A categorical approach to cyclic cohomology of quasi-Hopf algebras and Hopf algebroids
aa r X i v : . [ m a t h . K T ] S e p A CATEGORICAL APPROACH TO CYCLIC COHOMOLOGY OFQUASI-HOPF ALGEBRAS AND HOPF ALGEBROIDS.
IVAN KOBYZEV AND ILYA SHAPIRO
Abstract.
We apply categorical machinery to the problem of defining cyclic cohomol-ogy with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf alge-broids. In the case of the former, no definition was thus far available in the literature,and while a definition exists for the latter, we feel that our approach demystifies theseemingly arbitrary formulas present there. This paper emphasizes the importance ofworking with a biclosed monoidal category in order to obtain natural coefficients fora cyclic theory that are analogous to the stable anti-Yetter-Drinfeld contramodules forHopf algebras.
Monoidal category (18D10), abelian and additivecategory (18E05), cyclic homology (19D55), Hopf algebras (16T05).
Everything that’s extreme isdifficult. The middle parts aredone more easily. The verycentre requires no effort at all.The centre is equal toequilibrium. There’s no fight init.
Daniil Kharms,
Blue notebook Introduction
After its introduction by Tsygan and Connes independently in the 1980s, cyclic homol-ogy has been the subject of extensive research. An equivariant version for Hopf algebrasbegun with Connes-Moscovici [5, 6] and culminated with the definition of stable anti-Yetter-Drinfeld module coefficients by Hajac-Khalkhali-Rangipour-Sommerhauser [10, 11]and independently by Jara-Stefan [13], and stable anti-Yetter-Drinfeld contramodule co-efficients by Brzezinski [3].Beyond Hopf algebras one has a number of Hopf-like notions, relaxing the various axiomsof a Hopf algebra. One is naturally interested in extending an equivariant version ofcyclic homology, together with the relevant coefficients, to these more general settings.The problem turns out to be highly non-trivial as the definitions rely almost entirelyon the correct notion of coefficients; these are defined by complicated formulas that aresurprisingly difficult to generalize.
An approach to this problem is afforded by [12] where it is shown that the monoidal-categorical point of view provides a way to extract the necessary formulas from a con-ceptual model of the situation. It is demonstrated that indeed one obtains the usualdefinitions in the Hopf algebra case. Here we continue the application of the machinerydeveloped there. Another key observation for us is the importance of the existence ofinternal homomorphisms in our monoidal category; more precisely these are right adjointsto the tensor product. A category possessing internal homs is said to be biclosed whichrefers to the existence of both left and right internal homs. This property turns out to bethe sole ingredient required in order to define an analogue of stable anti-Yetter-Drinfeldcontramodule coefficients in the more general Hopf-like setting.In this paper we deal with two particular cases of Hopf-like objects. However, before webegin with these applications we examine the setting of Hopf algebras, namely that ofstable anti-Yetter-Drinfeld contramodule coefficients as in [3]. Our approach yields thesame definitions, but the transition from the categorical perspective to formulas is verystraightforward and serves both as motivation and contrast to the main two cases of thispaper considered subsequently.We begin with addressing the issue of defining cyclic cohomology for quasi-Hopf algebras,these were originally introduced by Drinfeld in [7]. Roughly speaking, we obtain quasi-Hopfalgebras by weakening the co-associativity condition in the definition of a Hopf algebrato that of co-associativity up to a prescribed isomorphism. An important classical fact isthat the category of representations of a quasi-Hopf algebra is still a monoidal category,and what is crucial for us is that this monoidal category is biclosed. From quite generalconsiderations one can then provide a conceptual description of coefficients, followed bythe unraveling of the definitions to obtain explicit and immensely unpleasant formulas.The complexity of the resulting expressions explains the absence of any definition of cyclichomology for quasi-Hopf algebras thus far.The second case that we consider is that of Hopf algebroids. Various related conceptspurporting to describe these objects exist and we choose to work with the definitions in [16]which follow those of [1]. Again, roughly, the generalization from Hopf algebras consistsof replacing the ground field k , over which everything is tensored, by a noncommutativering R . Thus the objects underlying representations of a Hopf algebroid consist not ofvector spaces over k , but of bimodules over R . It seems that Hopf algebroids provide anexample of a construct that is the most noncommutative in noncommutative geometry.We point out that a definition of cyclic cohomology with an analogue of stable anti-Yetter-Drinfeld contramodule coefficients has been given by Kowalzig in [15]. Our goal here isto explain the formulas that appear there as an inevitable consequence of the generalmachinery.The paper is organized as follows. In Section 2 we discuss some generalities as in [12, 22]concerning a conceptual definition of contramodule coefficients for a suitable monoidalcategory. A new phenomenon that arises in this paper is the difficulty of proving thata certain natural map is an isomorphism. It requires that we deal with weak centersas opposed to (strong) centers; this particular issue does not occur with Hopf algebras.Fortunately we are able to sidestep the problem by showing that stability, a condition thatis required in any case for a Hopf-type theory, guarantees that our objects of interest areindeed in the strong center. YCLIC COHOMOLOGY OF QUASI-HOPF ALGEBRAS AND HOPF ALGEBROIDS 3
Section 3 deals with the motivational case of contramodule coefficients for Hopf algebras.In Section 4 we recall the definition of quasi-Hopf algebras and give a prove that theircategories of modules are biclosed (this is a fact known to the experts in the field [4, 17, 19]).In Section 5 we unravel the categorical definitions of Section 2 to obtain definitions of anti-Yetter-Drinfeld contramodules for quasi-Hopf algebras. We remark that unlike the Hopfalgebra case, there are two different descriptions via formulas of the conceptually definedcategory of coefficients. Stability for type I contramodules is explicitly written down aswell.The Sections 6 and 7 address the definitions of Hopf algebroids, the biclosed property oftheir categories of modules (we provide a direct proof, but the fact can also be obtainedfrom [21]) and finally the translation into formulas of the definition of anti-Yetter-Drinfeldcontramodules and their stability.We remark that we handled the case of anti-Yetter-Drinfeld module analogues in ourfollow-up paper [14]. We did not want to increase the length of the exposition any furtherand modules, despite being more familiar, are actually technically more difficult and lessnatural/suitable from the categorical perspective on the subject of coefficients.
Acknowledgments : The authors wish to thank Masoud Khalkhali for stimulating ques-tions and discussions. Furthermore, we are grateful to the referee for the constructivecomments and useful references. The research of the second author was supported in partby the NSERC Discovery Grant number 406709.2.
Generalities
In this section we will establish the general formalism for cyclic cohomology with anti-Yetter-Drinfeld contramodule coefficients. Recall from [12] that the main ingredient inconstructing Hopf-cyclic cohomology is a symmetric 2-contratrace. We will briefly recallthe necessary definitions.Let M be a monoidal category. Consider an M -bimodule category M ∗ = Fun( M op , Vec) op ,where Vec denotes k -vector spaces, with actions defined by: M ⊲ F ( − ) := F ( − ⊗ M ) , F ( − ) ⊳ M := F ( M ⊗ − ) . Any element in Z M ( M ∗ ) is called a . In particular it requires the existenceof natural isomorphisms ι M : F ( − ⊗ M ) → F ( M ⊗ − ) , and a 2-contratrace is called symmetric, if the composition F ( M ⊗ M ′ ) → F ( M ′ ⊗ M ) → F ( M ⊗ M ′ ) is Id for all M, M ′ ∈ M .By [12, Proposition 3.9] a unital associative algebra object A in M and a symmetric 2-contratrace F provides us with a cocyclic object C • = F ( A • +1 ); its cyclic cohomologyrecovers exactly the classical Hopf-cyclic cohomology of an algebra A with coefficientsin a stable anti-Yetter-Drinfeld contramodule for the case M = H M , the category ofmodules over a Hopf algebra H . More precisely, F ( − ) = Hom H ( − , M ) where M is astable anti-Yetter-Drinfeld contramodule. This is explained in Section 3.A crucial element for our constructions is a biclosed monoidal category M . The property ofbeing biclosed implies in particular the existence of the following adjunctions for M, V, W ∈ I. KOBYZEV AND I. SHAPIRO M : Hom M ( W ⊗ V, M ) ≃ Hom M ( W, Hom l ( V, M )) , and Hom M ( V ⊗ W, M ) ≃ Hom M ( W, Hom r ( V, M )) , where Hom l ( V, M ) and Hom r ( V, M ) are left and right internal homomorphisms respec-tively.As in [22], we can introduce the contragradient M -bimodule category M op . Specifically,for M ∈ M op and V ∈ M , the actions are given by:(2.1) M ⊳ V := Hom r ( V, M ) , V ⊲ M := Hom l ( V, M ) . The natural object to consider is the center of this bimodule contragradient category: Z M ( M op ) (see [9] for the definition of the center of a bimodule category). However, insome situations (e.g. quasi-Hopf algebras or Hopf algebroids) it becomes too restrictive.If in the definition of a (strong) center element M ∈ M op we relax the condition that themaps τ : M ′ ⊲ M → M ⊳ M ′ are isomorphisms, we get a weak center. More formally: Definition 2.1.
The weak center w- Z M ( N ) of a M -bimodule category N consists ofobjects that are pairs ( N, τ N, − ) , where N ∈ N and τ N, − is a family of natural morphismssuch that for V ∈ M we have: τ N,V : V ⊲ N → N ⊳ V , satisfying the hexagon axiom as inthe usual definition of center.A morphism ( N, τ N, − ) → ( N ′ , τ N ′ , − ) is a morphisms t : N → N ′ in N satisfying τ N ′ ,V ◦ ( id V ⊲ t ) = ( t ⊳ id V ) ◦ τ N,V for all V ∈ M .Remark . One can give a reformulation of this definition in a categorical language.Consider an M -bimodule category N as a weak 2-category B with two objects L and R , where hom-categories are: B ( L ; L ) = B ( R ; R ) = M , B ( L ; R ) = N and B ( R ; L ) = ∅ .The composition of 1-cells given by the monoidal product of M and the M -actions on N .Also regard M as a weak 2-category with a single object. There are two inclusion functors l, r : M → B , mapping M identically to the endohom-categories of L and R respectively.Then the objects of w- Z M ( N ) are the lax natural transformations l → r .We need one more definition. Definition 2.3.
Let M be a biclosed monoidal category, M op a contragradient categoryas above. Then denote by Z ′M ( M op ) the full subcategory of the weak center that consistsof objects such that the identity map Id ∈ Hom M ( M, M ) is mapped to same via (2.2) Hom M ( M, M ) ≃ Hom M (1 , M ⊲ M ) → Hom M (1 , M ⊳ M ) ≃ Hom M ( M, M ) , where the map in the middle is postcomposition with τ and the isomorphisms come fromdefinitions of actions in M op . This condition is called stability . The next lemma shows that stability erases the difference between weak center and center.
Lemma 2.4.
In our current notation M ∈ Z ′M ( M op ) implies that M ∈ Z M ( M op ) .Proof. By Yoneda Lemma, the condition (2.2) implies that for any V ∈ M the chain ofmaps: Hom M ( V, M ) ≃ Hom M (1 , V ⊲ M ) → Hom M (1 , M ⊳ V ) ≃ Hom M ( V, M ) YCLIC COHOMOLOGY OF QUASI-HOPF ALGEBRAS AND HOPF ALGEBROIDS 5 is identity. For any T ∈ M consider:Hom M ( T ⊗ V, M ) ≃ Hom M ( T, V ⊲ M ) → Hom M ( T, M ⊳ V ) ≃ Hom M ( V ⊗ T, M ) ≃ Hom M ( V, T ⊲ M ) → Hom M ( V, M ⊳ T ) ≃ Hom M ( T ⊗ V, M ) . By the observation above and hexagon axiom this composition must be identity. Hence themap Hom M ( T, V ⊲ M ) → Hom M ( T, M ⊳ V ) is injective and the map Hom M ( V, T ⊲ M ) → Hom M ( V, M ⊳ T ) is surjective. Because this is true for any
T, V ∈ M , we can switch themand get that Hom M ( T, V ⊲ M ) τ ◦ −→ Hom M ( T, M ⊳ V )is bijective. This is true for any T ∈ M , so again by the Yoneda lemma, we get that τ M,V : V ⊲ M → M ⊳ V is an isomorphism. (cid:3)
Now we want to use this result to construct a symmetric 2-contratrace, and hence Hopf-cyclic cohomology.
Lemma 2.5.
Let M be a biclosed linear monoidal category. The category of Z ′M ( M op ) is equivalent to the category of representable left exact symmetric -contratraces on M via M ←→ Hom M ( − , M ) . Proof.
Follows from the previous Lemma. See [22] for details. (cid:3) Hopf algebras
We begin by considering the Hopf cyclic cohomology with contramodule coefficients in thecase of Hopf algebras with invertible antipode. Even though this case is well treated in[3], our constructions become especially clear in this setting. They serve a motivationalpurpose for the two cases of actual interest in this paper; namely that of quasi-Hopfalgebras and Hopf algebroids.Let H be a Hopf algebra (we will always assume the bijectivity of the antipode in thissection) over a field k . The category of H -modules, H M , is monoidal. It is well-knownthat the category is also biclosed. We are going to include a general proof (of a knownfact [4, 17, 19]) that the modules over a quasi-Hopf algebra form a biclosed category inSection 4. For any M, N ∈ H M , we can define the left internal Hom by(3.1) Hom l ( M, N ) = Hom k ( M, N ) , h · ϕ = h ϕ ( S ( h ) − ) , and right internal Hom by(3.2) Hom r ( M, N ) = Hom k ( M, N ) , h · ϕ = h ϕ ( S − ( h ) − ) . Using the adjunctions for
M, V, W ∈ H M :Hom H M ( W ⊗ V, M ) ≃ Hom H M ( W, Hom l ( V, M )) , and Hom H M ( V ⊗ W, M ) ≃ Hom H M ( W, Hom r ( V, M )) , we can introduce the contragradient H M -bimodule category H M op as in Section 2. Specif-ically, for M ∈ H M op and V ∈ H M , the action is given by:(3.3) M ⊳ V = Hom r ( V, M ) , and V ⊲ M = Hom l ( V, M ) . I. KOBYZEV AND I. SHAPIRO
The definition of a contramodule over a coalgebra was given first in [8], the definition of aY D -contramodules for a Hopf algebra was given in [3], we recall it below.
Definition 3.1.
Let ( H, ∆ , ε ) be a coalgebra over k . A vector space M is called a (right) H -contramodule, if there is a k -linear map µ : Hom k ( H, M ) → M , called contraaction,such that the following diagrams commute: Hom( H, Hom(
H, M )) Hom(
H, M )Hom(( H ⊗ H ) , M ) Hom( H, M ) M Hom(
H, µ ) µµη Hom(∆ , M ) where η ( f )( x ⊗ y ) = ( f ( x ))( y ) is a k -adjunction isomorphism; Hom( k, M ) Hom(
H, M ) M Hom( ε, M ) µ ≃ A morphism of contramodules ( M, µ ) → ( N, ν ) is a map of vector spaces t : M → N , suchthat the following diagram commutes: Hom(
H, M ) Hom(
H, N ) M N
Hom(
H, t ) νµ t Definition 3.2.
Let H be a Hopf algebra over k . A k -vector space M is called an anti-Yetter-Drinfeld contramodule ( aY D -contramodule), if • M is a left H -module, • M is a H -contramodule, • For all h ∈ H and any linear map f ∈ Hom(
H, M ) there is a compatibility condi-tion: (3.4) h ( µ ( f )) = µ ( h f ( S ( h ) − h )) . It is called stable ( saY D -contramodule), if for all m ∈ M we have µ ( r m ) = m where r m ( h ) = hm .A morphism of aY D -contramodules M → N is simultaneously a morphism of H -modulesand H -contramodules. YCLIC COHOMOLOGY OF QUASI-HOPF ALGEBRAS AND HOPF ALGEBROIDS 7
Because the category H M is a biclosed monoidal, we can define the contragradient H M -bimodule category H M op as in Section 2, and describe its center. Proposition 3.3.
The category of aY D -contramodules for a Hopf algebra H , with aninvertible antipode S , is isomorphic to Z H M ( H M op ) .Proof. Let (
M, µ ) be an aY D -contramodule, for any H -module V define τ : V ⊲M → M ⊳V by τ ( φ )( v ) = µ ( h φ ( hv )) , where φ ∈ Hom l ( V, M ) , v ∈ V, h ∈ H .Define θ : M ⊳ V → V ⊲ M by θ ( φ )( v ) = µ ( h φ ( S − ( h ) v )) , where φ ∈ Hom r ( V, M ) , v ∈ V, h ∈ H .Then ( x ∈ H )( θ ◦ τ ( φ ))( v ) = µ ( h ( τ φ )( S − ( h ) v )) = µ ( h µ ( x φ ( xS − ( h ) v )))= µ ( h φ ( h S − ( h ) v )) = µ ( h ε ( h ) φ ( v )) = φ ( v ) . And similarly τ θφ = φ , so that τ is an isomorphism (of vector spaces so far).Now for x ∈ H let us compare ( xτ φ )( v ) to ( τ xφ )( v ) using (3.4):( xτ φ )( v ) = x µ ( h φ ( hS − ( x ) v )) = µ ( h x φ ( S ( x ) hx S − ( x ) v ))= µ ( h x φ ( S ( x ) hε ( x ) v )) = µ ( h x φ ( S ( x ) hv )) = ( τ xφ )( v )so that τ is an isomorphism in H M .Let f ∈ Hom H ( V, V ′ ) then τ ( φ ◦ f )( v ) = µ ( h φ ( f ( hv ))) = µ ( h φ ( hf ( v ))) = ( τ φ ◦ f )( v )so that τ is functorial in V .Next, to verify the commutativity of(3.5) W ⊲ ( V ⊲ M ) Id⊲τ / / W ⊲ ( M ⊳ V ) Id / / ( W ⊲ M ) ⊳ V τ⊳Id / / ( M ⊳ W ) ⊳ V (cid:15) (cid:15) ( W ⊗ V ) ⊲ M O O τ / / M ⊳ ( W ⊗ V )is to check that taking f along the second row to w ⊗ v µ ( h f ( h w ⊗ h v )) is thesame as taking it the long way around, which is w ⊗ v µ ( x µ ( h f ( xw ⊗ hv ))).Those two are equal by the contraaction condition. Finally, to show that k ⊲ M → M ⊳ k is the identity observe that if, for m ∈ M we define a function, also called m : k → M , asfollows: m ( c ) = cm , then(3.6) τ m ( c ) = µ ( h m ( hc )) = µ ( h ǫ ( h ) cm ) = cm. Furthermore, if g : M → M ′ is a map of aY D -contramodules then( g ◦ τ φ )( v ) = g ( µ ( h φ ( hv ))) = µ ( h g ( φ ( hv ))) = τ ( g ◦ φ )( v )so that g is a morphism in Z H M ( H M op ). I. KOBYZEV AND I. SHAPIRO
What has been shown so far is that if M is an aY D -contramodule, then ( M, τ ) ∈Z H M ( H M op ) and any g : M → M ′ a morphism of aY D -contramodules induces a mor-phism between the corresponding central elements.Conversely, given a central element ( M, τ ) ∈ Z H M ( H M op ), take µ to be the composi-tion of τ : Hom l ( H, M ) → Hom r ( H, M ) and ev : Hom r ( H, M ) → M . Observe that theconsiderations (3.5) and (3.6) are of the if and only if kind, so that µ is a contraaction. Fur-thermore, the aY D condition is satisfied. More precisely, let x ∈ H and f ∈ Hom(
H, M ),note that τ ( f ( − h )) = ( τ f )( − h ) since r h : H → H is a morphism in H M , then µ ( x f ( S ( x ) − x )) = ev τ ( x f ( S ( x ) − x )) = ev τ (( x · f )( − x ))= ev x · τ ( f ( − x )) = ev x · ( τ f )( − x ) = ev x ( τ f )( S − ( x ) − x )= x ( τ f )( S − ( x ) x ) = x ( τ f )(1) = xev τ f = xµ ( f ) . Finally, suppose that f : M → M ′ is a map in the center, then we have H ⊲ M
Id⊲f (cid:15) (cid:15) τ / / M ⊳ H f⊳Id (cid:15) (cid:15) ev / / M f (cid:15) (cid:15) H ⊲ M ′ τ / / M ′ ⊳ H ev / / M where the left square commutes by the centrality of f and the right square commutesobviously. Since the top row is µ M and the bottom row is µ M ′ so f is a map of aY D -contramodules.Note that the constructions of central elements from aY D -contramodules and vice versaare inverses of each other. Indeed, the direction µ → τ → µ is obvious. The seconddirection τ → µ → τ follows from the observation that τ H ( f ( − h )) = ( τ H f )( − h ). Thus wehave proved the isomorphism of categories as claimed. (cid:3) Remark . Note that θ , the inverse of τ defined in the above proof is easily writtendown. This is not the case in the instance of quasi-Hopf algebras nor Hopf algebroids thusmaking it necessary to deal with the weak center, at least until the imposition of stability.We do not know if in the presence of an invertible antipode the weak center coincides withthe center for quasi-Hopf algebras and Hopf algebroids. The notion of stability, requiredfor cyclic cohomology, allows us to side-step the issue.In Definition 2.3 we let Z ′ H M ( H M op ) be the full subcategory of the weak center thatconsists of objects such that the identity map Id ∈ Hom H ( M, M ) is mapped to same via(3.7) Hom H ( M, M ) ≃ Hom H (1 , M ⊲ M ) → Hom H (1 , M ⊳ M ) ≃ Hom H ( M, M ) . Recall that Lemma 2.4 shows that Z ′ H M ( H M op ) is actually a full subcategory of the strongcenter, and so the middle map above is an isomorphism.We immediately obtain the following Corollary of Proposition 3.3. Corollary 3.5.
The category of saY D -contramodules for H is equivalent to Z ′ H M ( H M op ) .Proof. Let M be an aY D -contramodule, then to ensure that the condition (3.7) is satisfied,we need that τ ( Id )( m ) = µ ( h hm ) = µ ( r m ) = m YCLIC COHOMOLOGY OF QUASI-HOPF ALGEBRAS AND HOPF ALGEBROIDS 9 which is exactly the saY D condition. On the other hand, if M is a central element thatsatisfies (3.7), then it also satisfies that the chain of isomorphisms(3.8) Hom H ( V, M ) ≃ Hom H (1 , V ⊲ M ) ≃ Hom H (1 , M ⊳ V ) ≃ Hom H ( V, M )is identity for any V ∈ H M . Take V = H and note that r m ∈ Hom H ( H, M ) so that (3.8)implies that τ ( r m ) = r m and so µ ( r m ) = ev τ ( r m ) = ev ( r m ) = m , i.e., M is a stable aY D -contramodule. (cid:3) Let’s prove an easy lemma which will motivate the definition of the aY D -condition in thecase of Hopf algebroids.
Lemma 3.6.
Let H be a Hopf algebra with an invertible antipode, M be a left H -moduleand H -contramodule. Then equation (3.4) is equivalent to: (3.9) h ( µ ( f ( − S − ( h )))) = µ ( h f ( S ( h ) − )) for all h ∈ H , and f ∈ Hom k ( H, M ) .Proof. Assume that (3.4) holds. Then consider the left hand side of (3.9): h ( µ ( f ( − S − ( h )))) (i) = µ ( h f ( S ( h ) − h S − ( h )))= µ ( h f ( S ( h ) − ε ( h ))) = µ ( h f ( S ( h ) − )) , where (i) is equation (3.4).Now assume that (3.9) holds. Then: µ ( h f ( S ( h ) − h )) (i) = h µ ( f ( − S − ( h ) h ))= h µ ( f ( − ε ( h ))) = hµ ( f ) , where (i) is equation (3.9). (cid:3) Quasi-Hopf algebras
Let us remind the reader of all the necessary definitions following [7]. In this section k isa field. Definition 4.1.
A quasi-bialgebra is a collection ( A, ∆ , ε, Φ) , where A is an associative k -algebra with unity, ∆ : A → A ⊗ A and ε : A → k are homomorphisms of algebras, Φ ∈ A ⊗ A ⊗ A is an invertible elements, such that the following equalities hold: (4.1) ( id ⊗ ∆) (∆( a )) = Φ · ((∆ ⊗ id ) (∆( a ))) · Φ − ∀ a ∈ A (4.2) [( id ⊗ id ⊗ ∆)(Φ)] · [(∆ ⊗ id ⊗ id )(Φ)] = (1 ⊗ Φ) · [( id ⊗ ∆ ⊗ id )(Φ)] · (Φ ⊗ ε ⊗ id ) (∆( a )) = a, ( id ⊗ ε ) (∆( a )) = a, ∀ a ∈ A (4.4) id ⊗ ε ⊗ id (Φ) = 1 ⊗ Remark . In this paper we will use the Sweedler notation. Let’s denote(4.5) Φ = X ⊗ Y ⊗ Z, (4.6) Φ − = P ⊗ Q ⊗ R, here we mean the summation. In particular, the equality (4.1) can be written as: a ⊗ a ⊗ a = Xa P ⊗ Y a Q ⊗ Za R. We are interested in the category of left A -modules A M . It was proved in [7] that thiscategory is monoidal if a tensor product of two left A -modules M and N is defined by thesame formula as in the case of a bialgebra:(4.7) M ⊗ N = M ⊗ k N, a · ( m ⊗ n ) = a m ⊗ a n. The associativity morphism is no longer trivial as it was in the case of a bialgebra. If onesets the associativity morphism ( M ⊗ N ) ⊗ L → M ⊗ ( N ⊗ L ) to be the image of Φ inEnd k ( M ⊗ N ⊗ L ), then it becomes an isomorphism of left A -modules by (4.1). Consider k as an A -module by a · ε ( a )1 as in the bialgebra case. Then one defines a morphism λ M : k ⊗ M → M as the usual morphism of k -modules. Then λ M becomes an A -modulemorphism by (4.3). Similarly one can define a morphism ρ M : M ⊗ k → M . So k is a unitof a monoidal category A M . Remark . The equality (4.2) is equivalent to the pentagon axiom. Associativity andunit in the category respect each other by (4.4).Recall the definition of a quasi-Hopf algebra from [7].
Definition 4.4.
Let ( H, ∆ , ε, Φ) be a quasi-bialgebra. Then it is called a quasi-Hopfalgebra if there exist α, β ∈ H and anti-automorphism S : H → H , such that (4.8) S ( h ) αh = ǫ ( h ) α, (4.9) h βS ( h ) = ǫ ( h ) β. If one keeps notation as in Remark 4.2, then there should be equalities (4.10)
XβS ( Y ) αZ = 1 , (4.11) S ( P ) αQβR = 1 . Remark . We want to emphasize that the antipode S in the definition above is assumedto be invertible.It was shown in [7], that the category of H -modules for a quasi-Hopf algebra H , thatare finite dimensional as k -vector spaces, is rigid. The more general result, that withoutthe finite dimensionality condition the category of H -modules is closed, was proved in[4, 17, 19]. To fix notation and for the convenience of the reader we are going to includea sketch of a proof here. Proposition 4.6.
The category H M of left modules over a quasi-Hopf algebra H is abiclosed monoidal category. YCLIC COHOMOLOGY OF QUASI-HOPF ALGEBRAS AND HOPF ALGEBROIDS 11
Proof.
For any
M, N ∈ H M , we can define the left internal Hom by(4.12) Hom l ( M, N ) = Hom k ( M, N ) , h · ϕ = h ϕ ( S ( h ) − ) . Define also the evaluation morphism:(4.13) ev l : Hom l ( M, N ) ⊗ M → N, ϕ ⊗ m X ( ϕ ( S ( Y ) αZm )) . Notice that the morphism ev l is a morphism of left H -modules. Let’s define an adjunctionmap:(4.14) ζ l : Hom H M ( M ⊗ N, L ) → Hom H M ( M, Hom l ( N, L ))by the rule:(4.15) f ( m f ( P m ⊗ QβS ( R ) − )) . Observe that ζ l ( f ) is a morphism of H -modules. Construct also a map in the oppositedirection:(4.16) η l : Hom H M ( M, Hom l ( N, L )) → Hom H M ( M ⊗ N, L )by the rule:(4.17) η l ( g ) = ev l ◦ ( g ⊗ id ) . It is a morphism of H -modules as a composition of such. It will be useful to write theformula: η l ( g )( m ⊗ n ) = X ( g ( m )( S ( Y ) αZn )) . Note that ζ l and η l are mutually inverse and the category H M is left closed.For any M, N ∈ H M , we can define the right internal Hom by(4.18) Hom r ( M, N ) = Hom k ( M, N ) , h · ϕ = h ϕ ( S − ( h ) − ) . Define also the evaluation morphism:(4.19) ev r : M ⊗ Hom r ( M, N ) → N, m ⊗ ϕ R ( ϕ ( S − ( Q ) S − ( α ) P m )) . Let’s define an adjunction map:(4.20) ζ r : Hom H M ( N ⊗ M, L ) → Hom H M ( M, Hom r ( N, L ))by the rule:(4.21) f ( m f ( Y S − ( β ) S − ( X ) − ⊗ Zm )) . As above ζ r ( f ) is a morphism of H -modules.Construct also a map in the opposite direction:(4.22) η r : Hom H M ( M, Hom r ( N, L )) → Hom H M ( N ⊗ M, L )by the rule:(4.23) η r ( g ) = ev r ◦ ( id ⊗ g ) , the formula is: η r ( g )( n ⊗ m ) = R ( g ( m )( S − ( Q ) S − ( α ) P n )).Analogously to the case of left internal Hom one sees that η r and ζ r are mutually inverse.This means that the category H M is also right closed, and thus biclosed. (cid:3) Anti-Yetter-Drinfeld contramodules for a quasi-Hopf algebra
We are going to define the anti-Yetter-Drinfeld contramodules using the categorical ap-proach from Section 2. The category H M is biclosed monoidal. Using the adjunctions for M, V, W ∈ H M : Hom H M ( W ⊗ V, M ) ≃ Hom H M ( W, Hom l ( V, M )) , and Hom H M ( V ⊗ W, M ) ≃ Hom H M ( W, Hom r ( V, M )) , proved in Proposition 4.6, we can introduce the contragradient H M -bimodule category H M op as in Section 2. Specifically, for M ∈ H M op and V ∈ H M , the action is given by:(5.1) M ⊳ V = Hom r ( V, M ) , and V ⊲ M = Hom l ( V, M ) . For the Hopf-case we proved Proposition 3.3, that the center Z H M ( H M op ) is the sameas anti-Yetter Drinfeld contramodules. We want to generalize this fact to the quasi-Hopfcase, i.e., we need to change the notion of a contramodule due to the noncoassociativityphenomenon. We will only prove that aY D -contramodules coincide with the weak center,but it is enough for our purposes, as was explained in Section 2. Unlike the Hopf algebrasetting there are two types of contramodules. More precisely, there are two, in this casedifferent, ways of unraveling the definition of the center into formulas.We organize the material in the similar way to [18].5.1. Anti-Yetter-Drinfeld contramodules I.
We begin with a lemma explaining theorigin of the aY D -condition.
Lemma 5.1.
Let M ∈ H M op . Natural transformations τ ∈ Nat ( id ⊲ M, M ⊳ id ) are in1-1 correspondence with k -linear maps µ : Hom k ( H, M ) → M , such that (5.2) h ( µ ( f ( − S − ( h )))) = µ ( h f ( S ( h ) − )) . Proof.
Consider f ∈ Hom k ( H, M ), induce the H -module structure corresponding to theleft internal homomorphism. Given a morphism τ H : H ⊲ M → M ⊳ H , define: µ ( f ) = τ H ( f )(1), which we denote by ev ( τ H ( f )). Note that this is not a right “categorical”evaluation! Let x ∈ H and f ∈ Hom(
H, M ), observe that τ H ( f ( − h )) = ( τ H f )( − h ) since r h : H → H is a morphism in H M . Now we can check the formula: µ ( h f ( S ( h ) − )) = ev τ H ( h f ( S ( h ) − )) = ev τ H ( h · f ) = ev h · τ H ( f )= ev h ( τ H f )( S − ( h ) − ) = h ( τ H f )( S − ( h ))= h ev τ H ( f ( − S − ( h ))) = h ( µ ( f ( − S − ( h )))) . Conversely, given the map µ , for any V ∈ H M one can define the map τ V : V ⊲M → M ⊳V by the rule f µ ( x f ( x − )). It is a morphism of H -modules by (5.2).Given an H -homomorphism ϕ : V → W , consider the diagram:Hom l ( W, M ) −◦ ϕ (cid:15) (cid:15) τ W / / Hom r ( W, M ) −◦ ϕ (cid:15) (cid:15) Hom l ( V, M ) τ V / / Hom r ( V, M ) . YCLIC COHOMOLOGY OF QUASI-HOPF ALGEBRAS AND HOPF ALGEBROIDS 13
It commutes, so we have naturality.These correspondences are mutually inverse. The only nontrivial direction follows fromthe observation that τ H ( f ( − h )) = ( τ H f )( − h ). (cid:3) Now we need to find a replacement for a contramodule condition. Let’s consider a hexagonaxiom of the weak center:
V ⊲ ( W ⊲ M ) V ⊲ ( M ⊳ W ) (
V ⊲ M ) ⊳ W ( V ⊗ W ) ⊲ M M ⊳ ( V ⊗ W ) ( M ⊳ V ) ⊳ W.id ⊲ τ W τ V ⊳ idτ V ⊗ W There are three associativity isomorphisms and we need explicit formulas for them:
V ⊲ ( W ⊲ M ) → ( V ⊗ W ) ⊲ M, ( v ( w f v ( w ))) ( v ⊗ w Xf S ( Z ) v ( S ( Y ) w )); V ⊲ ( M ⊳ W ) → ( V ⊲ M ) ⊳ W, ( v ( w f v ( w ))) ( w ( v Y f S ( Z ) v ( S ( X ) w ))); M ⊳ ( V ⊗ W ) → ( M ⊳ V ) ⊳ W, ( v ⊗ w f ( v ⊗ w )) ( w ( v Zf ( S ( Y ) v ⊗ S ( X ) w )) . Assume that V = H = W in the hexagon diagram. After evaluating the resulting mor-phisms at 1, we obtain the equality: µ ( h Y µ ( x f S ( Z ) h ( xS ( X )))) = Zµ ( h f S ( Z ) h S ( Y ) ( S ( Y ) h S ( X ))) . (5.3) Remark . Notice, that if H is a Hopf algebra, then the condition (5.3) becomes: µ ( h µ ( x f h ( x ))) = µ ( h f h ( h )) , which is exactly the right contramodule condition as used in [3] (see Definition 3.1).The unitality condition τ k = id M gives the following relation. For f ∈ Hom l ( k, M ),(5.4) f (1) = µ ( x ε ( x ) f (1)) . Definition 5.3.
Let H be a quasi-Hopf algebra. A pair ( M, µ ) , where M is a left H -moduleand µ : Hom( H, M ) → M is a k -linear map, is called a left-right anti-Yetter-Drinfeldcontramodule of type I, if it satisfies the equalities (5.2) , (5.3) and (5.4) .A morphism of two aY D -contramodules ( M, µ ) → ( M ′ , µ ′ ) is an H -morphism f : M → M ′ , which commutes with µ and µ ′ . Theorem 5.4.
The category of aY D -contramodules of type I for a quasi-Hopf algebra H is isomorphic to w- Z H M ( H M op ) .Proof. We have seen that an object in the center gives an aY D -contramodule. Assumethat we have a morphism of two central objects: f : ( M, τ ) → ( M ′ , τ ′ ). Then we have: H ⊲ M
Id⊲f (cid:15) (cid:15) τ H / / M ⊳ H f⊳Id (cid:15) (cid:15) ev / / M f (cid:15) (cid:15) H ⊲ M ′ τ ′ H / / M ′ ⊳ H ev / / M ′ where the left square commutes by the centrality of f and the right square commutesobviously. Since the top row is µ and the bottom row is µ ′ so f is a map of aY D -contramodules.Conversely, let ( M, µ ) be an aY D -contramodule. By Lemma 5.1, there is a natural trans-formation τ : id ⊲ M → M ⊳ id . Formula (5.3) gives the hexagon axiom for τ . Equality(5.4) gives the condition τ k = id .We observed that the constructions of central elements from aY D -contramodules and viceversa are inverses of each other, thus we have proved the isomorphism of categories. (cid:3) Remark . For a Hopf algebra with an invertible antipode the center Z H M ( H M op ) andthe weak center w- Z H M ( H M op ) coincide (one can easily write the formula for the τ − -see the proof of Proposition 3.3 and Remark 3.4). In the quasi-Hopf case we can neitherprove nor disprove the analogous fact.5.2. Anti-Yetter-Drinfeld contramodules II.
We can introduce aY D -contramodulesof type II using the actual categorical evaluation and adjunctions for internal homomor-phisms.
Lemma 5.6.
Let M ∈ H M op . Natural transformations τ ∈ Nat ( id ⊲ M, M ⊳ id ) are in1-1 correspondence with k -linear maps ν : Hom k ( H, M ) → M , such that (5.5) h · ( ν ( f )) = ν ( h f ( S ( h ) − h )) . Proof.
Consider the morphism τ H : H ⊲ M → M ⊳ H . Using the adjunction it is in one-to-one correspondence with the morphism: ˆ τ H : H ⊗ Hom l ( H, M ) → M . Let’s define ν bythe rule: ν ( f ) = ˆ τ H (1 ⊗ f )for any f ∈ Hom k ( H, M ) with the necessary H -module structure.The right action of H on itself is a morphism in H M , so, because ˆ τ is natural, we havethe formula: ˆ τ ( x ⊗ f ) = ˆ τ (1 ⊗ f ( − · x )). Using this we get the following equalities: h · ν ( f ) = h · ˆ τ (1 ⊗ f ) = ˆ τ ( h · (1 ⊗ f )) = ˆ τ ( h ⊗ h f ( S ( h ) − ))= ˆ τ (1 ⊗ h f ( S ( h ) − h )) = ν ( h f ( S ( h ) − h )) . Conversely, given a k -linear map ν : Hom k ( H, M ) → M , equip the vector space Hom k ( H, M )with an H -module structure in the following way: h · f = h f ( S ( h ) − h ) . Denote this H -module by Hom ∆ ( H, M ). Then (5.5) implies that ν is a morphism in thecategory H M . Consider also a map: θ V : V ⊗ Hom l ( V, M ) → Hom ∆ ( H, M ) , v ⊗ f ( x f ( xv )) . One can easily check that it is a morphism in H M . Now define ˆ τ V as a composition ν ◦ θ V . Clearly, ˆ τ V is natural in V . These correspondences are mutually inverse. For agiven v ∈ V consider a morphism H → V by the rule h h · v . By naturality of ˆ τ V , it isuniquely defined from ν . (cid:3) YCLIC COHOMOLOGY OF QUASI-HOPF ALGEBRAS AND HOPF ALGEBROIDS 15
Remark . It will be useful to write explicitly the reconstruction formula of τ for a given M ∈ H M and a map ν : Hom k ( H, M ) → M satisfying (5.5). For V ∈ H M , the map τ V : Hom l ( V, M ) → Hom r ( V, M ) is constructed by the rule:(5.6) f ν ( x Z f ( S ( Z ) xY S − ( β ) S − ( X ) − )) . We can see it from the proof of Lemma 5.6 and the adjuction.Analogously, one can observe that ν = ev r ◦ τ H , where ev r is a composition of maps(unit ⊗ id ) : Hom r ( H, M ) → H ⊗ Hom r ( H, M ) and ev r .As for type I from the hexagon axiom we get: ν ( h Z Y ν ( x Z f S ( Z ) κ ( h ) ( κ ( x ) S ( X ))))= Zν ( h Z f S ( Z )( κ ( h )) S ( Y ) ( S ( Y )( κ ( h )) S ( X ))) , (5.7)where we used the notation κ ( x ) = S ( Z ) xY S − ( β ) S − ( X ).The unitality condition τ k = id M applied with the rule (5.6) gives the following relation.For f ∈ Hom l ( k, M ), f (1) = ν ( x ε ( S ( Z ) xY S − ( β ) S − ( X )) Z f (1)) . For a quasi-Hopf algebra the equality ε ◦ S = ε holds (for the proof see [7]). So we cansimplify: ε ( S ( Z ) xY S − ( β ) S − ( X )) Z = Zε ( Y ) ε ( X ) ε ( β ) ε ( x ) = ε ( β ) ε ( x ) . Finally, we have:(5.8) f (1) = ν ( x ε ( β ) ε ( x ) f (1)) . Definition 5.8.
Let H be a quasi-Hopf algebra. A pair ( M, ν ) , where M is a left H -moduleand ν : Hom( H, M ) → M is a k -linear map, is called a left-right anti-Yetter-Drinfeldcontramodule of type II, if it satisfies the equalities (5.5) , (5.7) and (5.8) . Theorem 5.9.
The category of aY D -contramodules of type II for a quasi-Hopf algebra H is isomorphic to w- Z H M ( H M op ) .Proof. Similar to Theorem 5.4. (cid:3)
Remark . As we mentioned before, the difference between aY D -contramodules of typeI and the ones of type II comes from the difference between the naive evaluation and thecategorical evaluation in the category of H -modules for a quasi-Hopf algebra H . The twostructure maps µ and ν are not the same. For the convenience of the reader we providethe explicit formulas relating the two structures.Given ( M, µ ), an aY D -contramodule of type I, one keeps the H -module structure on M unchanged, but as an aY D -contramodule of type II ( M, ν µ ) is given by the formula:(5.9) ν µ ( f ) = Rµ ( h f ( hS − ( Q ) S − ( α ) P )) , for any f ∈ Hom(
H, M ) . Conversely, given (
M, ν ), an aY D -contramodule of type II, one can define (
M, µ ν ), an aY D -contramodule of type I by the formula:(5.10) µ ν ( f ) = ν ( h ( Z · f )( hY S − ( β ) S − ( X ))) , for any f ∈ Hom(
H, M ) . Cohomology theory with contramodule coefficients.
Let’s begin with the dis-cussion of stability. Recall the Definition 2.3 of Z ′ H M ( H M op ). We want to write anexplicit formula for this condition. First for a given aY D -contramodule M of type I andan element m ∈ M we can define a k -linear map: r ′ m : H → M , by the rule:(5.11) x βxS − ( Q ) S − ( α ) P m.
Definition 5.11.
Let M be an aY D contramodule of type I. It is called stable ( saY D -contramodule), if for all m ∈ M we have (5.12) Rµ ( r ′ m ) = m. We immediately obtain the following Corollary of Theorem 5.9.
Corollary 5.12.
The category of saY D -contramodules for H is equivalent to Z ′ H M ( H M op ) .Proof. The condition (2.2) that id M maps to the same under the chain of maps is literallythe equation (5.12) if we use the explicit formulas for adjunctions and τ and notice that ε ( P ) QβS ( R ) = β (cid:3) Using Lemma 2.5, we can construct Hopf-cyclic cohomology for quasi-Hopf algebras with saY D -contramodule coefficients. 6.
Hopf algebroids
First we will give all the necessary definitions following [1]. Let k be a commutative ringand R an algebra over k (not necessary commutative). A ring A together with a ringmap η : R → A is called an R -ring. Categorically, R -rings are monoids in the category of R -bimodules.We will use the following easy observation. Giving the ring map η : R ⊗ k R op → A isequivalent to giving two (so called, source and target) maps: s := η ( − ⊗ R ) : R → A and t := η (1 R ⊗ − ) : R op → A .Similarly R -corings are defined as comonoids in the category of R -bimodules. So an R -coring is a triple ( C, ∆ , ǫ ), where C is an R -bimodule, and ∆ : C → C ⊗ R C and ǫ : C → R are R -bimodule maps satisfying coassiativity and counit conditions.Bialgebroids are generalization of bialgebras, but now algebra and coalgebra structuresare defined in different monoidal categories. Definition 6.1.
Let R l be a k -algebra. A left R l -bialgebroid B l consists of R l ⊗ k R opl -ring ( B l , s l , t l ) and R l -coring ( B l , ∆ l , ǫ l ) on the same k -module B l , such that(1) the bimodule structure in the R l -coring ( B l , ∆ l , ǫ l ) is related to the R l ⊗ k R opl -ringstructure via r · b · r ′ := s l ( r ) t l ( r ′ ) b, r, r ′ ∈ R l , b ∈ B l (2) the coproduct ∆ l : B l → B l ⊗ R l B l corestricts to a k -algebra map from B l to B l × lR l B l .Here B l × lR l B l is the Takeuchi product defined by: B l × lR l B l := { X i b i ⊗ R l b ′ i | X i b i t l ( r ) ⊗ R l b ′ i = X i b i ⊗ R l b ′ i s l ( r ) ∀ r ∈ R l } . YCLIC COHOMOLOGY OF QUASI-HOPF ALGEBRAS AND HOPF ALGEBROIDS 17 (3) The left counit is a left character of the R l -ring ( B l , s l ) , i.e., ǫ l ( bb ′ ) = ǫ l ( bs l ( ǫ l b ′ )) = ǫ l ( bt l ( ǫ l b ′ )) , ∀ b, b ′ ∈ B l So a bialgebroid is a quintuple of data ( B l , R l , s l , t l , ∆ l , ǫ l ), but we will usually write it asjust B l .Analogously one defines right bialgebroids. Definition 6.2.
Let R r be a k -algebra. A right R r -bialgebroid B r consists of R r ⊗ k R opr -ring ( B r , s r , t r ) and R r -coring ( B r , ∆ r , ǫ r ) on the same k -module B r . The following conditionholds:(1) R r -bimodule structure in the coring ( B r , ∆ r , ǫ r ) is given by: r · b · r ′ := bt r ( r ) s r ( r ′ ) , r, r ′ ∈ R r , b ∈ B r . (2) The coproduct ∆ r : B r → B r ⊗ R r B r corestricts to a k -algebra map from B r to B r × rR r B l . Here one changes the Takeuchi product: B r × rR r B r := { X i b i ⊗ R r b ′ i | X i s r ( r ) b i ⊗ R r b ′ i = X i b i ⊗ R r t r ( r ) b ′ i ∀ r ∈ R r } . (3) The right counit is a right character of the R r -ring ( B r , s r ) , i.e., ǫ r ( bb ′ ) = ǫ r ( s r ( ǫ r b ) b ′ ) = ǫ r ( t r ( ǫ r b ) b ′ ) , ∀ b, b ′ ∈ B r . We will use the following Sweedler’s notation for coproducts:(6.1) ∆ l ( b ) = b ⊗ R l b , ∆ r ( b ) = b ⊗ R r b . Let us consider a left bialgebroid B l . In particular B l is a ring, so we can consider acategory of left modules over it B l M . Any left B l -module M is also a R l -bimodule via r · m · r ′ := s l ( r ) t l ( r ′ ) m, r, r ′ ∈ R l , m ∈ M. Given
M, N ∈ B l M , we have an R l -bimodule M ⊗ R l N . It can be supplied with the left B l -module structure via the left coproduct ∆ l : b · ( m ⊗ R l n ) = b m ⊗ R l b n, ∀ b ∈ B l Remark . Schauenburg proved [20] that for a R l ⊗ k R opl -ring ( B l , s l , t l ), the followingstructures are equivalent: • structure of a left algebroid on B l • a monoidal structure on the category B l M , such that the forgetful functor B l M → R l M R l is strictly monoidal.Similarly, any right B r -module M is an R r -bimodule via r · m · r ′ := ms r ( r ′ ) t r ( r ) , r, r ′ ∈ R r , m ∈ M. If M, N ∈ M B r , we can induce a right B r -module structure on M ⊗ R r N via right comul-tiplication ∆ r . Given an R r ⊗ k R opr -ring ( B r , s r , t r ), the structures are equivalent: • structure of a right algebroid on B r • a monoidal structure on the category of right B r -modules M B r , such that theforgetful functor M B r → R r M R r is strictly monoidal.Finally we can recall the definition of a Hopf algebroid. Definition 6.4.
A Hopf algebroid is given by a triple ( H l , H r , S ) ,where H l = ( H l , R l , s l , t l , ∆ l , ǫ l ) is a left R l -bialgebroid, H r = ( H r , R r , s r , t r , ∆ r , ǫ r ) is a right R r -bialgebroid on the same k -algebra H and S : H → H is a k -module map satisfying the following axioms:(1) s l ◦ ǫ l ◦ t r = t r , s r ◦ ǫ r ◦ t l = t l , t l ◦ ǫ l ◦ s r = s r , t r ◦ ǫ r ◦ s l = s l . (2) Mixed coassociativity: (∆ l ⊗ R r id H )∆ r = ( id H ⊗ R l ∆ r )∆ l and (∆ r ⊗ R l id H )∆ l = ( id H ⊗ R r ∆ l )∆ r .(3) For r ∈ R l , r ′ ∈ R r and h ∈ H , we have S ( t l ( r ) ht r ( r ′ )) = s r ( r ′ ) S ( h ) s l ( r ) . (4) m ◦ ( S ⊗ R l id H ) ◦ ∆ l = s r ◦ ǫ r and m ◦ ( id H ⊗ R r S ) ◦ ∆ r = s l ◦ ǫ l , where m is amultiplication in the k -algebra H .Remark . From property (1) it follows that R l is isomorphic to R opr .Mixed coassociativity can be written in Sweedler’s notation as:( h ) ⊗ R l ( h ) ⊗ R r h = h ⊗ R l ( h ) ⊗ R r ( h ) and ( h ) ⊗ R r ( h ) ⊗ R l h = h ⊗ R r ( h ) ⊗ R l ( h ) . Let H = ( H l , H r , S ) be a Hopf algebroid. The map S is called the antipode. From now onwe will always assume that the antipode S is invertible. As was proved in [1, Proposition4.4], the antipode is an antihomomorphism of the ring H . Remark . For a Hopf algebroid H and any h ∈ H we have the following identities:(6.2) S − ( h ) h = t r ǫ r ( h ) , h S − ( h ) = t l ǫ l ( h ) , see [16, Lemma 4].For r ∈ R l we have(6.3) t r ǫ r t l ( r ) = S − ( t l ( r )) , s r ǫ r s l ( r ) = S ( s l ( r )) , see [16, 1.8]. Lemma 6.7.
For a Hopf algebroid H , and h ∈ H , r ∈ R l , r ′ ∈ R r one has: (6.4) t r ( r ′ ) S − ( h ) t l ( r ) = S − ( s l ( r ) hs r ( r ′ )) Proof.
Because the antipode is a unital map, property (3) of Definition 6.4 implies: S ( t l ( r )) = s l ( r ) and S ( t r ( r ′ )) = s r ( r ′ ). Because the antipode is an antihomomorphism,we have: S − ( s l ( r ) hs r ( r ′ )) = S − ( s r ( r ′ )) S − ( h ) S − ( s l ( r )) = t r ( r ′ ) S − ( h ) t l ( r ) . (cid:3) A structure of a left module over the Hopf algebroid H is a structure of a left moduleover the underlying k -algebra H . We want to study the category of left H -modules H M .Firstly recall that it is a monoidal category, because H l is a left bialgebroid, and a modulestructure is given by left comultiplication ∆ l . We will denote the monoidal structuresimply by ⊗ .If M, N are R -bimodules, we will denote by Hom( M, N ) R a morphism of right R -modules,and by Hom R ( M, N ) a morphism of left R -modules. We need the following easy lemma: Lemma 6.8. If M, N ∈ H M , then f ∈ Hom(
M, N ) R l is equivalent to f ∈ Hom R r ( M, N ) .Similarly, f ∈ Hom(
M, N ) R r is equivalent to f ∈ Hom R l ( M, N ) . YCLIC COHOMOLOGY OF QUASI-HOPF ALGEBRAS AND HOPF ALGEBROIDS 19
Proof. If f ∈ Hom(
M, N ) R l , then for any r ∈ R l and m ∈ M , f ( t l ( r ) m ) = t l ( r ) f ( m ).Now let’s consider a ∈ R r , then f ( s r ( a ) m ) = f ( t l ǫ l s r ( a ) m ) (here we used property (1) inDefinition 6.4). By R l -right linearity, f ( t l ǫ l s r ( a ) m ) = t l ǫ l s r ( a ) f ( m ). Switching back weget s r ( a ) f ( m ). Three other implications are proved analogously. (cid:3) The closeness of the category of left modules over a Hopf algberoid was proved in [21].This result is crucial for most of the constructions in the paper. To fix notation and forthe convenience of the reader we are going to include a proof here.
Proposition 6.9.
