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Generalized (anti) Yetter-Drinfeld modules as components of a braided T-category

If H is a Hopf algebra with bijective antipode and \alpha, \beta \in Aut_{Hopf}(H), we introduce a category_H{\cal YD}^H(\alpha, \beta), generalizing both Yetter-Drinfeld and anti-Yetter-Drinfeld modules. We construct a braided T-category {\cal YD}(H) having all these categories as components, which if H is finite dimensional coincides with the representations of a certain quasitriangular T-coalgebra DT(H) that we construct. We also prove that if (\alpha, \beta) admits a so-called pair in involution, then_H{\cal YD}^H(\alpha, \beta) is isomorphic to the category of usual Yetter-Drinfeld modules_H{\cal YD}^H.


Introduction
Let H be a Hopf algebra with bijective antipode S and α, β ∈ Aut Hopf (H). We introduce the concept of an (α, β)-Yetter-Drinfeld module, as being a left H-module right H-comodule M with the following compatibility condition: This concept is a generalization of three kinds of objects appeared in the literature. Namely, for α = β = id H , we obtain the usual Yetter-Drinfeld modules; for α = S 2 , β = id H , we obtain the so-called anti-Yetter-Drinfeld modules, introduced in [7], [8], [10] as coefficients for the cyclic cohomology of Hopf algebras defined by Connes and Moscovici in [5], [6]; finally, an (id H , β)-Yetter-Drinfeld module is a generalization of the object H β defined in [4], which has the property that, if H is finite dimensional, then the map β → End(H β ) gives a group anti-homomorphism from Aut Hopf (H) to the Brauer group of H. It is natural to expect that (α, β)-Yetter-Drinfeld modules have some properties resembling the ones of the three kinds of objects we mentioned. We will see some of these properties in this paper (others will be given in a subsequent one), namely the ones directed to our main aim here, which is the following: if we denote by H YD H (α, β) the category of (α, β) -Yetter-Drinfeld modules and we define YD(H) as the disjoint union of all these categories, then we can organize YD(H) as a braided T-category (or braided crossed group-category, in the original terminology of Turaev, see [16]) over the group G = Aut Hopf (H) × Aut Hopf (H) with multiplication (α, β) * (γ, δ) = (αγ, δγ −1 βγ). We also prove that the subcategory YD(H) f d consisting of finite dimensional objects has left and right dualities, and that, if H is finite dimensional, then YD(H) coincides with the representations of a certain quasitriangular T-coalgebra DT (H) that we construct. Our second aim is to prove that, if α, β ∈ Aut Hopf (H) such that there exists a so-called pair in involution (f, g) corresponding to (α, β), then H YD H (α, β) is isomorphic to H YD H . This result is independent on the theory concerning YD(H), but we can give it a very short proof using the results obtained during the construction of YD(H).

Preliminaries
We work over a ground field k. All algebras, linear spaces, etc. will be over k; unadorned ⊗ means ⊗ k . Unless otherwise stated, H will denote a Hopf algebra with bijective antipode S. We will use the versions of Sweedler's sigma notation: (2) . For unexplained concepts and notation about Hopf algebras we refer to [11], [12], [13], [15]. By α, β, γ... we will usually denote Hopf automorphisms of H. [0] , and denote, for a ∈ A, as an element in H ⊗ A ⊗ H. We can consider the Yetter-Drinfeld datum (H, A, H) as in [3] (the second H is regarded as an H-bimodule coalgebra), and the Yetter-Drinfeld category A YD(H) H , whose objects are k-modules M endowed with a left A-action (denoted by a ⊗ m → a · m) and a right H-coaction (denoted by m → m (0) ⊗m (1) ) satisfying the equivalent compatibility conditions for all a ∈ A and m ∈ M . Recall now from [9] the construction of the (left) diagonal crossed product H * ⊲⊳ A, which is an associative algebra constructed on H * ⊗ A, with multiplication given by , for all p ∈ H * and a ∈ A, see [9]. In the case when H is finite dimensional, by results in [1], [3] it follows that the category A YD(H) H is isomorphic to the category H * ⊲⊳A M of left modules over H * ⊲⊳ A.
Example 2.5 For β ∈ Aut Hopf (H), define H β as in [4], that is H β = H, with regular right H-comodule structure and left H-module structure given by Example 2.6 Take l an integer and define α l = S 2l ∈ Aut Hopf (H). The compatibility in An object in H YD H (S 2l , id) will be called an l − YD-module. Hence, a 0 − YD-module is a Yetter-Drinfeld module and a 1 − YD-module is an anti-Yetter-Drinfeld module. The right-left version of l − YD-modules has been introduced in [14].
Example 2.7 Let α, β ∈ Aut Hopf (H) and assume that there exist an algebra map f : H → k and a group-like element g ∈ H such that Definition 2.8 If α, β ∈ Aut Hopf (H) such that there exist f, g as in Example 2.7, we will say that (f, g) is a pair in involution corresponding to (α, β) (in analogy with the concept of modular pair in involution due to Connes and Moscovici) and the (α, β)-Yetter-Drinfeld modules k and V constructed in Example 2.7 will be denoted by f k g and respectively f V g .
Proof. It is easy to see that the compatibility conditions for the two categories are the same.
In particular, the category of anti-Yetter-Drinfeld modules coincides with H(S 2 ,id) YD(H) H , which improves the remark in [7] that anti-Yetter-Drinfeld modules are entwined modules.
Assume now that H is finite dimensional; then A(α, β) becomes a D(H)-bicomodule algebra, with structures In particular, A(H) becomes a D(H)-bicomodule algebra, improving the remark in [7] that β = id we recover the result in [7] that the category of anti-Yetter-Drinfeld modules is isomorphic to A(H) M). The correspondence is given as follows.
where {e i }, {e i } are dual bases in H and H * .

