THE AFFINENESS CRITERION FOR WEAK YETTER-DRINFEL’D MODULES

In this paper, we introduce the concept of total quantum integrals in the case of weak Hopf algebras and study the affineness criterion for weak Yetter-Drinfel’d modules, which is a generalization of the results studied by Menini and Militaru (J. Algebra, 247 (2002), 467-508). Mathematics Subject Classification (2010): 16W30


Introduction
The integrals for a Hopf algebra H and the more general ones were introduced by Doi ( [7]), stating that the existence of an integral is the necessary and sufficient condition for the existence of a natural transformation between two functors linking the categories of relative Hopf-modules M H A and right H-comodules M H .The categorical point of view towards integrals associated to a Doi-Koppinen datum (H, A, C) was introduced by Caenepeel et al. ( [4]) to prove separability theorems.
In [9], the authors weakened the conditions for a total A-integral in the sense of Caenepeel.The integrals cover the integrals introduced by Doi and the classic integral by Larson and Sweedler [8].As a major application, the quantum integrals associated to quantum Yetter-Drinfel'd modules H YD A were introduced.
Weak bialgebras and weak Hopf algebras given in [2] generalize the ordinary bialgebras and Hopf algebras by weakening the comultiplication of the unit and the multiplication of the counit.Comultiplication is allowed to be non-unital, ∆(1 H ) = 1 1 ⊗1 2 = 1 H ⊗1 H but the comultiplication is coassociative.In exchange for coassociativity, the multipicativity of the counit is repaced by a weaker condition: ε(hg) = ε(h1 1 )ε(1 2 g), implying that the unit representation is not necessarily one-

Preliminaries
We always work over a fixed field k and follow Sweedler's book [10] for the terminologies on coalgebras and comodules.For the comultiplication ∆ in a coalgebra C, we use the Sweedler-Heyneman's notation, ∆(c) = c 1 ⊗ c 2 , for all c ∈ C. All algebras, linear spaces etc. will be over k.All maps are k-linear and ⊗ means ⊗ k unless otherwise specified, etc. Definition 2.1.Let H be both an algebra and a coalgebra.Then H is called a weak bialgebra in [2] if it satisfies the following conditions: (xy) = (x) (y), Moreover, if there exists a k-linear map S : H −→ H called antipode, satisfying the following axioms for all x ∈ H, Then the weak bialgebra H is called a weak Hopf algebra.
For any weak Hopf algebra H, we define two maps ε t , ε s : H → H by the formulas and denote by H t the image ε t (H), and denote by H s the image ε s (H).
Let H be a weak Hopf algebra.Recall from [2] that the following properties hold, for all h, g ∈ H, (W1) H t and H s are two sub-algebras of H, Definition 2.2.Let H be a weak Hopf algebra over the field k.Recall from [3] that a left H-comodule algebra is an algebra A together with a multiplicative left We use the standard notation ρ l (a) = a (−1) ⊗ a (0) , for any a ∈ A. For the coassociativity of the left comodule, we write ((id Similarly, a right H-comodule algebra is an algebra A together with a multiplicative right H-coaction ρ r : A → A ⊗ H satisfying the condition We use the notation ρ r (a) = a (0) ⊗ a (1) .For the coassociativity of the right comodule, we write (( An H-bicomodule algebra is an algebra A, that is an H-bimodule, such that A is a left and a right H-comodule algebra.

The affineness criterion for weak Yetter-Drinfel'd modules
Recall form [6], a weak Yetter-Drinfel'd module is a right A-module and a left The category of weak Yetter-Drinfel'd modules and k-linear maps that preserve the A-action and H-coaction is denoted H YD A .
Lemma 3.1.Let A be an H-bicomodule algebra.For all a ∈ A, we have Proof.Similar to [10].
Proof.Assume first that Eq.(3.5) holds.Then for a ∈ A, m ∈ M , Conversely, if Eq.(3.1) holds, then An important object of H YD A is the Verma structure (A, •, ρ), where • is the multiplication on A and the left H-coaction ρ is given by for all a ∈ A.
Recall from [1], see also [5], that H YD A can be viewed as a category of weak Doi-Koppinen modules associated to the weak Doi-Koppinen datum (H ⊗ H op , A, H), where Here we shall check which is a condition for being a module coalgebra.Indeed, Definition 3.3.Let H be a weak Hopf algebra with a bijective antipode S and A an H-bicomodule algebra.A k-linear map γ : for all g, h ∈ H.A quantum integral γ : for all h ∈ H.
Proposition 3.4.Let H be a weak Hopf algebra with a bijective antipode S and A an H-bicomodule algebra.Assume that there exists a total quantum integral γ : H → Hom(H, A).Then ρ : A → H A splits in H YD A .
Proof.We can prove that the map for all h ∈ H, a ∈ A, is a left H-colinear retraction of ρ.In particular, Since we have Since λ is an H-colinear map, we have for all g ∈ H and a ∈ A. We define now for all h ∈ H, a ∈ A. Then, for a ∈ A, we have i.e., λ still is a retraction of ρ.Now, for h ∈ H, a, b ∈ A, we have We can define the coinvariants of A as then B is a subalgebra of A and will be called the subalgebra of quantum coinvariants.