The category of left modules over a Hopf algebroid H M is a biclosedmonoidal category.Proof. For any
M, N ∈ H M , we can define the left internal Hom by(6.5) Hom l ( M, N ) = Hom(
M, N ) R l , h · ϕ = h ϕ ( S ( h ) − ) . Notice that we used the structure of right comultiplication here.First of all, we need to prove that this is well defined. For any a ∈ R r consider, by Lemma6.8 and property (3) in Definition 6.4: h s r ( a ) ϕ ( S ( h ) − ) = h ϕ ( s r ( a ) S ( h ) − ) = h ϕ ( S ( h t r ( a )) − ) . So at least the combination of symbols h ϕ ( S ( h ) − ) makes sense. Now let’s check if ϕ ∈ Hom(
M, N ) R l implies that h ϕ ( S ( h ) − ) is also a morphism of right R l -modules. Todo that we observe: t l ( r ) h ⊗ R r h = s r ǫ r t l ( r ) h ⊗ R r h = h ⊗ R r t r ǫ r t l ( r ) h = h ⊗ R r S − ( t l ( r )) h by property (1) in Definition 6.4, right Takeuchi property, and (6.3) respectively.Finally, using that S is an antihomomorphism, we have: t l ( r ) h ϕ ( S ( h ) − ) = h ϕ ( S ( h ) t l ( r ) − ) , so h ϕ ( S ( h ) − ) ∈ Hom(
M, N ) R l .An adjunction map(6.6) ζ l : Hom H M ( M ⊗ N, L ) → Hom H M ( M, Hom l ( N, L ))is defined similarly to the case of Hopf algebras:(6.7) f ( m f ( m ⊗ − )) . Take f ∈ Hom H M ( M ⊗ N, L ). Observe that ζ l ( f ) is a morphism of H -modules: h · ζ l ( f )( m ) = h · f ( m ⊗ − ) = h ( f ( m ⊗ S ( h ) − )) = f (( h ) m ⊗ ( h ) S ( h ) − ) (i) = f ( h m ⊗ ( h ) S (( h ) ) − ) (ii) = f ( h m ⊗ R l s l ǫ l ( h ) − )= f ( t l ǫ l ( h ) h m ⊗ R l − ) (iii) = ζ l ( f )( hm ) . Where (i) is mixed coassociativity, (ii) property (4) of Definition 6.4, (iii) R l -bimodulestructure and counit.Define the evaluation morphism:(6.8) ev l : Hom l ( M, N ) ⊗ M → N, ϕ ⊗ m ϕ ( m ) . Notice that the morphism ev l is the morphism of left H -modules. Indeed,ev l ( h · ( ϕ ⊗ m )) = ev l ( h · ϕ ⊗ h · m ) = ( h ) ϕ ( S (( h ) ) h m )= h ϕ ( S (( h ) )( h ) m ) = h ϕ ( s r ǫ r ( h ) m )= h s r ǫ r ( h ) ϕ ( m ) = h · ev l ( ϕ ⊗ m ) . Using the evaluation we can construct a map that is to be the inverse of ζ l :(6.9) η l : Hom H M ( M, Hom l ( N, L )) → Hom H M ( M ⊗ N, L )by the rule:(6.10) η l ( g ) = ev l ◦ ( g ⊗ id ) . It is a morphism of H -modules as a composition of such. To see that these maps aremutually inverse we can forget about additional structure and use the usual tensor-homadjunction theorem for R l -bimodules. Hence we proved that the category H M is leftclosed.For any M, N ∈ H M , we can define the right internal Hom by(6.11) Hom r ( M, N ) = Hom R l ( M, N ) h · ϕ = h ϕ ( S − ( h ) − ) , unlike the case of left internal homs, the underlying structure is a morphism of left R l -modules.As for the left hom, we need to check that this is well defined. For any a ∈ R r consider: h t r ( a ) ϕ ( S − ( h ) − ) = h ϕ ( t r ( a ) S − ( h ) − ) = h ϕ ( S − ( h s r ( a )) − )by Lemma 6.8 and Lemma 6.7 respectively. So the notation h ϕ ( S − ( h ) − ) makes sense.Now we check that if ϕ ∈ Hom R l ( M, N ) then h ϕ ( S − ( h ) − ) is also a morphism of left R l -modules. First, make an observation: h ⊗ R r s l ( r ) h = h ⊗ R r t r ǫ r s l ( h ) = s r ǫ r s l ( r ) h ⊗ R r h = S ( s l ( r )) h ⊗ R r h by property (1) in Definition 6.4, right Takeuchi property, and (6.3) respectively. Because S − is an antihomomorphism, we have: s l ( r ) h ϕ ( S − ( h ) − ) = h ϕ ( S − ( h ) s l ( r ) − ) , so h ϕ ( S − ( h ) − ) ∈ Hom R l ( M, N ).Let’s define an adjunction map:(6.12) ζ r : Hom H M ( N ⊗ M, L ) → Hom H M ( M, Hom r ( N, L ))by the rule:(6.13) f ( m f ( − ⊗ m )) . Define also the evaluation morphism:(6.14) ev r : M ⊗ Hom r ( M, N ) → N, m ⊗ ϕ ϕ ( m ) . YCLIC COHOMOLOGY OF QUASI-HOPF ALGEBRAS AND HOPF ALGEBROIDS 21
Check that the morphism ev r is the morphism of left H -modules:ev r ( h · ( m ⊗ ϕ )) = ev r ( h · m ⊗ h · ϕ ) = ( h ) ϕ ( S − (( h ) ) h m )= h ϕ ( S − (( h ) )( h ) m ) = h ϕ ( t r ǫ r ( h ) m )= h t r ǫ r ( h ) ϕ ( m ) = h · ev r ( m ⊗ ϕ ) . As above one can construct the inverse of ζ r using the evaluation, we skip the details. So H M is right closed. (cid:3) At the end of the section let’s observe the following useful fact.
Lemma 6.10.
Let ϕ ∈ Hom l ( M, N ) , and ψ ∈ Hom r ( M, N ) . Then: t l ( r ) · ϕ = ϕ ( s l ( r ) − ) , s l ( r ) · ϕ = s l ( r ) ϕ ( − ) s l ( r ) · ψ = ψ ( t l ( r ) − ) , t l ( r ) · ψ = t l ( r ) ψ ( − ) . Proof.
To prove this, we need to recall that ∆ r is both R l - and R r -bimodule map [1]. So∆ r ( t l ( r )) = 1 ⊗ R r t l ( r )1 and ∆ r ( s l ( r )) = s l ( r )1 ⊗ R r . Then the proof of the firstformula is as follows: t l ( r ) · ϕ = 1 ϕ ( S ( t l ( r )1 ) − ) = 1 ϕ ( S (1 ) s l ( r ) − ) = ϕ ( s l ( r ) − )where the last equality is due to ∆ r being unital, while the second formula is obtainedthus: s l ( r ) · ϕ = s l ( r )1 ϕ ( S (1 ) − ) = s l ϕ ( − ) . The two others are proved similarly. (cid:3) Anti-Yetter-Drinfeld contramodules for Hopf algebroids
In this section we are going to describe the coefficients for the Hopf-cyclic cohomologytheory for Hopf algebroids. We will discuss only aYD contramodules and proceed similarlyto the case of quasi-Hopf algebras. The category H M is biclosed monoidal. Using theadjunctions for M, V, W ∈ H M :Hom H M ( W ⊗ V, M ) ≃ Hom H M ( W, Hom l ( V, M )) , and Hom H M ( V ⊗ W, M ) ≃ Hom H M ( W, Hom r ( V, M )) , proved in Proposition 6.9, we can introduce the contragradient H M -bimodule category H M op as in Section 2. Specifically, for M ∈ H M op and V ∈ H M , the action is given by:(7.1) M ⊳ V = Hom r ( V, M ) , and V ⊲ M = Hom l ( V, M ) . We want to understand the center Z H M ( H M op ).First of all let us recall the notion of contramodules over bialgebroids (as it was definedin [2], generalizing the classical contramodules for coalgebras [8]). Definition 7.1.
A right contramodule over a left R l -bialgebroid B l is a right R l -module M together with a right R l -module map: µ : Hom( B l , M ) R l → M, called the contraaction, such that the diagrams commute: Hom( B , Hom( B , M ) R l ) R l Hom( B , M ) R l Hom(
B ⊗ R l B , M ) R l Hom( B , M ) R l M Hom( B , µ ) R l µµ ∼ = Hom(∆ l , M ) R l and Hom( R l , M ) R l Hom( B , M ) R l M Hom( ǫ l , M ) R l ∼ = µ where B l is a R l -bimodule via r · b · r ′ = s l ( r ) t l ( r ′ ) b . The right R l -module structureon Hom( B l , M ) R l is defined by f ( − ) · r = f ( r · − ) = f ( s l ( r ) − ) . The isomorphism Hom( B , Hom( B , M ) R l ) R l → Hom(
B ⊗ R l B , M ) R l is simply a tensor-hom adjunction. Let’s write the formulas explicitly; the first commutative diagram gives:(7.2) µ ( x µ ( y ϕ ( x ⊗ y ))) = µ ( h ϕ ( h ⊗ R l h )) , for ϕ ∈ Hom(
B ⊗ R l B , M ) R l . The second diagram is:(7.3) µ ( x m · ǫ l ( x )) = m, for any m ∈ M. As was observed in [2] or [15], we can equip a right contramodule M with a structure ofa left R l -module via: r · m = µ ( x m · ǫ l ( xs l ( r ))) , for any m ∈ M, r ∈ R l . Under this action the morphism µ is an R l -bimodule map, if we define a left R l -modulestructure on Hom( B l , M ) R l by r · f ( − ) = f ( − s l ( r )) (see loc.cit. ) Let us state the definitionof an anti-Yetter-Drinfeld contramodule over a Hopf algebroid. Notice that the definitionis slightly different from the one used in [15]. Definition 7.2.
