smooth complex vector bundle over G with a G -representation. Then a section u : G → E is G -invariant if and only if it is C ∞ c ( G ) -invariant.Proof. Recall that a section u of the bundle E is G invariant, if g · u ( x ) = u ( y ) for all arrow x g −→ y , while u is C ∞ c ( G ) invariant if a · u = ε l ( a ) u for all a ∈ C ∞ c ( G ). That G -invariance ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 17 implies C ∞ c ( G )-invariance is clear. For the converse, pick an arrow x g −→ y and a bisection( U, σ ) such that g ∈ σ ( U ) [MM03]. Then choose any function a ∈ C ∞ c ( G ) with support in σ ( U ). Note that on a bisection U , we have a ( t | U ) − = ε l ( a ) and a · u = a ( t | U ) − u = ε l ( a ) u .Hence the lemma follows. (cid:3) Now take B = C ∞ ( G ) and Ω = Ω( G ), the C -valued smooth functions and forms on G ,respectively. Lemma 3.15.
The differential d on G satisfies d ( a · ω ) = d ( ε l ( a )) ∧ ω + a · d ( ω ) (3.5) for a ∈ C ∞ c ( G ) and ω ∈ Ω( G ) . Hence [ a − ε l ( a ) , d ] = 0 for all a ∈ C ∞ c ( G ) , thus implying H = C ∞ c ( G ) (see (3.2) for H ).Proof. As observed above in the proof of Lemma 3.14, on a bisection U , we have a ( t | U ) − = ε l ( a ) and a · u = a ( t | U ) − u = ε l ( a ) u . Now (3.5) follows from Leibniz rule and locality of d .The last statement follows from (3.5) and the fact that s l ≡ t l . (cid:3) Denote by Ω( G ) G the G -invariant forms. Then forms on the “orbit or leaf space” arecaptured as follows. Proposition 3.16.
The pair (Ω( G ) , d ) is a C ∞ c ( G ) -covariant differential calculus, and wehave (Ω( G ) G , d ) = (Ω( G ) C ∞ c ( G ) , d ) as differential graded algebras.Proof. Since G acts by local diffeomorphisms, it follows that d is G -invariant. So d descendsto Ω( G ) G . The proposition now follows from Lemma 3.14 and Lemma 3.15. (cid:3) Definition 3.17. i) We say that an H -covariant differential calculus (Ω , d ) over an H -module algebra B hastotal dimension n if Ω k = 0 , for all k > n , and Ω n = 0 .ii) If in addition, there exists a ( B, B ) -bimodule and an H -module isomorphism vol : Ω n → B , then we say that Ω is orientable.iii) If Ω is a ∗ -calculus over a ∗ -algebra, then a ∗ -orientation is an orientation which is also ∗ -preserving, meaning vol .iv) A ∗ -orientable calculus is one which admits a ∗ -orientation.v) Let τ be a state on B , i.e., a unital linear functional τ : B → C such that τ ( b ∗ b ) ≥ .We call the functional τ ◦ vol the integral associated to τ and denote it by R τ .vi) We say that the integral is closed if R τ ( dω ) = 0 for all ω ∈ Ω n − . Definition 3.18.
An ´etale groupoid G is oriented if G is oriented in the ordinary senseand G acts by orientation preserving local diffeomorphisms. Proposition 3.19.
With B = C ∞ ( G ) and Ω = Ω( G ) , orientation in the sense of Defini-tion 3.17 coincide with groupoid orientation on G .Proof. This follows from Proposition 3.16. (cid:3)
Lemma 3.20.
Assume that (Ω , d ) is ∗ -oriented with orientation vol and of total dimension n . Then (Ω , d ) is ∗ -oriented. Proof.
Since vol is assumed to be H -linear, it restricts to Ω , which in turn shows thatΩ nH = 0 so that it also has total dimension 2 n . The lemma now follows from Lemma 2.22and Proposition 3.11. (cid:3) Complex structures.
The setup below is due to [ ´B17] and we follow it closely. Weshall omit the proofs of some of the results here as they are essentially given in [ ´B17].
Definition 3.21. An H -covariant almost complex structure for an H -covariant ∗ -differentialcalculus (Ω , d ) over an H -module ∗ -algebra B is an N -algebra grading ⊕ ( k,l ) ∈ N Ω ( k,l ) for Ω such thati) the H -action preserves the N -grading;ii) Ω n = ⊕ k + l = n Ω ( k,l ) , for all n ∈ N ;iii) → Ω preserves the N -grading, where the N -grading on Ω is given by Ω ( k,l ) =Ω ( l,k ) . Let ∂ and ∂ be the unique homogeneous operators of order (1 ,
0) and (0 ,
1) respectively,defined by ∂ | Ω ( k,l ) = proj Ω ( k +1 ,l ) ◦ d ∂ | Ω ( k,l ) = proj Ω ( k,l +1) ◦ d, (3.6)where proj Ω ( k,l +1) and proj Ω ( k,l +1) are the projections from Ω ( k + l +1) onto Ω ( k +1 ,l ) and Ω ( k,l +1) ,respectively.As in [ ´B17], we have: Lemma 3.22. If ⊕ ( k,l ) ∈ N Ω ( k,l ) is an H -covariant almost complex structure for an H -covariant ∗ -differential calculus (Ω , d ) over an H -module ∗ -algebra B , then the following two conditionsare equivalent:i) d = ∂ + ∂ ;ii) the triple ( ⊕ ( k,l ) ∈ N Ω ( k,l ) , ∂, ∂ ) is an H-covariant double complex.Proof. The proof of the equivalence is in [ ´B17]. All we have to show is the H -covariant partin ii). Observe that proj Ω ( k +1 ,l ) and proj Ω ( k,l +1) are H -linear. Then for h ∈ H ,[ h − s l ε l ( h ) , ∂ | Ω ( k,l ) ] = [ h − t l ε l ( h ) , ∂ | Ω ( k,l ) ] = 0 . Thus we get (3.1) for ∂ | Ω ( k,l ) , and similarly for ∂ | Ω ( k,l ) , hence the covariance. (cid:3) Definition 3.23.
When the conditions in Lemma 3.22 hold for an almost complex structure,then we say that the almost complex structure is integrable.
We also call an integrable almost complex structure a complex structure and the doublecomplex ( ⊕ ( k,l ) ∈ N Ω ( k,l ) , ∂, ∂ ) its Dolbeault double complex. Note that ∂ ( ω ∗ ) = ( ∂ω ) ∗ , ∂ ( ω ∗ ) = ( ∂ω ) ∗ , ω ∈ Ω , (3.7)as they follow from the integrability condition. Lemma 3.24.
Suppose that (Ω , d ) admits an H -covariant complex structure. Then (Ω , d ) admits a complex structure. We call this a transverse complex structure on B . ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 19
Remark 3.25.
Strictly speaking, we haven’t defined what complex structure (or any otherstructures) means on an algebra without any equivariance. The idea is to forget the “ H -covariant” part and take the rest as the corresponding definition. In the present situation, acomplex structure is a bigrading that satisfies Conditions i) and ii) in Definition 3.21 with d = ∂ + ∂ .Proof of Lemma 3.24. Condition i) in Definition 3.21 implies that (Ω , d ) admits an N -algebra grading by (Ω ) ( k,l ) = Ω ( k,l )0 , ( k, l ) ∈ N . Condition ii) follows automatically, whileCondition iii) follows from that fact that H -linear. ∂ and ∂ restrict to the space ofinvariant forms as in Proposition 3.11. Finally, d = ∂ + ∂ then follows automatically. (cid:3) As in [CW91], we define:
Definition 3.26.
The foliation F on a foliated manifold ( M, F ) is transversely holomorphicif it carries a transverse complex structure in the sense of Definition 3.12. If the foliation F is transversely holomorphic, the normal bundle N ( M, F ) of F has acomplex structure corresponding to the complex structure on N . Therefore any complexvalued basic k -form can be represented as a sum of the k -forms of pure type ( r, s ) corre-sponding to the decomposition of k -forms on the complex manifold N . Let Ω k C ( M, F ) denotethe space of complex valued basic k -forms on the foliated manifold ( M, F ), and denote byΩ ( r,s ) C ( M, F ) the space of complex valued basic forms of pure type ( r, s ). Then we haveΩ k C ( M, F ) = ⊕ r + s = k Ω ( r,s ) C ( M, F ). The exterior derivative d : Ω k C ( M, F ) → Ω k +1 C ( M, F ) de-composes into two components d = ∂ + ∂ , where ∂ is of bidegree (1 ,
0) and ∂ is of bidegree(0 , ∂ : Ω ( r,s ) → Ω ( r +1 ,s ) and ∂ : Ω ( r,s ) → Ω ( r,s +1) .Keeping in mind Definition 3.26 and the case for orbifolds (see [BBF + Definition 3.27.
An ´etale groupoid G is holomorphic if G is a complex manifold and G acts by local biholomorphic transformations. This fits into our framework as follows:
Proposition 3.28.
An ´etale groupoid G is holomorphic if and only if (Ω( G ) , d ) admits a C ∞ c ( G ) -covariant complex structure.Proof. First observe that an almost complex structure on G is also given by a bundlemap J : T ∗ ( G ) → T ∗ ( G ) (and its extension to the exterior algebra bundle) such that J ◦ J = − Id T ∗ ( G ) . The bidegree decomposition is a consequence of this fact. Since bundlemaps are sections of the HOM-bundle, G is almost complex if and only if (Ω( G ) , d ) admitsa C ∞ c ( G )-covariant almost complex structure, by Lemma 3.14. Since integrability is same inboth sense, we have the proposition proved. (cid:3) The orbit space inherits a complex structure:
Corollary 3.29. If G is holomorphic, then (Ω( G ) G , d ) admits a complex structure.Proof. This follows from Proposition 3.28 and Lemma 3.24. (cid:3) Hermitian and K¨ahler structures
We fix an H -covariant ∗ -differential calculus (Ω , d ) over an H -module ∗ -algebra B of totaldimension 2 n .As in [ ´B17], the following is a non-commutative generalization of an almost symplecticform. Definition 4.1.
An almost symplectic form for Ω is a central real H -invariant 2-form σ ( h · σ = s l ε l ( h ) · σ for all h ∈ H ) such that, the Lefschetz operator L : Ω → Ω , ω σ ∧ ω satisfies the following condition: the maps L n − k : Ω k → Ω n − k (4.1) are isomorphisms for all ≤ k < n . Since σ is a central real form, L is a ( B, B )-bimodule morphism and ∗ -preserving ( L L ). Moreover, the H -invariance condition implies that L is also an H -module morphism.Indeed, we have h · ( σ ∧ ω ) = h · σ ∧ h · ω = ι B ( ε l ( h )) σ ∧ h · ω = σ ∧ ι B ( ε l ( h ))( h · ω )) = σ ∧ ( s l ε l ( h ) h ) · ω = σ ∧ h · ω . Definition 4.2.
A symplectic form is a d -closed almost symplectic form. Buachalla, [ ´B17], introduced Hermitian structure which is a symplectic form compatiblewith a complex structure.
Definition 4.3.
An Hermitian structure for Ω is a pair (Ω ( · , · ) , σ ) , where Ω ( · , · ) is an H -covariant complex structure, and σ is an almost symplectic form, called the Hermitian form,such that σ ∈ Ω (1 , . We have:
Lemma 4.4.
Suppose that (Ω ( · , · ) , σ ) is an Hermitian structure for (Ω , d ) . Then σ inducesa Hermitian structure on (Ω , d ) .Proof. By definition, σ ∈ Ω . The H -linearity of L shows that σ is an almost symplecticform for (Ω , d ). Finally, σ ∈ (Ω (1 , ) = (Ω ) (1 , , by Lemma 3.24. (cid:3) We say that an almost complex structure is of diamond type if Ω ( a,b ) = 0 whenever a > n or b > n . Supposing a > n and observing that the isomorphism L a + b − n maps Ω n − b,n − a ontoΩ ( a,b ) , we see that the existence of an Hermitian structure implies that the complex structurehas to be of diamond type. Definition 4.5.
The Hodge map associated to an Hermitian structure is the morphismuniquely defined by ⋆ ( L j ( ω )) = ( − k ( k +1)2 i a − b [ j ]![ n − j − k ]! L n − j − k ( ω ) ω ∈ P ( a,b ) ⊂ P k . (4.2)Observe that ⋆ is an H -module morphism. Hence it descends to Ω . ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 21
Lemma 4.6.
We havei) ⋆ ( ω ) = ( − k ω for all ω ∈ Ω k ,ii) ⋆ is an isomorphism,iii) ⋆ (Ω ( a,b ) ) = Ω ( n − b,n − a ) ,iv) ⋆ is a ∗ -preserving. Given an Hermitian structure (Ω ( · , · ) , κ ), we first recover the Hermitian metric associatedto it: Definition 4.7.
Define g : Ω ⊗ B Ω → B by g ( ω ⊗ η ) = 0 for ω ∈ Ω k , η ∈ Ω l , k = l , and g ( ω ⊗ η ) = vol( ω ∧ ∗ ( η ∗ )) (4.3)for ω, η ∈ Ω k .A metric on the orbit space should be an invariant one as is showed in the following lemma. Lemma 4.8.
For ω, η ∈ Ω k and h ∈ H , it holds that g ( h · ω ⊗ h · η ) = h · g ( ω ⊗ η ) , (4.4) so that g is H -covariant.Proof. We compute g ( h · ω ⊗ h · η ) = g ( h · ω ⊗ S ( h ) ∗ · η )= vol( h · ω ∧ ∗ ( S ( h ) ∗ · η ) ∗ )= vol( h · ω ∧ ∗ (( S ( S ( h ) ∗ )) ∗ ) · η ∗ )= vol( h · ω ∧ ∗ ( h · η ∗ )= vol( h · ω ∧ h · ∗ ( η ∗ ))= vol( h · ( ω ∧ η ∗ ))= h · vol( ω ∧ ∗ ( η ∗ ))= h · g ( ω ⊗ η ) , (cid:3) Proposition 4.9.
The following decompositions are orthogonal with respect to h , i :i) The degree decomposition Ω = ⊕ k Ω k ;ii) The bidegree decomposition Ω k = ⊕ ( a,b ) Ω ( a,b ) ;iii) The Lefschetz decomposition Ω k = ⊕ j ≥ L j ( P k − j ) . Proposition 4.9 immediately implies the following.
Corollary 4.10.
We have g ( ω ⊗ η ) = g ( η ⊗ ω ) ∗ for ω, η ∈ Ω . We recall from [MM03]:
Definition 4.11.
The foliation F on a foliated manifold ( M, F ) is transversely Riemannianif it carries a transverse Riemannian structure in the sense of Definition 3.12. The metric on N ( M, F ) is induced from a bundle-like metric on M . Recall from [CW91]: Definition 4.12.
The foliation F on a foliated manifold ( M, F ) is transversely Hermitianif it carries a transverse Hermitian structure in the sense of Definition 3.12. The operator ⋆ : Λ k ( M, F ) → Λ q − k ( M, F ) defined via the transverse part of the bundle-like metric of F extends to Λ k C ( M, F ) → Λ q − k C ( M, F ), where q is the complex codimensionof F .Being motivated by this, we make the following definition. Definition 4.13.
An ´etale groupoid G is Hermitian if G admits a G -invariant Hermitianstructure. Again, algebraically we have the following proposition.
Proposition 4.14.
An ´etale groupoid G is Hermitian if and only if (Ω( G ) , d ) admits a C ∞ c ( G ) -covariant Hermitian structure.Proof. The proof of the statement that G is Hermitian implies that (Ω( G ) , d ) admits a C ∞ c ( G )-invariant Hermitian structure is straightforward. For the converse, we recover theHermitian metric as in Definition 4.7, and Lemma 4.8 shows that it is G -invariant. Com-patibility follows from Proposition 4.9. (cid:3) Corollary 4.15. If G is Hermitian, then (Ω( G ) G , d ) admits a Hermitian structure.Proof. This follows from Proposition 4.14 and Lemma 4.4. (cid:3)
The Hermitian structure is said to be positive definite if g ( ω ⊗ ω ) > ω ∈ Ω. In that case, we define an inner product (positive definite, Hermitian) on Ω bysetting h ω, η i = τ g ( ω ⊗ η ) = Z τ ω ∧ ⋆ ( η ∗ ) (4.5)for ω, η ∈ Ω and a fixed faithful state τ on B . We denote the corresponding norm of ω by k ω k . Moreover, Lemma 4.8 shows that g induces a metric on Ω that takes values in B .Applying τ , we get an inner-product on Ω which is really the restriction of h· , ·i to Ω . Fromnow on, we assume that the Hermitian structure to be positive definite. Proposition 4.16.
The Hodge map ⋆ is unitary. We now define the Laplacians.
Definition 4.17. i) The codifferential is defined as d ∗ := − ⋆ d⋆ ;ii) the holomorphic codifferential is defined as ∂ ∗ := − ⋆ ∂⋆ ;iii) the anti-holomorphic codifferential is defined as ∂ ∗ = − ⋆ ∂⋆ . Observe that for ω ∈ Ω, d ∗ ( ω ∗ ) = ( d ∗ ω ) ∗ , ∂ ∗ ( ω ∗ ) = ( ∂ ∗ ω ) ∗ and ∂ ∗ ( ω ∗ ) = ( ∂ ∗ ω ) ∗ . (4.6)Now, it is natural to define the d -, ∂ - and ∂ - Laplacians, respectively as∆ d := ( d + d ∗ ) , ∆ ∂ := ( ∂ + ∂ ∗ ) , ∆ ∂ := ( ∂ + ∂ ∗ ) . (4.7) ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 23
Proposition 4.18.
The operator adjoints of d , ∂ and ∂ are d ∗ , ∂ ∗ and ∂ ∗ , respectively. The following will be used later.
Corollary 4.19.
The Laplacians ∆ d , ∆ ∂ and ∆ ∂ are symmetric. We have:
Lemma 4.20.
The operator d ∗ (respectively ∂ , ∂ ) and hence ∆ d (respectively ∆ ∂ , ∆ ∂ )descends to Ω .Proof. Since ⋆ is H -linear, we have for h ∈ H ,[ h − s l ε l ( h ) , d ∗ ] = [ h − t l ε l ( h ) , d ∗ ] = 0 . Hence d ∗ descends to Ω . (cid:3) Given the Laplacians ∆ d , ∆ ∂ and ∆ ∂ , we define the d -harmonic, ∂ -harmonic and ∂ -harmonic forms to be, respectively H d := ker(∆ d ) , H ∂ := ker(∆ ∂ ) , H ∂ := ker(∆ ∂ ) . (4.8) Proposition 4.21.
We havei) ∆ d ω = 0 if and only if dω = 0 and d ∗ ω = 0 ;ii) ∆ ∂ ω = 0 if and only if ∂ω = 0 and ∂ ∗ ω = 0 ;iii) ∆ ∂ ω = 0 if and only if ∂ω = 0 and ∂ ∗ ω = 0 .Proof. We only prove i), the other proofs being similar. Clearly, ∆ d ω = 0 if dω = 0 and d ∗ ω = 0. Now h ∆ d ω, ω i = k dω k + k d ∗ ω k . Thus if ∆ d ω = 0, then the both terms on right-hand side must vanish, i.e., dω = 0 and d ∗ ω = 0. (cid:3) According to [ ´B17], K¨ahler structures are defined as follows.
Definition 4.22.
A K¨ahler structure is an Hermitian structure (Ω ( · , · ) , κ ) such that the Her-mitian form κ is d -closed. Such a form is called a K¨ahler form. Proposition 4.23.
We havei) H k∂ = ⊕ a + b = k H ( a,b ) ∂ and H k∂ = ⊕ a + b = k H ( a,b ) ∂ , where H ( a,b ) ∂ = { ω ∈ Ω ( a,b ) | ∆ ∂ ω = 0 } . Similarly, define H ( a,b ) ∂ ;ii) if the Hermitian structure is K¨ahler, then both decompositions coincide with H kd = ⊕ a + b = k H ( a,b ) d . In particular, H kd = H k∂ = H ∂ k . The proof in [ ´B17] does not use equivariance. Hence the above proposition also holds for(Ω , d ). Proposition 4.24.
The Hodge map ⋆ and the map α α ∗ commute with the Laplacian ∆ d . Hence, in the K¨ahler case, they also commute with ∆ ∂ and ∆ ∂ . Lemma 4.25.
A K¨ahler structure (Ω ( · , · ) , κ ) on (Ω , d ) induces via κ a K¨ahler structure on (Ω , d ) .Proof. Since κ is automatically d | Ω -closed, the lemma follows from Lemma 4.4. (cid:3) Following [CW91], we have:
Definition 4.26.
The foliation F on a foliated manifold ( M, F ) is transversely K¨ahler if itcarries a transverse K¨ahler structure in the sense of Definition 3.12. The K¨ahler form of N defines a basic (1 , M, F ) which is called the transverseK¨ahler form of the foliation F . Motivated by this and the case for orbifolds, we define: Definition 4.27.
An ´etale groupoid G is K¨ahler if G admits a G -invariant K¨ahler structure. The following is routine:
Proposition 4.28.
An ´etale groupoid G is K¨ahler if and only if (Ω( G o ) , d ) admits a C ∞ c ( G ) -covariant K¨ahler structure.Proof. This follows from Proposition 4.14 and Proposition 3.16. (cid:3)
Corollary 4.29. If G is K¨ahler, then (Ω( G ) G , d ) admits a K¨ahler structure.Proof. This follows from Proposition 4.28 and Lemma 4.25. (cid:3)
Theorem 4.30.
The following relations hold: ∂∂ ∗ + ∂ ∗ ∂ = 0 , ∂ ∗ ∂ + ∂∂ ∗ = 0 , ∆ d = 2∆ ∂ = 2∆ ∂ . (4.9)5. The Hodge decomposition
In this section, we prove the Hodge decomposition theorem. We remark that in [ ´B17], thecosemisimplicity is used to prove the theorem for quantum homogeneous spaces. What weprove below corresponds to, classically, Hodge decomposition for G . Ideally, one should useonly the compactness for G without any equivariance. This is what we do. To descend tothe space of invariant forms, we need something more. More about it below (see Definition5.7). Following [War83], we make the following definition. Definition 5.1.
For η ∈ Ω k , a weak solution to ∆ d ( ω ) = η is a bounded linear functional l : Ω k → C such that l (∆ d ( φ )) = h η, φ i , for all φ ∈ Ω k . (5.1)The next definition is equivalent to the ellipticity of the Laplacian in the classical situation. Definition 5.2.
The Hermitian structure is said to be d -regular if the following are satisfied:i) Let η ∈ Ω k , and let l be a weak solution of ∆ d ( ω ) = η . Then there exists an element ω ∈ Ω k such that l ( ν ) = h ω, ν i for every ν ∈ Ω k .ii) For a sequence { η n } in Ω k such that k η n k ≤ c and k ∆ d ( η n ) k ≤ c for all n and for someconstant c > , there exists a Cauchy subsequence of { η n } in Ω k . ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 25
A sufficient condition for regularity is provided in Theorem 7.2. We now show that, as inthe classical situation, regularity is sufficient for the decomposition to hold.
Theorem 5.3.
Assume that the Hermitian structure is regular. Then for each k with ≤ k ≤ n , the space H kd of d -harmonic forms is finite dimensional and we have the followingorthogonal direct sum decomposition of Ω k called the Hodge decomposition: Ω k = ∆ d (Ω k ) ⊕ H kd = ( dd ∗ ⊕ d ∗ d )(Ω k ) ⊕ H kd = d (Ω k − ) ⊕ d ∗ (Ω k +1 ) ⊕ H kd . (5.2) Proof.
We closely follow [War83]. If H kd were not finite dimensional, then H kd would containan infinite orthonormal sequence. But by condition ii) in Definition 5.2, this orthonormalsequence would contain a Cauchy subsequence, which is impossible. Thus H kd is finite di-mensional.Observe that it is sufficient to prove the first equality.Let ω , · · · , ω l be an orthonormal basis of H kd . Then an arbitrary form η ∈ Ω k can uniquelybe written as η = ν + l X i =1 h η, ω i i ω i , (5.3)where ν lies in ( H kd ) ⊥ . Thus we have an orthogonal direct sum decompositionΩ k = ( H kd ) ⊥ ⊕ H kd . (5.4)The theorem will be proved by showing that ( H kd ) ⊥ = ∆ d (Ω k ). We let P denote the projec-tion operator of Ω k onto H kd so that P ( η ) is the harmonic part of η .It can be shown that ∆ d (Ω k ) ⊂ ( H kd ) ⊥ . Indeed, if ω ∈ Ω k and η ∈ H kd , then h ∆ d ( ω ) , η i = h ω, ∆ d ( η ) = 0 . Conversely, we claim that ( H kd ) ⊥ ⊂ ∆ d (Ω k ) . (5.5)In order to prove (5.5), we first need the following inequality.We claim that there is a constant c > k η k ≤ c k ∆ d k for all η ∈ ( H kd ) ⊥ . (5.6)Suppose the contrary. Then there exists a sequence η j ∈ ( H kd ) ⊥ with k η j k = 1 and k ∆ d ( η j ) k →
0. By condition ii) in Definition 5.2, a subsequence of the η j , which for con-venience we can assume to be { η j } itself, is Cauchy. Thus lim j →∞ h η j , ψ i exists for each ψ ∈ Ω k . We define a linear functional l on Ω k be setting l ( ψ ) = lim j →∞ h η j , ψ i for ψ ∈ Ω k . (5.7) Now l is clearly bounded, and l (∆ d ( φ )) = lim j →∞ h η j , ∆ d ( φ ) i = lim j →∞ h ∆ d ( η j ) , φ i = 0 , (5.8)so l is weak solution of ∆ d ( η ) = 0. By condition i) in Definition 5.2, there exists η ∈ Ω k such that l ( ψ ) = h η, ψ i . Consequently, η j → η . Since k η j k = 1 and η j ∈ ( H kd ) ⊥ , it followsthat k η k = 1 and ( H kd ) ⊥ . But ∆ d ( η ) = 0, so η ∈ H kd , which is a contradiction. Thus theinequality in (5.6) is proved.Now we shall use (5.6) to prove (5.5). Let η ∈ ( H kd ) ⊥ . We define a linear functional l on∆ d (Ω k ) by setting l (∆ d ( φ )) = h η, φ i for all φ ∈ Ω k . (5.9)This l is well-defined; for if ∆ d ( φ ) = ∆ d ( φ ), then φ − φ ∈ H kd , so that h η, φ − φ i = 0.Also l is a bounded linear functional on ∆ d (Ω k ), for let φ ∈ Ω k and let ψ = φ − P ( φ ). Thenusing the above inequality, we obtain that | l (∆ d ( φ )) | = | l (∆ d ( φ )) | = | h η, ψ i | ≤ k η kk ψ k≤ c k η kk ∆ d ( ψ ) k = c k η kk ∆ d ( φ ) k . (5.10)By the Hahn-Banach theorem, l extends to a bounded linear functional on Ω k . Thus l is aweak solution of ∆ d ( ω ) = η . By condition i) in Definition 5.2, there exists ω ∈ Ω k such that∆ d ( ω ) = η . Hence (5.5) is proved. Consequently, we have( H kd ) ⊥ = ∆ d (Ω k ) , (5.11)and the Hodge decomposition is proved. (cid:3) Similarly, ∂ -regularity and ∂ -regularity lead to Hodge decompositions for ∆ ∂ and ∆ ∂ , withfinite dimensional harmonic spaces H ( a,b ) ∂ , H ( a,b ) ∂ , respectively. Moreover, if the Hermitianstructure is K¨ahler, then d -regularity coincide with ∂ -regularity and ∂ -regularity. In thissituation, H ( a,b ) ∂ = H ( a,b ) ∂ .From now on, we assume d -, ∂ - and ∂ -regularity. Corollary 5.4.
We have ker( d ) = H d ⊕ d (Ω) , ker( ∂ ) = H ∂ ⊕ ∂ (Ω) , ker( ∂ ) = H ∂ ⊕ ∂ (Ω) , (5.12) and H kd = H kd , H ( a,b ) ∂ = H ( a,b ) ∂ H ( a,b ) ∂ = H ( a,b ) ∂ . (5.13) where H kd is the k -th cohomology of (Ω , d ) , H ( a,b ) ∂ is the a -th cohomology of (Ω ( · ,b ) , ∂ ) and H ( a,b ) ∂ is the b -th cohomology of (Ω ( a, · ) , ∂ ) . Corollary 5.5.
Let (Ω ( · , · ) , κ ) be K¨ahler. Then for a d -closed form ω of type ( a, b ) , thefollowing conditions are equivalent:i) The form ω is d -exact;ii) the form ω is ∂ -exact;iii) the form ω is ∂ -exact;iv) the form ω is ∂∂ -exact. ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 27
Proof.
We add another equivalent condition: v ) The form ω is orthogonal to H ( a,b ) . TheK¨ahler condition says that we don’t have to specify with respect to which differential operator( d , ∂ or ∂ ) harmonicity is considered.Using Hodge decomposition, we see that v ) is implied by any of the other conditions.Moreover, iv ) implies i )- iii ). Thus it suffices to show that v ) implies iv ).If ω ∈ Ω ( a,b ) is d -closed (and thus ∂ -closed) and orthogonal to the space of harmonic forms,then Hodge decomposition with respect to ∂ yields that ω = ∂ ( η ). Now applying Hodgedecomposition with respect to ∂ to the form η yields that η = ∂ ( ν ) + ∂ ∗ ( ν ′ ) + ν ′′ for some harmonic ν ′′ . Thus ω = ∂∂ ( ν ) + ∂∂ ∗ ( ν ′ ). Using ∂∂ ∗ = − ∂ ∗ ∂ and ∂ ( ω ) = 0we conclude ∂∂ ∗ ∂ ( ν ′ ) = 0. Since h ∂∂ ∗ ∂ ( ν ′ ) , ∂ ( ν ′ ) i = k ∂ ∗ ∂ ( n ′ ) k , it follows that ∂∂ ∗ ( ν ′ ) = − ∂ ∗ ∂ ( ν ′ ) = 0. Therefore, ω = ∂∂ ( ν ). (cid:3) Corollary 5.6.
