Anti-Yetter-Drinfeld Modules for Quasi-Hopf Algebras

We apply categorical machinery to the problem of defining anti-Yetter-Drinfeld modules for quasi-Hopf algebras. While a definition of Yetter-Drinfeld modules in this setting, extracted from their categorical interpretation as the center of the monoidal category of modules has been given, none was available for the anti-Yetter-Drinfeld modules that serve as coefficients for a Hopf cyclic type cohomology theory for quasi-Hopf algebras. This is a followup paper to the authors' previous effort that addressed the somewhat different case of anti-Yetter-Drinfeld contramodule coefficients in this and in the Hopf algebroid setting.


Introduction
It is an interesting fact that the theory of coefficients in Hopf cyclic theories began with what is now known as anti-Yetter-Drinfeld modules in [9,6,7] that followed [2,3]. It was not until [1] that anti-Yetter-Drinfeld contramodules were introduced. The latter, in retrospect, seem a lot more natural, though they involve notions that are less so. In this followup paper to [10] we make the modifications necessary to deal with the definitions of anti-Yetter-Drinfeld modules for quasi-Hopf algebras which generalize the Hopf algebras by relaxing the coassociativity condition to coassociativity up to a specified isomorphism. This isomorphism complicated matters sufficiently that a direct generalization of the formulaic approach used for Hopf algebras is not possible. Similar to what we do here, but less laborious calculations, have been performed in [11] where the categorical notion of the center of a monoidal category of modules over a quasi-Hopf algebra has been unwound into formulas. Our task is complicated by two factors: we deal with a certain bimodule category over the category of modules over a quasi-Hopf algebra and we allow not only finite dimensional representations. The main theme of [10] was exploiting the fact that the category of modules is biclosed, i.e., it possesses internal Homs. This allows for a definition of a natural bimodule category over it, the center of which is what we are looking for. The justification for the importance of the center is the observation that its elements (or rather the ones satisfying an additional stability condition) can be used to quickly manufacture a functor called a symmetric 2contratrace [8] and thus define a cyclic cohomology theory. As mentioned above, historically anti-Yetter-Drinfeld modules appeared before their contramodule versions, but in this paper we rely on the conceptual definition of generalized anti-Yetter-Drinfeld contramodules as in [10] to obtain the module version. In particular the question of stability is explicitly reduced to the contramodule case, though it is immediate that it is equivalent to the stability in [8], though not completely analogous to what is possible to do in the Hopf algebra case; see Remark 2.7. The paper is organized as follows. In Section 2 we review and augment some generalities from [10] that we use subsequently. Section 3 is a review of the basics of quasi-Hopf algebras and recalls the definition of internal Homs, from [10], for them. Finally in Section 4 we unravel the conceptual definitions of Section 2 into formulas. An interesting observation is the appearance of two distinct ways of writing down the formulas; these are identical for Hopf algebras but very different here. This mirrors a similar phenomenon that occurs in the contramodule case.
Acknowledgments: The authors wish to thank Masoud Khalkhali for stimulating questions and discussions. The research of the second author was supported in part by the NSERC Discovery Grant number 406709.

Generalities
In this section we will extend the general formalism of [10] from generalized anti-Yetter-Drinfeld contramodule coefficients to their module variant. Recall from [8] that the main ingredient in constructing Hopf-cyclic cohomology is a symmetric 2-contratrace. It is with a view towards this goal that we undertake the following. Let M be a biclosed monoidal category, i.e., it possesses internal Homs. More precisely, the property of being biclosed implies in particular the existence of the following adjunctions for M, V, W ∈ M: A natural object to consider in this situation is the center of a bimodule category N ; roughly speaking a category with a left and a right action of our monoidal category M. However, here it becomes too restrictive. If in the definition of a (strong) center element N ∈ N we relax the condition that the maps τ : N ′ ⊲ N → N ⊳ N ′ are isomorphisms, we get a weak center. More formally:  [5]).
We need one more definition. We have from [10]: We called such M stable and denoted the full subcategory containing them by Z ′ M (M op ). These are exactly the generalized stable anti-Yetter-Drinfeld contramodules and the functor Hom M (−, M ) is a symmetric 2-contratrace. Recall that for an algebra A ∈ M the collection Hom M (A ⊗•+1 , M ) is naturally a cocyclic vector space.
2.1. The module variant modifications. Assume as above that M is biclosed and suppose further that there exists a tensor auto-equivalence (−) # of M together with natural identifications: Observe that should such a functor exist we would immediately have natural identifications: Consider # M, an M-bimodule category with the right and left M-module structures given by: Lemma 2.4. We have a functor between weak centers: Proof. The weak center structure on 1 ◭ M is obtained as follows: Recall a definition of stability for generalized anti-Yetter-Drinfeld modules from [8]: is identity then M is called stable. It is immediate that the following definition is equivalent to this one.
We have the following analogue of Lemma 2.3: We will denote the full subcategory of such M by Z ′ M ( # M). Proof. By definitions if M is stable then σ 1◭M = Id and so by Lemma 2.3 and the proof of Lemma 2.4 the centrality map τ : V # ⊗ M → M ⊗ V is such that its right dual is an isomorphism, i.e., D(τ ) is an isomorphism. If D reflects isomorphisms then τ is an isomorphism as well.
The M of the lemma above are exactly the generalized stable anti-Yetter-Drinfeld modules. We note that the functor Hom M (M ⊗−, 1) is a symmetric 2-contratrace in this case. This follows immediately from its isomorphism to Hom M (−, DM ) as DM ∈ Z ′ M (M op ). Remark 2.7. The current approach to general stability of aY D modules, via stability of aY D contramodules (implicitly as in [8] or explicitly as in Definition 2.5), may seem unsatisfactory but it is the only way in general. In particular situations one may do better. More precisely, it may happen that for M ∈ w-Z M ( # M) the map σ DM may have a predual, i.e., a σ M such that Id ◭ σ M = σ 1◭M . This is the case for Hopf algebras where σ M (m) = m 1 m 0 , but this question remains open in the quasi-Hopf algebra case.