An anti-Yetter-Drinfeld ( aY D ) contramodule M over a Hopf algebroid H is a left H -module and a right H l -contramodule, such that both underlying R l -bimodulestructures coinside: r · m · r ′ = s l ( r ) t l ( r ′ ) m , and the following aY D -condition holds: (7.4) h ( µ ( f ( − S − ( h )))) = µ ( h f ( S ( h ) − )) . Let’s explicitly write the condition that µ is R l -bilinear. Right R l -linearity is:(7.5) µ ( f ( s l ( r ) − )) = t l ( r ) µ ( f ( − )) , f ∈ Hom( H , M ) R l , r ∈ R l . YCLIC COHOMOLOGY OF QUASI-HOPF ALGEBRAS AND HOPF ALGEBROIDS 23
Left R l -linearity gives:(7.6) µ ( f ( − s l ( r ))) = s l ( r ) µ ( f ( − )) , f ∈ Hom( H , M ) R l , r ∈ R l . Remark . Let us observe that all expressions in equation (7.4) are well defined, if µ is R l -bilinear. We just need to check the linearity of the left hand side. Namely, we have byLemma 6.7 and Lemma 6.8 with formula (7.5) respectively: h ( µ ( f ( − S − ( h s r ( a ))))) = h ( µ ( f ( − t r ( a ) S − ( h )))) = h t r ( a ) µ ( f ( − S − ( h ))) . We are going to show that the category of aY D -contramodules is the weak center of abimodule category H M op . Lemma 7.4.
Let M ∈ H M op . Natural transformations τ ∈ Nat ( id ⊲ M, M ⊳ id ) are in 1-1correspondence with right R l -module maps µ : Hom( H , M ) R l → M (formula (7.5) ), whichare also left R l -module maps (formula (7.6) ), such that the property (7.4) holds.Proof. Consider f ∈ Hom( H , M ) R l , induce the H -module structure corresponding to theleft internal homomorphism. Given a morphism τ H : H ⊲ M → M ⊳ H , define: µ ( f ) = τ H ( f )(1), which we denote by ev ( τ H ( f )). Let x ∈ H and f ∈ Hom( H , M ) R l , observe that τ H ( f ( − h )) = ( τ H f )( − h ) since r h : H → H is a morphism in H M .First, let’s check that µ satisfies (7.5): µ ( f ( s l ( r ) − )) (i) = µ ( t l ( r ) · f ( − )) = ev ( τ ( t l ( r ) · f )) (ii) = ev ( t l ( r ) · τ ( f )) (i) = ev ( t l ( r )( τ ( f ))) = t l ( r ) τ ( f )(1) = t l ( r ) µ ( f ) , where (i) is Lemma 6.10 and (ii) holds since τ is H -linear.Now we check that µ satisfies (7.6): µ ( f ( − s l ( r ))) = ev τ ( f ( − s l ( r ))) = ev τ ( f )( − s l ( r ))= τ ( f )( s l ( r )) (i) = s l ( r ) τ ( f )(1) = s l ( r ) µ ( f ) , where (i) is due to the left R l -linearity of τ .Now we can check the aY D -condition: µ ( h f ( S ( h ) − )) = ev τ ( h f ( S ( h ) − )) = ev τ (( h · f )( − )) = ev h · τ ( f ( − ))= ev h ( τ f )( S − ( h ) − ) = h ( τ f )( S − ( h ))= h ev τ ( f ( − S − ( h ))) = h µ ( f ( − S − ( h ))) . Conversely, given a map µ : Hom( H , M ) R l → M , satisfying (7.4), (7.5) and (7.6). For any V ∈ H M one can define a map: τ V : V ⊲ M → M ⊳ V, by the rule f µ ( x f ( x − )) . We check that it is well-defined. First of all, for a given v ∈ V and f ∈ Hom(
V, M ) R l ,a map x f ( xv ) is in Hom( H , M ) R l . Indeed, f ( t l ( r ) xv ) = t l ( r ) f ( xv ), because f is R l -linear.Then we need to understand that the target, a map v µ ( x f ( xv )), is in Hom R l ( V, M ).It follows directly from formula (7.6).
Finally, we should check, that the constructed map τ is a morphism of H -modules:( h · τ ( f ))( v ) = h τ f ( S − ( h ) v ) = h µ ( x f ( xS − ( h ) v )) (7.4) = µ ( x ( h · f )( v )) = ( τ ( h · f ))( v ) . Given an H -homomorphism ϕ : V → W , consider a diagram:Hom l ( W, M ) −◦ ϕ (cid:15) (cid:15) τ W / / Hom r ( W, M ) −◦ ϕ (cid:15) (cid:15) Hom l ( V, M ) τ V / / Hom r ( V, M ) . It commutes, so we have naturality.These correspondences are mutually inverse. The only nontrivial direction follows fromthe observation that τ H ( f ( − h )) = ( τ H f )( − h ). (cid:3) Finally, we can formulate the following:
Proposition 7.5.
The category of aY D -contramodules for a Hopf algebroid H is isomor-phic to w- Z H M ( H M op ) .Proof. Given a central element (
M, τ ) ∈ Z H M ( H M op ), define µ as in Lemma 7.4. Itsatisfies aYD condition, so we just need to check that M is a right H -contramodule.By definition of the center and because all associativity maps are trivial, the diagram(7.7) W ⊲ ( V ⊲ M ) Id⊲τ / / W ⊲ ( M ⊳ V ) / / ( W ⊲ M ) ⊳ V τ⊳Id / / ( M ⊳ W ) ⊳ V (cid:15) (cid:15) ( W ⊗ V ) ⊲ M O O τ / / M ⊳ ( W ⊗ V )commutes for any V, W ∈ H M . Taking f along the second row to w ⊗ v µ ( h f ( h w ⊗ h v )) is the same as taking it the long way around, which is w ⊗ v µ ( x µ ( h f ( xw ⊗ hv ))). In particular, if we take H for V and W and assume v = w = 1,we will obtain the contraaction condition. Next, we have that R l ⊲ M → M ⊳ R l is theidentity.Define: m ( r ) = t l ( r ) m , then(7.8) µ ( h m ( ǫ l ( hs l ( r )))) = µ ( h m ( h · r )) = τ m ( r ) = m ( r ) = t l ( r ) m. The first equality is just an H -module structure of R l .Finally, suppose that f : M → M ′ is a map in the center, then we have H ⊲ M
Id⊲f (cid:15) (cid:15) τ / / M ⊳ H f⊳Id (cid:15) (cid:15) ev / / M f (cid:15) (cid:15) H ⊲ M ′ τ / / M ′ ⊳ H ev / / M YCLIC COHOMOLOGY OF QUASI-HOPF ALGEBRAS AND HOPF ALGEBROIDS 25 where the left square commutes by the centrality of f and the right square commutesobviously. Since the top row is µ M and the bottom row is µ M ′ so f is a map of aY D -contramodules.Conversely, consider an aY D -contramodule M . By Lemma 7.4 it provides us with naturalmorphisms τ V : V ⊲ M → M ⊳ V for every V ∈ H M . Observe that the considerations (7.7)and (7.8) are of the if and only if kind, so µ being a contraaction implies that the diagram(7.7) commutes and τ R l = id .If φ : M → M ′ is a map of aY D -contramodules then( φ ◦ τ f )( v ) = φ ( µ ( h f ( hv ))) = µ ( h φ ( f ( hv ))) = τ ( φ ◦ f )( v )so that φ is a morphism in Z H M ( H M op ). What has been shown so far is that if M isan aY D -contramodule, then ( M, τ ) ∈ Z H M ( H M op ) and any g : M → M ′ a morphism of aY D -contramodules induces a morphism between the corresponding central elements.As we noted in Lemma 7.4, the constructions of central elements from aY D -contramodulesand vice versa are inverses of each other, thus we have proved the isomorphism of cate-gories. (cid:3) Remark . Similar to Remark 5.5 we cannot prove nor disprove whether the centercoincides with the weak center in the case of Hopf algebroids.Recall the following:
Definition 7.7. An aY D -contramodule M is called stable ( saY D ), if µ ( r m ) = m , wherethe map r m : H → M , is given by h hm . By Lemma 2.4, Z ′ H M ( H M op ) is the full subcategory of the center that consists of objectssuch that the identity map Id ∈ Hom H ( M, M ) is mapped to same via(7.9) Hom H ( M, M ) ≃ Hom H (1 , M ⊲ M ) ≃ Hom H (1 , M ⊳ M ) ≃ Hom H ( M, M ) . We immediately obtain the following Corollary of Proposition 7.5.
Corollary 7.8.
The category of saY D -contramodules for a Hopf algebroid H is isomorphicto Z ′ H M ( H M op ) .Proof. Let M be an aY D -contramodule, then to ensure that the condition (7.9) is satisfied,we need that τ ( Id )( m ) = µ ( h hm ) = µ ( r m ) = m which is exactly the saY D condition. On the other hand, if M is a central element thatsatisfies (7.9), then it also satisfies that the chain of isomorphisms(7.10) Hom H ( V, M ) ≃ Hom H (1 , V ⊲ M ) ≃ Hom H (1 , M ⊳ V ) ≃ Hom H ( V, M )is identity for any V ∈ H M . Take V = H and note that r m ∈ Hom H ( H , M ) so that (7.10)implies that τ ( r m ) = r m and so µ ( r m ) = ev τ ( r m ) = ev ( r m ) = m , i.e., M is a stable aY D -contramodule. (cid:3) Using Lemma 2.5, we can construct Hopf-cyclic cohomology for Hopf algebroids with saY D -contramodule coefficients.
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