Let (Ω ( · , · ) , κ ) be K¨ahler. Then there exists a decomposition H kd = ⊕ a + b = k H ( a,b ) ∂ = ⊕ a + b = k H ( a,b ) ∂ . (5.14) The decomposition does not depend on the chosen K¨ahler structure.
For foliated manifolds there are different ways of proving the decomposition; see for ex-ample [CW91, EKA90, PR96]. To use averaging as in [ ´B17], it turns out that the correctgeneralization of compact lie groups are proper ´etale groupoids. For proper ´etale groupoids,there are Haar systems and cut off functions, by which one can average sections to makethem invariant; see for example [Har15]. Motivated by this, we make the following definition.
Definition 5.7.
We say that H acts on (Ω , d ) properly (or (Ω , d ) is a proper H -module)if there is a graded C -linear morphism π : Ω → Ω which is a self-adjoint idempotent withrange Ω . So we are actually capturing orbit spaces for proper ´etale groupoids or orbifolds. Note thatif the Hopf algebroid is assumed to be semisimple, i.e., there is an integral (see [B¨oh09]), thenit acts properly on any module. See Proposition 7.3 for a sufficient condition (or rather theactual projection, the algebraisation of which is the above definition) for such a projectionto exist.
Corollary 5.8.
For a d -regular Hermitian structure on (Ω , d ) which is also a proper H -module, any ω ∈ Ω k can be written as ω = ∆ d ( η ) + ν , (5.15) where η ∈ Ω k and ν ∈ H kd ∩ Ω k . Hence Hodge decomposition hold for (Ω , d ) under d -regularity. Corollary 5.8 implies that the same proof as in Corollary 5.5 goes through and implies ananalogue of Corollary 5.5 for (Ω , d ) under the properness assumption. Proof of Corollary 5.8.
The result follows from Hodge decomposition once we show that ∆ d commutes with π . Now let ω ∈ Ω. Then h η, ∆ d π ( ω ) i = h ∆ d ( η ) , π ( ω ) i = h π ∆ d ( η ) , ω i = h ∆ d ( η ) , ω i = h η, ∆ d ( ω ) i = h π ( η ) , ∆ d ( ω ) i = h η, π ∆ d ( ω ) i for all η ∈ Ω . Hence ∆ d π ( ω ) = π ∆ d ( ω ). Here we use that ∆ d is self-adjoint and it preservesΩ . (cid:3) Formality
In this section we prove an analogue of the classical result that says compact K¨ahlermanifolds are formal. For foliated manifolds this was shown in [EKA90, CW91] and fororbifolds in [BBF + Definition 6.1.
Two differential graded algebras ( X, d X ) and ( Y, d Y ) are equivalent if thereexists a sequence of differential graded algebra quasi-isomorphisms ( C , d C ) · · · ( C n , d C n )( X, d X ) ( C , d C ) · · · ( Y, d Y ) . (6.1) Definition 6.2.
A differential grade algebra ( X, d X ) is called formal if ( X, d X ) is equivalentto a differential graded algebra ( Y, d Y ) with d Y = 0 . We note that (
X, d X ) is formal if and only if ( X, d X ) is equivalent to its cohomology dif-ferential graded algebra ( H · ( X, d X ) , d = 0).Now in our setup, suppose that (Ω , d ) admits an H -covariant complex structure. Introducethe operator d c : Ω k → Ω k +1 defined as d c = −√− ∂ − ∂ ). Lemma 3.22 then implies that dd c = − d c d = 2 √− ∂∂ and ( d c ) = 0. By Lemma 3.24, we see that d c descends to Ω . Weprove below an analogue of the dd c -lemma in the classical situation. Lemma 6.3.
Suppose (Ω , d ) admits a d -regular K¨ahler (d-regular Hermitian which is K¨ahler)structure. Let ω ∈ Ω k be a d c -exact and d -closed form. Then there exists a form η ∈ Ω k − with ω = dd c ( η ) . The same holds for (Ω , d ) if (Ω , d ) is a proper H -module.Proof. We write ω = d c ( φ ) and consider the Hodge decomposition φ = d ( η ) + ν + d ∗ ( ψ ).The property of being K¨ahler implies that the harmonic part ν is also ∂ -closed and ∂ -closed.Hence d c ( φ ) = d c d ( η ) + d c d ∗ ( ψ ).It suffices to show d c d ∗ ( ψ ) = 0. We now use 0 = d ( ω ) = dd c d ∗ ( ψ ) and d c d ∗ = − d ∗ d c as inthe proof of Corollary 5.5. Hence,0 = h dd ∗ d c ( ψ ) , d c ( ψ ) i = k d ∗ d c ( ψ ) k ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 29 and thus d c d ∗ ( ψ ) = − d ∗ d c ( ψ ) = 0.Now for the last statement, we observe that because of Corollary 5.8 and Lemma 4.25,the same proof as above gives the dd c -lemma for (Ω , d ). (cid:3) Lemma 6.3 implies the following: if ω ∈ Ω k is a d c -closed and d -exact form for a d -regular K¨ahler structure on (Ω , d ), then ω = d c d ( η ) for some η ∈ Ω k − . To see this, oneintroduces the operator I : Ω → Ω defined by I( ω ) = P a,b i a − b proj Ω ( a,b ) ( ω ) and observesthat d c = I − d I so that ω is d c -closed and d -exact if and only if I( ω ) is d -closed and d c -exact. Then using Lemma 6.3 one writes I( ω ) = dd c ( η ) and hence ω = (I − d I − d )(I( η )) =( − k − d c d (I( η )) d c d ( ν ), where ν = ( − k − I( η ).We next consider the sub differential graded algebra (Ω c , d ) ⊂ (Ω , d ) consisting of all d c -closed forms. Since dd c = − d c d , we get that d (Ω c ) ⊂ Ω c . Lemma 6.4.
For a d -regular K¨ahler structure on (Ω , d ) , the inclusion j : (Ω c , d ) → (Ω , d ) is a differential graded algebra quasi-isomorphism. If the H -action is proper, then the sameconclusion holds for (Ω , d ) .Proof. Let ω ∈ (Ω k ) c be a d -exact form. Then by Lemma 6.3, we get that ω = dd c ( η ) forsome η ∈ Ω k − . Injectivity of j ∗ is now clear because d c ( η ) is already d c -closed.By Corollary 5.4, any cohomology class in H kd can be represented by a d -harmonic form ω ∈ Ω k . By Proposition 4.23, any d -harmonic form is also ∂ -harmonic and ∂ -harmonic.Thus ω is d c -closed and hence ω is in the range of j ∗ . This gives the surjectivity of j ∗ .The last statement is obtained by the same proof and corresponding results for (Ω , d ). (cid:3) Since dd c = − d c d , it follows that d induces a natural differential d : H kd c → H k +1 d c , where H kd c is the k -th cohomology of (Ω c , d c ). Lemma 6.5.
For a d -regular K¨ahler structure on (Ω , d ) , the natural projection p : (Ω c , d ) → ( H d c , d ) is a differential graded algebra quasi-isomorphism. If the H -action is proper, thenthe same holds for (Ω , d ) .Proof. Let ω ∈ Ω k be d -closed and d c -exact. Then Lemma 6.3 implies that ω = dd c ( η ). Inparticular, ω is in the image of d : (Ω k − ) c → (Ω k ) c . Hence p ∗ is surjective.Let an element in the cohomology of ( H d c , d ) be represented by the d c -closed form ω .Then d ( ω ) is d -exact and d c -closed. Thus d ( ω ) = dd c ( η ) by Lemma 6.3. Hence ω − d c ( η )is both d c -closed and d -closed and represents the same class as ω in H d c . This proves thesurjectivity of p ∗ . (cid:3) Corollary 6.6.
For a d -regular K¨ahler structure on (Ω , d ) , the differential d is trivial on H d c .Proof. If ω is d c -closed, then d ( ω ) is d -exact and d c -closed, and thus it is of the form d ( ω ) = d c d ( η ) for some η . So 0 = [ d ( ω )] ∈ K k +1 d c . (cid:3) If the H -action is proper then the above corollary holds for (Ω , d ). Theorem 6.7.
Any given (Ω , d ) is a formal differential graded algebra if it admits a d -regular K¨ahler structure. The same conclusion holds for (Ω , d ) if the H -action is assumedto be proper.Proof. By Lemma 6.4 and Lemma 6.5 respectively, j : (Ω c , d ) → (Ω , d ) and p : (Ω c , d ) → ( H d c , d ) are differential graded algebra quasi-isomorphisms. Thus, the diagram(Ω c , d )(Ω , d ) ( H d c , p j it follows that (Ω , d ) is equivalent to a differential graded algebra with a trivial differential. (cid:3) A sufficient condition for d -regularity In this section we establish a sufficient condition for an Hermitian structure to be d -regularin the sense of Definition 5.2. We also prove that the projection φ as in Definition 5.7 com-mutes with ∆ d .Recall from (4.5) that for a positive definite Hermitian structure, an inner product is givenby h ω, η i = τ g ( ω ⊗ η ) = Z τ ω ∧ ⋆ ( η ∗ ) . (7.1) Definition 7.1.
Define the Hilbert space of forms L (Ω) to be the completion of Ω withrespect to the inner product given by (7.1) . Then ∆ d becomes a non-negative (see the proof of Proposition 4.21) densely defined sym-metric (see Corollary 4.19) operator on L (Ω). Thus ∆ d has a canonical self-adjoint exten-sion called the Friedrichs extension which we again denote by ∆ d . Moreover, assume that ∩ k dom(∆ kd ) = Ω. Theorem 7.2.
Assume that ∆ d has purely discrete spectrum, in the sense that there is anorthonormal basis { ω j } for the Hilbert space L (Ω) consisting of forms ω j ∈ Ω which areeigenforms for ∆ d : ∆ d ( ω j ) = λ j ω j , for some scalar λ j such that λ < λ < · · · → ∞ as j → ∞ . Then the Hermitian structure is d -regular.Proof. We have to show that conditions i) and ii) in Definition 5.2 are satisfied. For i),suppose we are given η ∈ Ω and a weak solution l of ∆ d ( ω ) = η . Write η = P j c j ω j .Observe that (5.1) implies that c = 0 and c j λ j = l ( ω j ). Hence ω = P ∞ c j λ j ω j is the form weare looking for. All we have to show is that ω ∈ Ω, i.e., ω is “smooth”. As in the classicalsituation this follows from the basic estimate: introduce the norms k v k k = k ∆ kd ( v ) k + k v k on H k = dom(∆ kd ). Then these spaces become Hilbert spaces with respect to these norms ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 31 [Hig06]. Now ω j ∈ Ω = ∩ k H k , hence any finite linear combination of ω j ’s is in H k , for all k .Observe that for m > n large enough so that λ n > k m X n c j λ j ω j k k = m X n | c j | λ kj λ j + m X n | c j | λ j < m X n | c j | λ kj + m X n | c j | = k m X n c j ω j k k . Since η ∈ Ω = ∩ k H k , we get that ω ∈ H k , for all k , hence smooth. Thus we proved thatcondition i) holds.For ii), fix λ ∈ ρ (∆ d )-the resolvent set, and observe that the resolvent ( λ − ∆ d ) − is acompact self-adjoint operator. By hypothesis, k ( λ − ∆ d )( η n ) k ≤ c ( | λ | + 1) for all n . So, bycompactness, { η n = ( λ − ∆ d ) − ( λ − ∆ d )( η n ) } has a norm-convergent subsequence, hence thesubsequence is Cauchy.Thus the Hermitian structure is d -regular. (cid:3) Now let L (Ω ) be the closure of Ω in L (Ω), and let P be the orthogonal projection onto L (Ω ). The following is extracted from Proposition 1.17 of [Sch12]. Proposition 7.3.
Suppose ∆ d | Ω is essentially self-adjoint on L (Ω ) . Then P takes dom(∆ d ) into dom(∆ d ) , and ∆ d P ( ω ) = P ∆ d ( ω ) for all ω ∈ dom(∆ d ) . Moreover, P takes Ω into Ω .Hence P | Ω gives a projection in the sense of Definition 5.7.Proof. Let ω be in dom(∆ d ). Then h ∆ d | Ω ( η ) , P ( ω ) i = h P ∆ d | Ω ( η ) , ω i = h ∆ d | Ω ( η ) , ω i = h η, ∆ d ( ω ) i = h P ( η ) , ∆ d ( ω ) i = h η, P ∆ d ( ω ) i for all η ∈ Ω . We use that ∆ d is symmetric and ∆ d preserves Ω . So P ( ω ) ∈ dom((∆ d | Ω ) ∗ )and (∆ d | Ω ) ∗ ( P ( ω )) = P ∆ d ( ω ). By hypothesis, ∆ d | Ω is essentially self-adjoint, so we have(∆ d | Ω ) ∗ = ∆ d | Ω . But ∆ d is closed, hence ∆ d | Ω ⊂ ∆ d . Therefore we have the firststatement. The last statement follows from the assumption that ∩ k dom(∆ kd ) = Ω. (cid:3) A further weakening condition can be given for Proposition 7.3 to hold. Namely, wedetermine when ∆ d | Ω is essentially self-adjoint. For this we consider the strongly continuousone-parameter unitary group U ( t ) = e it ∆ d . Lemma 7.4.
Assume that D := { ω ∈ Ω | ( i ∆ d ) n ( ω ) n ! → as n → ∞} is dense in L (Ω ) .Then U ( t ) takes L (Ω ) into L (Ω ) . Proof.
Pick ω from the dense set D above. Observe that U ( t ) ω − ω = R t dds ( U ( s ) ω ) ds = R t iU ( s )∆ d ( ω ) ds . So for η ∈ L (Ω ) ⊥ , h U ( t ) ω, η i = h U ( t ) ω − ω, η i = h Z t iU ( s )∆ d ( ω ) ds, η i = Z t h iU ( s )∆ d ( ω ) , η i = Z t h iU ( s )∆ d ( ω ) − i ∆ d ( ω ) , η i (since ∆ d takes Ω into Ω )= Z t Z s h U ( r )( i ∆ d ) ( ω ) , η i drds (by repeating the steps above)= ... (inductively)= Z t · · · Z t n h U ( t )( i ∆ d ) n ( ω ) , η i dtdt · · · dt n = Z σ h U ( t ) ( i ∆ d ) n ( ω ) n ! , η i , where σ is the standard simplex. Now the result follows from the density assumption on D . (cid:3) Before we go onto the next proposition, we observe that U ( t ) takes Ω and hence D intoΩ because of the assumption that Ω consists of “smooth vectors” and Lemma 7.4. Wefollow Proposition 6.3 of [Sch12]. Proposition 7.5.
Under the hypothesis of Lemma 7.4, the operator ∆ d | Ω is essentiallyself-adjoint.Proof. Suppose that τ ∈ { , − } and η ∈ ker((∆ d | Ω ) ∗ − τ iI ). Let ω ∈ D . Lemma 7.4 andremarks made above imply that U ( t ) ω ∈ Ω . Now, ddt h U ( t ) ω, η i = h i ∆ d U ( t ) ω, η i = h iU ( t ) ω, τ iη i = τ h U ( t ) ω, η i . Thus the function g ( t ) = h U ( t ) ω, η i is real analytic (because ω is smooth) and satisfies g ′ = τ g . Hence g ( t ) = g (0) e τt and so h ω, U ( − t ) η i = h ω, e τt η i . Since D is dense in L (Ω ),we get that U ( − t ) η = e τt η . So t → U ( − t ) η is differentiable at t = 0 and ddt | t =0 U ( − t ) η = τ η = − i ∆ d ( η ). Because ∆ d is self-adjoint, it follows that η = 0. (cid:3) A noncommutative example
So far we have focused on a single example, that of coming from ´etale groupoids. Wehave also mentioned Hopf algebras and weak Hopf algebras and built our framework usingthese as guiding examples. In this section we describe another example, namely, the Connes-Moscovici Hopf algebroid, which is over a noncommutative base, thus providing wider scopeof our framework. Before we plunge into the Connes-Moscovici Hopf algebroid, we describea special case, namely the following.
ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 33
The enveloping Hopf algebroid of an algebra.
Given an arbitrary C -algebra A , let H = A ⊗ C A op . The left bialgebroid structure over A is given as s l ( a ) = a ⊗ C , t l ( b ) = 1 ⊗ C b ; (8.1a)∆ l ( a ⊗ b ) = ( a ⊗ C ⊗ A (1 ⊗ C b ) , ε l ( a ⊗ C b ) = ab ; (8.1b)and the right bialgebroid structure over A op is given as s r ( b ) = 1 ⊗ C b, t r ( a ) = a ⊗ C
1; (8.2a)∆ r ( a ⊗ C b ) = ( a ⊗ C ⊗ A op (1 ⊗ C b ) , ε r ( a ⊗ C b ) = ba ; (8.2b)for a, b ∈ A . Finally, the antipode S ( a ⊗ C b ) = b ⊗ C a (8.3)makes H into a Hopf algebroid. If A is a ∗ -algebra then H is Hopf ∗ -algebroid. Then an H -covariant differential calculus on A is just a differential calculus on A , Definition 3.6 issatisfied with H being C ! Covariant complex and further structures are then described asin Remark 3.25. So we get back the usual (non-covariant) structures. We now come to The Connes-Moscovici Hopf algebroid.
Let Q be a Hopf algebra over C with antipode T satisfying T = id and A a Q -module algebra. Consider H = A ⊗ C Q ⊗ C A with multiplicationgiven by ( a ⊗ C q ⊗ C b )( a ′ ⊗ C q ′ ⊗ C b ′ ) = a ( q a ′ ) ⊗ C q q ′ ⊗ C ( q b ′ ) b. (8.4)for a, b, a ′ , b ′ ∈ A and q, q ′ ∈ Q . A left bialgebroid structure over A , known as the Connes-Moscovici bialgebroid, is given as s l ( a ) = a ⊗ C ⊗ C , t l ( b ) = 1 ⊗ C ⊗ C b ; (8.5a)∆ l ( a ⊗ C q ⊗ C b ) = ( a ⊗ C q ⊗ C ⊗ A (1 ⊗ C q ⊗ C b ); (8.5b) ε l ( a ⊗ C q ⊗ C b ) = aε ( q ) b ; (8.5c)for a, b ∈ A and q ∈ Q . ε is the counit of Q and we have used Sweedler notation for thecoproduct of Q . This much is in the literature, see for example [B¨oh09]. We now put a rightbialgebroid structure on H over A op as s r ( b ) = 1 ⊗ C ⊗ C b, t r ( a ) = a ⊗ C ⊗ C
1; (8.6a)∆ r ( a ⊗ C q ⊗ C b ) = ( a ⊗ C q ⊗ C ⊗ A op (1 ⊗ C q ⊗ C b ); (8.6b) ε r ( a ⊗ C q ⊗ C b ) = T ( q )( ba ); (8.6c)for a, b ∈ A and q ∈ Q . We only check the Takeuchi condition, leaving the rest tedious butstraightforward checking of right bialgebroid axioms to the reader. Given a, b, c ∈ A and q ∈ Q , we have s r ( a )( b ⊗ C q ⊗ C ⊗ A op (1 ⊗ C q ⊗ C c )= { (1 ⊗ C ⊗ C a )( b ⊗ C q ⊗ C } ⊗ A op (1 ⊗ C q ⊗ C c )=( b ⊗ C q ⊗ C a ) ⊗ A op (1 ⊗ C q ⊗ C c )= { ( b ⊗ C q ⊗ C ⊗ C ⊗ C T ( q ) a ) } ⊗ A op (1 ⊗ C q ⊗ C c )=( b ⊗ C q ⊗ C ⊗ A op (1 ⊗ C q ⊗ C c )( T ( q ) a ⊗ C ⊗ C b ⊗ C q ⊗ C ⊗ A op ( q T ( q ) a ⊗ C q ⊗ C c )=( b ⊗ C q ⊗ C ⊗ A op ( a ⊗ C q ⊗ C c ) (we use that T = id )=( b ⊗ C q ⊗ C ⊗ A op { ( a ⊗ C ⊗ C ⊗ C q ⊗ C c ) } =( b ⊗ C q ⊗ C ⊗ A op t r ( a )(1 ⊗ C q ⊗ C c ) , thus proving the Takeuchi condition. Now we define the antipode S as S ( a ⊗ C q ⊗ C b ) = T ( q ) b ⊗ C T ( q ) ⊗ C T ( q ) a. (8.7)Again, the antipode axioms are straightforward to check. As an example we show that µ ( S ⊗ id H )∆ l = s r ε r holds: µ ( S ⊗ id H )∆ l ( a ⊗ C q ⊗ C b )= S ( a ⊗ C q ⊗ C ⊗ C q ⊗ C b )=( T ( q )1 ⊗ C T ( q ) ⊗ C T ( q ) a )(1 ⊗ C q ⊗ C b )=(1 ⊗ C T ( q ) ⊗ C T ( q ) a )(1 ⊗ C q ⊗ C b )= T ( q )1 ⊗ C T ( q ) q ⊗ C T ( q ) bT ( q ) a =1 ⊗ C T ( q ) q ⊗ C T ( q )( ba )=1 ⊗ C ⊗ C T ( q )( ba )= s r ε r ( a ⊗ C q ⊗ C b ) . Theorem 8.1.
With the structures described above, H becomes a Hopf algebroid, which wecall the Connes-Moscovici Hopf algebroid. Furthermore, if Q is a Hopf ∗ -algebra and A is a Q -module ∗ -algebra then H becomes a Hopf ∗ -algebroid in our sense. Remark 8.2.
Observe that taking Q = C gives the enveloping Hopf algebroid back and A = C reduces H to a Hopf algebra. Thus it is a simultaneous generalization of the casesdiscussed above. Remark 8.3.
We have used T = id to make H into a Hopf algebroid. We think that itis possible to remove this condition by introducing a “modular pair in involution”, that inturn produces a “twisted antipode” for Q , hence for H . Since our intention was to producea genuinely noncommutative example, we do not investigate this in this paper. We end this section by a proposition.
Proposition 8.4.
Let (Ω , d ) be a Q -covariant differential calculus on A . Then (Ω , d ) canbe made into an H -covariant differential calculus on A in the sense of Definition 3.6. Fur-thermore, if Q is a Hopf ∗ -algebra, A is a Q -module ∗ -algebra and (Ω , d ) is a Q -covariant ∗ -differential calculus then it can be made into an H -covariant ∗ -differential calculus in thesense of Definition 3.7. ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 35
Proof.
We define the H -action on Ω as follows:( a ⊗ C q ⊗ C b ) · ω = a ( q · ω ) b. (8.8)The only non-trivial part to check is that (3.1) holds. This is easy because H contains1 ⊗ C Q ⊗ C (cid:3) Further directions and comments
We end this paper by discussing some directions that we have not touched upon.
Comparison with Connes’ approach.
In [Con82, Con85, Con86], the approach taken tostudy singular spaces, in particular, the leaf space of a foliation is as follows. One models thesingular space by a groupoid G and then considers the convolution algebra C ∞ c ( G ) as thefunction algebra of the space in question. We have considered the groupoid here also, but assymmetries. To consider noncommutative complex geometry on the singular space, we needa differential calculus on the algebra C ∞ c ( G ). Here there are many choices and it is a priorinot clear what is the correct choice to make. In fact, if one takes a discrete group and viewit as a groupoid then the convolution algebra is the group algebra and we don’t know whata choice of differential calculus would be (neither the universal one nor a bicovariant one),let alone the study of noncommutative complex structure and the meaning of it. So beforemoving onto arbitrary groupoids, one needs to answer the following question. Question 9.1.
Construct (or even classify) differential calculi on the group algebra C Γ of adiscrete group Γ . Are there any complex structures on it? If so, what does it mean to havea complex structure on C Γ ? Comparison with Fr¨ohlich et al.’s approach.
In [FGR97], they study spectral dataassociated to Hermitian, K¨ahler structure. [ ´B17] already mentions this and it is beingtaken up by him and collaborators [ ´BDS19]. We sketch this in our set up. Note that H isrepresented on L (Ω) by unbounded operators with common domain Ω. We first show thatthese operators are closable, by exhibiting densely defined adjoint operators. Taking ideasfrom [KP11], we exploit the ( A r , A r )-bimodule structure on Ω ⊗ B Ω which is given by (2.26)via θ − : A r → A opl ; explicitly, a · ( ω ⊗ η ) · a = S ( s r ( a )) · ω ⊗ s r ( a ) · η, (9.1)for a , a ∈ A r and ω, η ∈ Ω. We assume that the faithful state τ used to define theinner-product (4.5) is right invariant, i.e., τ ( h · b ) = τ ( ε r ( h ) · b ) , (9.2)for h ∈ H and b ∈ B . We have the following lemma. Lemma 9.2.
For ω, η ∈ Ω and h ∈ H , τ g ( ω ⊗ S ( h ) · η ) = τ g ( h · ω ⊗ η ) (9.3) holds, where g is as in Definition 4.7. Thus h h · ω, η i = h ω, ( S ( h )) ∗ · η i . Proof.
The proof is essentially contained in [KP11]. We compute τ g ( ω ⊗ S ( h ) · η ) = τ g ( ω ⊗ s r ε r ( h ) S ( h ) η ) (2.12)= τ ( ε r ( h ) · g ( ω ⊗ S ( h ) · η )) (4.4)= τ ( h · g ( ω ⊗ S ( h ) · η )) (9.2)= τ g ( h · ω ⊗ h S ( h ) · η ) (2.6)= τ g ( h · ω ⊗ ε l ( h ) · η )= τ g ( t l ε l ( h ) h · ω ⊗ η )= τ g ( h · ω ⊗ η ) . The last statement follows from the definition of H -action on Ω. (cid:3) Thus H is represented by closable operators having a common dense domain. We denotethe adjoint of h ∈ H by h † so that h † = ( S ( h )) ∗ on Ω. From now on, let us allow a notationalabuse of denoting by h both the operator on Ω and its closure in L (Ω). At this point, wemake an additional regularity assumption (similar to assumption in Lemma 7.4): Assumption.
Given h ∈ H , D h = { ω ∈ Ω | P ∞ k h n ω k n ! < ∞} is dense in L (Ω). Lemma 9.3.
For h ∈ H with h = h † and ω ∈ D h , define U h by U h ( ω ) = X n i n n ! h n ω, which is well defined by the above Assumption . Then U h extends to a unitary operator on L (Ω) denoted by e ih .Proof. The result follows from the observations that for such an h , D h = D − h and that U h U − h = U − h U h = id . (cid:3) Recall the subset H of H from Definition 3.3. We have the following lemma. Lemma 9.4.
For h ∈ H , the commutator [∆ d , e ih ] extends to a bounded operator on L (Ω) .Proof. Observe that e ih ∆ d − ∆ d e ih = Z dds ( e ish ∆ d e i (1 − s ) h ) ds = Z ( ihe ish ∆ d e i (1 − s ) h − ie ish ∆ d e i (1 − s ) h h ) ds = Z i ( e ish h ∆ d e i (1 − s ) h − e ish ∆ d he i (1 − s ) h ) ds = Z i ( e ish [ h, ∆ d ] e i (1 − s ) h ) ds. Since h ∈ H , [ h, ∆ d ] = [ s l ε l ( h ) , ∆ d ] extends to a bounded operator. As e ith is unitary, theintegrand is bounded. Hence the result follows. (cid:3) Combining Lemma 9.3 and Lemma 9.4, we get the following proposition.
ENERALIZED SYMMETRY IN NONCOMMUTATIVE COMPLEX GEOMETRY 37
Proposition 9.5.
Let A be the ∗ -algebra generated by operators of the form ae ih b with a, b ∈ A l and h ∈ H in B ( L (Ω)) . Then ( A , L (Ω) , ∆ d ) forms a spectral triple. If we assume that ∆ d has purely discrete spectrum then we get a spectral triple of compacttype. We also note that [ ´B17] computes the spectrum for the concrete examples. In ourabstract setup, we propose a way of doing it generally. We have already assumed an analogue(or rather a corollary) of classical Sobolev embedding (see the remarks before Theorem 7.2).It would be interesting to know the answer of the following: Question 9.6.
If we assume an analogue of Relich’s lemma ( H k ֒ → H k +2 is compact inthe notation of the Proof of Theorem 7.2) then does it follow that ∆ d has purely discretespectrum? See [Hig06] for the setup and more on abstract pseudo-differential calculi whichhas motivated this question. This would give a uniform way of proving that the Laplacian ∆ d has purely discretespectrum in the setting of noncommutative differential calculi. Further examples.
As examples for our framework, we have mentioned ´etale groupoids,Hopf algebras, weak Hopf algebras and the Connes-Moscovici Hopf algebroid. There isanother class of examples coming from Lie-Rinehart algebras and associated jet spaces; see[KP11]. It would be interesting to know the answer of the following
Question 9.7.
Investigate if these examples fit into our framework. If so, what is themeaning of having a complex structure on a Lie-Rinehart algebra?
On this note, we mention a result from an ongoing work that produces a left bialgebroidthat is not of the form dealt with in this paper. Let X be the finite set { , · · · , n } . Proposition 9.8.
There is a left bialgebroid H over C ( X ) such that the action on C ( X ) liftsto an action on the space of universal one forms in the sense of Definition 3.3. Moreover,it is not of the form C ( X ) Q for any Hopf algebra Q . Finally, we ask a question which is not directly related to this work but interesting inits own right. In [GJ18], it is shown that a coaction of a compact quantum group on analgebra can be lifted to a differential calculus (at least in the classical situation) under somesuitable (unitarity of the coaction, technically, see also (4.4)) conditions, like one expectsfrom a group action. So we ask
Question 9.9.
Is the above true for unitary action (i.e., (4.4) is satisfied) of Hopf algebroids?
We have shown above that if we have the action on the full differential calculus, then,under some more conditions, the action becomes unitary. So we are seeking a converse ofthis.
Acknowledgement
The first author is grateful to Aritra Bhowmick for several discussions on foliations, toYuri A. Kordyukov for answering many questions, to Edwin J. Beggs and Shahn Majid forhelpful comments, and finally to R´eamonn ´O Buachalla for countless discussions and hisinterest in this work. He also thanks the second author for answering many questions on complex geometry and the third author for introducing him to the theory of foliations. Thesecond author is partially supported by J.C. Bose National Fellowship. The third authoris partially supported by J.C. Bose National Fellowship and Research Grant awarded byD.S.T. (Govt. of India).
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