Now, we will construct functors connecting
for all n ∈ N, a, a ∈ A. In this way, we have constructed a covariant functor called the induction functor We now prove that the above functors are an adjoint pair.Proof.The unit and the counit of the adjointness are given by for all N ∈ M B , n ∈ N, and Let A be an H-bicomodule algebra.Notice that ) Lemma 3.6.Let A be an H-bicomodule algebra.Then Proof.We can construct the desired map as follows Notice that θ(a) ∈ (H A) coH , we check it as follows From Lemma 3.6 it follows that the adjunction map β H A can be viewed as a map in H YD A via It is easy to see that the flip map τ : Applying Proposition 3.5 for N = V ⊗ B ∼ = B ⊗ V , we obtain the following isomorphisms in M B : Let us define Let us define now for all a, b ∈ A. The map ξ is surjective, as β and τ are.We will prove that ξ is a morphism in H YD A .where A ⊗ A and H A are weak Yetter-Drinfel'd modules via Eq.(3.8),Eq.(3.9), and Eq.(3.10),Eq.(3.11), respectively.Indeed, such that the adjunction map β M1 for M 1 is bijective.As g is k-split and there exists a total quantum integral γ : H → Hom(H, A), we obtain that g also splits dimensional and irreducible.Weak Hopf algebras provide a good framework for studying symmetries of certain quantum field theories.The examples of weak SHUANGJIAN GUO Hopf algebras are groupoid algebras, face algebras and generalized Kac algebras.It has turned out that many classical results of bialgebras and Hopf algebras can be generalized to weak bialgebras and weak Hopf algebras.The main purpose of this paper is to define the more general concept of an integral associated to weak Yetter-Drinfel'd modules, which generalizes the integral introduced by Menini and Militaru in ordinary Hopf algebra setting.The paper is organized as follows: In Section 3, we introduce weak Yetter-Drinfel'd modules and prove that there exists some equivalence between the category H YD A of weak Yetter-Drinfel'd modules and the category M B of left Bmodule, where B = A coH = {a ∈ A|ρ(a) = S −1 (1 (1) )1 (−1) ⊗ a1 (0) } (see Proposition 3.5), and study the affineness criterion for weak Yetter-Drinfel'd modules (see Theorem 3.7).

Lemma 3 . 2 .
Let M be a right A-module and a left H-comodule.Then the compatibility relation Eq. (3.1) is equivalent to

Proposition 3 . 5 .
Let H be a weak Hopf algebra with a bijective antipode S and A an H-bicomodule algebra.Then the induction functor − ⊗ B A : M B → H YD A is a left adjoint of the coinvariant functor (−) coH : H YD A → M B .
where H A is a right A-module in the usual way, i.e. (h a)b = (S −1 (1 (1) )h1 (−1) ⊗ a1 (0) )b = S −1 (1 (1) )h1 (−1) ⊗ ab1 (0) = h ab, for all h ∈ H and a, b ∈ A. On the other hand, the mapu : H A → A H, h ⊗ a → S −1 (a (1) )ha (−1) ⊗ a (0)is a splitting surjection of right A-module, where the first H A has the usual right A-module structure and the second H A has the right A-module given in 53 Eq.(3.10).The right inverse of u is given byv : H A → A H, h ⊗ a → S −2 (a (1) )hS(a (−1) ) ⊗ a (0) .Hence, we can view the second H A as a right A-module direct summand of the first H A. So we obtain that H A, with the right A-module structure given in Eq.(3.10), is still projective as a right A-module.It follows that there exitsζ : H A → A ⊗ A such that ξ • ζ = id H A since A ⊗ A → H A is surjective.Hence, ξ splits in the category of right A-modules.In particular ξ is a k-splitepimorphism in H YD A .Let now M ∈ H YD A .Then A ⊗ A ⊗ M ∈ H YD A via the structures arising from A ⊗ A, that is, (a ⊗ b ⊗ m) • c = ac ⊗ b ⊗ m; ρ A⊗A⊗M (a ⊗ b ⊗ m) = S −1 (a (1) )a (−1) ⊗ a (0) ⊗ b ⊗ m,for all a, b, c ∈ A and m ∈ M .On the other hand, H A ⊗ M ∈ H YD A via the structures arising from H A, that is,(h a ⊗ m) • b = S −1 (b (1) )hb (−1) ⊗ ab (0) ⊗ m; ρ H A⊗M (h a ⊗ m) = h 1 ⊗ h 2 ⊗ a ⊗ m, for all a, b ∈ A, h ∈ H and m ∈ M .We obtain that ξ ⊗ id M : A ⊗ A ⊗ M → H A ⊗ M is a k-split epimorphism in H YD A .Applying H YD A = H M(H ⊗ H op ) A , we obtain that the mapf : H A ⊗ M → M, h a ⊗ m → m (0) γ(S −2 (a (1) )hS(a (−1) ))(m (−1) )a (0)is a k-split epimorphism in H YD A .Hence, the compositiong = f • (ξ ⊗ id M ) : A ⊗ A ⊗ M → M, a ⊗ b ⊗ m → m (0) γ(S −2 (b (1) )S(b (−1) ))(m (−1) )b (0) a is a k-split epimorphism in H YD A .We note that the structure of A ⊗ A ⊗ M as an object in H YD A is of the form A ⊗ V , for the k-module V = A ⊗ M .To conclude, we have constructed a k-split epimorphism inH YD A A ⊗ A ⊗ M = M 1 g −→ M −→ 0