Recalling Quasi-Hopf algebras
Let us remind the reader of all the necessary definitions following [4]. In this section k is a field. Definition 3.1. 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: Remark 3.2. In this paper we will use the Sweedler notation. Let's denote here we mean the summation. In particular, the equality (3.1) can be written as: We are interested in the category of left A-modules A M. It was proved in [4] that this category is monoidal if a tensor product of two left A-modules M and N is defined by the same formula as in the case of a bialgebra: The associativity morphism is no longer trivial as it was in the case of a bialgebra. Recall the definition of a quasi-Hopf algebra from [4]. Remark 3.5. We want to emphasize that the antipode S in the definition above is assumed to be invertible.

Anti-Yetter-Drinfeld modules for a quasi-Hopf algebra
Yetter-Drinfeld modules for a quasi-Hopf algebra were described in [11]. The consideration of Y D modules as centers of a monoidal category H M was crucial to write down the complicated formulas. In this section we are going to define the anti-Yetter-Drinfeld modules using the categorical approach from [8]. Unlike the Hopf-case, there are two ways to define aY D modules. Remark 4.1. Observe that as required according to Section 2 we have a very trivial identi- Furthermore, the D functor being essentially a vector space duality functor reflects isomorphisms and so as long as we insist on stability in the sense of Definition 2.5 we do not need to worry about the difference between the weak and the strong center.
If H is a Hopf algebra it was proved in [8] that the center Z H M ( # H M) is the same as anti-Yetter-Drinfeld modules. We are going to use this fact as a guide and give a description of aY D modules in the quasi-Hopf case.

4.1.
Anti-Yetter-Drinfeld modules I. We organize the material similar to [11].  By the hexagon axiom of the center, the following diagram is commutative:  Consider the following diagram: The left square commutes by the naturality of τ and the right square commutes by the definition of the center. If we start with 1 ⊲ m in the upper left corner we will get the equality: This condition is exactly the same as in the Hopf case. and (4.5).
. This is a morphism in the category by construction of M ⊗ r H and V # . We can formulate a Lemma similar to Lemma 4.2.  [1] , such that Proof. First assume that τ is given. Then for m ∈ M we define λ(m) := ( τ H (m))(1). From the fact that ( τ H (m)) is a right internal homomorphism we get (4.6). Conversely, given a k-linear map λ : M → M ⊗H we can consider it as an H-homomorphism M → M ⊗ r H by (4.6). Now for any V ∈ H M we define τ V by: Everything is constructed naturally and the one-to-one correspondence is clear.
Here we used that ε is algebra map and the formula (3.3). For a quasi-Hopf algebra the equality ε • S = ε holds (for the proof see [4]). So we can simplify ε(κ) = ε(α)ε(Q)ε(P ). Using (3.4), we get the final equality: (4.9) m = ε(m [1] )m [0] ε(α). And as before we have the following Theorem with a proof that is very similar to the type I case and so is omitted.
Theorem 4.9. The category of aY D-modules of type II for a quasi-Hopf algebra H is equivalent to w-Z H M ( # H M). Remark 4.10. Type I and type II aY D modules are different, though of course are equivalent as categories. The difference between them is like the difference between two maps: H ⊗ Hom r (H, V ) → V , where the first map is h ⊗ f → f (h) (naive evaluation) and the second one is ev r (h ⊗ f ) (actual evaluation). In the Hopf case these two evaluations are the same, but it is no longer true for a quasi-Hopf algebra. The reader is also invited to read the proof that in the Hopf case formula (4.3) is equivalent to (4.6) ([8, Lemma 2.2]).