A braided T-category YD(H)
Let α, β ∈ Aut Hopf (H) and consider the objects H α , H β as in Example 2.5. In [4] was considered the object M = H α ⊗ H β , with the following structures: for all h, x, y ∈ H, where by · we denoted both the actions of H on H α and H β given as in Example 2.5. Then it was noticed in [4] that M satisfies a compatibility condition which says that M ∈ H YD H (id, βα).
On the other hand, it was noticed in [7] that the tensor product between an anti-Yetter-Drinfeld module and a Yetter-Drinfeld module becomes an anti-Yetter-Drinfeld module. The next result can be seen as a generalization of both these facts.
Proof. Obviously (α,β) N is a left H-module and right H-comodule, so we check the compatibility condition. We compute: Proof. We prove that c M,N is H-linear. We compute: so the two terms are equal. The fact that c M,N is H-colinear is similar and left to the reader. We prove now (3.2). First note that, due to Remark 3.3, we have M ( N P ) = M ⊗N P and M (N ⊗ P ) = M N ⊗ M P . We compute: and we are done. The proof of (3.3) is easier and left to the reader, and similarly the last statement of the Proposition.
Proof. We first prove that M * is indeed an object in H YD H (α −1 , αβ −1 α −1 ). We compute: On k we have the trivial module and comodule structure, and with these k ∈ H YD H . We want to prove that b M and d M are H-module maps. We compute: Also they are H-comodule maps: Similarly, one can prove: Consequently, if we consider YD(H) f d , the subcategory of YD(H) consisting of finite dimensional objects, we obtain:

Theorem 3.8 YD(H) f d is a braided T-category with left and right dualities over G, the left (respectively right) duals being given as in Proposition 3.6 (respectively Proposition 3.7).
Assume now that H is finite dimensional. We will construct a quasitriangular T-coalgebra over G, denoted by DT (H), with the property that the T-category Rep(DT (H)) of representations of DT (H) is isomorphic to YD(H) as braided T-categories.
Moreover, the structure of DT (H) was constructed in such a way that, via the isomorphisms H * ⊲⊳H(α,β) M ≃ H YD H (α, β) from Section 2, we obtain: Theorem 3.10 Rep(DT (H)) and YD(H) are isomorphic as braided T-categories over G.
Both F and G act as identities on morphisms.
Proof. One checks, by direct computation, that F and G are functors, inverse to each other. Alternatively, we can give a very short proof using results from the previous section. By Example 2.7, we have f k g ∈ H YD H (α, β). By Proposition 3.6, we get ( f k g ) * ∈ H YD H ((α, β) −1 ). Then, one can check that actually F (M ) = ( f k g ) * ⊗ M ∈ H YD H and G(N ) = ( f k g ) ⊗ N ∈ H YD H (α, β). Also, one can see that ( f k g ) * ⊗ f k g = f k g ⊗ ( f k g ) * = k as objects in H YD H , hence F • G = G • F = id, using the associativity of the tensor product.
As we have noticed before, for any α ∈ Aut Hopf (H) we have that (ε, 1) is a pair in involution corresponding to (α, α), hence we obtain: Also, as a consequence of the theorem, we obtain the following result (a right-left version was given in [14]), which might be useful for the aria of applicability of anti-Yetter-Drinfeld modules: