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On Doi-Hopf modules and Yetter-Drinfeld modules in symmetric monoidal categories.

1 Introduction

Various notions of modules appear in Hopf algebra theory. Doi-Hopf modules and Yetter-Drinfeld modules are defined as vector spaces with an action and a coaction, with a compatibility relation that is different in both cases. In [15], it was shown that Yetter-Dr nfeld modules are spec al cases of Do -Hopf modules. This allows to transport properties of Doi-Hopf modules to Yetter-Drinfeld modules, and, in particular, it leads to generalizations of the Drinfeld double construction. Similar results have been applied for Hopf-group coalgebras [8], weak Hopf algebras [16], quasi-Hopf algebras in [10] and weak [pi]-coalgebras in [20].

The aim of this paper is two-fold. First, we will generalize this result to Hopf algebras in braided monoidal categories. Some of the results mentioned above appear as special cases. It also leads to new results, if we apply it to Hopf-group coalgebras and monoidal Hom-Hopf algebras, as these are Hopf algebras in a suitable symmetric monoidal category, see [14, 13]. On the other hand, we want to construct non-trivial examples of entwining structures in monoidal categories or, equivalently, of monoidal wreath structures, see [9].

In Section 2, we recall the diagrammatic notations for the structure of a braided Hopf algebra and for the (co)action of a (co)algebra on an object in a module category. In Section 3 we introduce the notion of (right) entwining structure in a monoidal category and show that giving an entwining structure is equivalent to giving a (co)algebra structure in a suitable monoidal category of transfer morphisms through a (co)algebra (Proposition 3.2). Then we show that particular examples of entwining structures can be obtained from lax Doi-Koppinen structures, abbreviated as DK-structures. These are triples (B, A, C) consisting of an algebra A, a coalgebra C and an object B which is at the same time an algebra and a coalgebra and that acts on C and coacts on A in such a way that the structure morphisms are respectively coalgebra and algebra morphisms in C. In the situation where B is a bialgebra, C is a B-module coalgebra and A is a B-comodule algebra, we recover the classical notion of DK structure. Furthermore, particular examples of lax DK structures can be obtained from lax Yetter-Drinfeld structures (abbreviated YD structures) over a lax Hopf algebra. A lax Hopf algebra is an object B admitting an algebra and a coalgebra structure in C such that IdB has an inverse [S.sub.B] in the convolution algebra Hom(B, B) that is an anti-algebra and an anti-coalgebra endomorphism of B. Actually, a simple inspection shows that this condition is not needed in the proof of the fact that any lax YD structure induces a lax DK structure (Proposition 3.9). In turn, we need this extra condition in the proof of our main result, namely Theorem 4.6.

We point out that our definition of an entwined module is given in the framework of module categories. This new approach will be used in Section 5, where we will show that the category of Doi-Hopf modules over a DK (monoidal) structure can be identified with the category of weak Doi-Hopf modules introduced in [2]. Unfortunately, in the weak case Theorem 4.6 does not apply since we don't know whether a weak Hopf algebra is a lax Hopf algebra in a suitable symmetric monoidal category. Nevertheless, Theorem 4.6 can be used as a source of inspiration for the definition of a weak Yetter-Drinfeld module over a weak bialgebra and then, moving backwards, we can prove a weak version of Theorem 4.6. Finally, note that the theory performed in the weak case gives particular non-trivial examples of entwining structures in monoidal categories of bimodules, and this achieves our second aim.

2 Preliminaries

2.1 Hopf algebras in braided monoidal categories

A monoidal category is a category C together with a functor [cross product] : C x C [right arrow] C, called the tensor product, an object [1.bar] [member of] C called the unit object, and natural isomorphisms a : [member of] [omicron] ([cross product] x Id) [right arrow] [cross product] [omicron] (Id x [cross product]) (the associativity constraint), Z : [cross product] [omicron] ([1.bar] x Id) [right arrow] Id (the left unit constraint) and r : [cross product] [omicron] (Id x [1.bar]) [right arrow] Id (the right unit constraint). In addition, a has to satisfy the pentagon axiom, and l and r have to satisfy the triangle axiom. We refer to [19, XI.2] for a detailed discussion. In the sequel, for any object X [member of] C we will identify [1.bar] [cross product] X [congruent to] X [congruent to] X [cross product] [1.bar] using [l.sub.x] and [r.sub.X]. In addition, all the results will be proved for strict monoidal categories (i.e., for monoidal categories for which all a, l are r are the identity morphisms). Then the results remain valid in the case of an arbitrary monoidal category, since every monoidal category is equivalent to a strict one, see [19] for more detail. Also

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

will be the diagrammatic notation for the following morphisms in C : [Id.sub.x] : X [right arrow] X, f : X [right arrow] Y, [mu] : X [cross product] Y [right arrow] Z and v : X [right arrow] Y [cross product] Z.

In a monoidal category C we can define algebras and coalgebras. An algebra in C is an object A of C endowed with a multiplication [[m.bar].sub.A] : A [cross product] A [right arrow] A and unit morphism [[[eta].bar].sub.A] : [1.bar] [right arrow] A such that [[m.bar].sub.A] is associative up to the associativity constraint a of C and [[m.bar].sub.A] [omicron] ([[eta].bar] [cross product] [Id.sub.A]) = [[m.bar].sub.A] [omicron] ([Id.sub.A] [cross product] [[[eta].bar].sub.A]) = [Id.sub.A]. We will write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly, a coalgebra in C is an object C of C together with a comultiplication [[[DELTA].bar].sub.c] : C [right arrow] C [cross product] C and a counit [[[epsilon].bar].sub.c] : C [right arrow] 1 such that [[DELTA].sub.c] is coassociative up to the coassociativity constraint a and ([[[epsilon].bar].sub.c] [cross product] [Id.sub.c]) [omicron] [[DELTA].sub.c] = [Id.sub.c] = ([Id.sub.c] [cross product] [[[epsilon].bar].sub.c]) [omicron] [[DELTA].sub.c]. We use the diagrammatic notation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The switch functor [tau] : C x C [right arrow] C x C is defined by [tau](X, Y) = (Y, X). A prebraiding on a monoidal category C is a natural transformation c : [cross product] [right arrow] [cross product] [omicron] [tau], satisfying the conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

(see [19, XIII.1]), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any two objects X and Y of C.

A prebraiding c is called a braiding if it is a natural isomorphism. In this case we denote [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and call C a braided monoidal category. One of the main properties of a braiding c on a monoidal category C is given by the equality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

which holds for any objects X, Y and Z of C. It is considered as the categorical version of the Yang-Baxter equation.

The naturality of c means that (g [cross product] f)[c.sub.M,N] = [c.sub.U,V] (f [cross product] g), for any f : M [right arrow] U, g : N [right arrow] V in C. In particular, for T [right arrow] C and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.3)

In a similar way, for a morphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] between X and Y [cross product] Z, we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.4)

For a braided monoidal category C, let [C.sup.in] be equal to C as a monoidal category, with the mirror-reversed braiding [[c.bar].sub.X,Y] = [c.sup.-1.sub.Y,X]. It is well known that [C.sub.in] is also a braided monoidal category. We call C symmetric if C = [C.sup.in], as braided monoidal categories. When C is symmetric we denote [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If A and A' are two algebras in a braided monoidal category then there are two algebra structures in C on A [cross product] A'. Namely, we denote by A [[cross product].sub.+] A' the object A [cross product] A' of C endowed with the multiplication ([[m.bar].sub.A] [cross product] [[m.bar].sub.A']) [omicron] ([Id.sub.A] [cross product] [c.sub.A',A] [cross product] [Id.sub.A']) and tensor product unit morphism. Then, with this structure, A [[cross product].sub.+] A' becomes an algebra in C (see, for instance, [21, Lemma 2.1]). The second algebra structure in C on A [cross product] A, denoted in what follows by A [[cross product].sub.-] A', is obtained by considering A and A algebras in [C.sub.in]. Since [C.sup.in] = C as a monoidal category we obtain that A [[cross product].sub.-] A' is an algebra in C with the multiplication ([[m.bar].sub.A] [cross product] [[m.bar].sub.A']) [omicron] ([Id.sub.A] [cross product] [c.sup.-1.sub.A,A'] [cross product] [Id.sub.A']) and tensor product unit morphism.

There is a notion of opposite algebra in a braided monoidal category. More precisely, if A is an algebra in a braided monoidal category C then [A.sup.op+] is the object A endowed with the new multiplication [[m.bar].sub.A] [omicron] [c.sub.A,A] and original unit morphism. Then one can easily see that [A.sup.op+] is an algebra in C, called the c-opposite algebra associated to A. Replacing C by [C.sup.in] we obtain [A.sup.op-], the object A endowed with the multiplication [[m.bar].sub.A] [omicron] [c.sup.-1.sub.A,A] and the same unit morphism as that of A, and we call it the [c.sup.-1]-opposite algebra associated to A. For further use, we denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.5)

Likewise, if C and C' are two coalgebras in a braided monoidal category then C [cross product] C has two coalgebra structures in C. We denote by C [[cross product].sup.+] C the coalgebra in C having the comultiplication given by ([Id.sub.c] [cross product] [C.sub.C,C] [cross product] [Id.sub.c']) [omicron] ([[[DELTA].bar].sub.c] [cross product] [[[DELTA].bar].sub.c]) and the tensor product counit morphism. Analogously, C [[cross product].sup.-] C' is the coalgebra in C with comultiplication ([Id.sub.c] [cross product] [c.sup.-1.sub.C',C] [cross product] [Id.sub.c']) [omicron] ([[[DELTA].bar].sub.c] [cross product] [[[DELTA].bar].sub.c]) and the tensor product counit morphism as counit.

Next, we recall the notions of a bialgebra and Hopf algebra in a braided monoidal category. A bialgebra in C is a fivetuple (B,[[m.bar].sub.B],[[[eta].bar].sub.B],[[[DELTA].bar].sub.B],[[[epsilon].bar].sub.B]), such that (B,[[m.bar].sub.B],[[[eta].bar].sub.B) is an algebra and (B, [[[DELTA].bar].sub.B],[[[epsilon].bar].sub.B]) is a coalgebra such that [[[DELTA].bar].sub.B] : B [right arrow] B [cross product] B and [[[epsilon].bar].sub.B] : B [right arrow] [1.bar] are algebra morphisms. B [cross product] B has the tensor product algebra structure (using the braiding on C), and [1.bar] is considered as an algebra in C with the multiplication and unit map both equal to the identity morphism of [1.bar]. For later reference, we give explicit formulas for the axioms of a bialgebra [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

A Hopf algebra in a braided monoidal category C is a bialgebra B in C together with a morphism [S.bar] : B [right arrow] B in C (the antipode) satisfying the axioms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)

It is well-known, see [21, Lemma 2.3], that the antipode [S.bar] of a Hopf algebra B in a braided monoidal category C is an anti-algebra and anti-coalgebra morphism, in the sense that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

2.2 Modules and comodules in module categories

Let C be a monoidal category. A right C-category (also known as a module category over C) is a quadruple (D, *, [PSI], r), where D is a category, * : D x C [right arrow] V is a functor, and [PSI] : * [omicron] (* x Id) [right arrow] * [omicron] (Id x [cross product]) and r : * [omicron] (Id x [1.bar]) [right arrow] Id are natural isomorphisms such that the diagrams

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

commute, for all M [member of] D and X, Y, Z [member of] C. Obviously C is itself a right C-category, with * = [cross product], and [PSI] and r the natural identities (recall that we assumed that C is strict). In fact, the above mentioned coherence theorem can be extended to C-categories, and this enables us to assume, without loss of generality, that [PSI] and r are natural identities.

Let D be a right C-category, and consider an algebra A in C. A right module in D over A is an object M [member of] D together with a morphism [v.sub.M] : M [??] A [right arrow] M such that [v.sub.M] [omicron] ([Id.sub.M] * [[[eta].bar].sub.A]) = [r.sub.M] and such that the diagram

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

commutes. Let [D.sub.A] be the category of right modules and right linear maps in V over A. The right module structure on M [member of] [D.sub.A] will be written symbolically as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] When [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] will be a shorter notation for the right A-module structure morphism of M in C. Furthermore, in this case one can define left A-modules, too. For simplicity, we denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the morphism that defines on an object R of C a left A-module structure in C.

We can also define the dual notion of right comodule in a right C-category V over a coalgebra C in C. The category of right comodules and right colinear maps in D over C will be denoted as [D.sup.C]. The right comodule structure on M [member of] [D.sup.C] will be written as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. When C = V we denote the morphism [[rho].sub.m] by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Likewise, we can define left C-comodules in a monoidal category C. In this case we will denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the morphism that defines on an object R of C a a left C-comodule structure in C.

Finally, if A, B are algebras in a monoidal category C (respectively if C, D are coalgebras in C) then we can define (A, B)-bimodules (respectively (C, D)-comodules) in C. They are left A and right B-modules (respectively, left C and right D-comodules) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

3 Entwining structures defined by lax Doi-Koppinen and lax Yetter-Drinfeld structures

Throughout this section, C is a monoidal or a braided monoidal category. As we have seen, we can assume that C is strict, without loss of generality. Entwined structures over a field were introduced by Brzezinski in [5]; in [18], this notion was generalized to symmetric monoidal categories. In fact the assumption that C has a symmetry is not needed.

Definition 3.1. A right entwining structure in C is a triple (A, C, [psi]), where A is an algebra in C, C is a coalgebra in C, and [psi] : C [cross product] A [right arrow] A [cross product] C is a morphism in C, which we denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], such that the following equalities hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

We call an entwining structure (A, C, [psi]) in C trivial if A = [1.bar] and [psi] = [Id.sub.c], or if C = [1.bar] and [psi] = [Id.sub.A].

Next, we show that any entwining structure in C can be viewed as a trivial entwining structure in a different monoidal category. The next result is essentially due to Schauenburg [22] and has its roots in a paper of Tambara [23]. Also, our result is slightly more general than in [22] because we do not assume that the entwining map [psi] is an isomorphism in C.

Let us start by presenting some concepts and constructions. Similar to [22, Definition 4.2] we define the category of transfer morphisms through an algebra (or a coalgebra) as follows.

For an algebra A in C, we consider the category [T.sub.A] of right transfer morphisms through A. The objects are pairs (X, [[psi].sub.X], A) with X [member of] C and [[psi].sub.X,A] : X [cross product] A [right arrow] A [cross product] X a morphism in C such that (3.1.a) and (3.1.b) are fulfilled (with C replaced by X). A morphism in [T.sub.A] between (X, [[psi].sub.X,A]) and (Y, [[psi].sub.Y,A]) is a morphism [mu] : X [right arrow] Y in C such that ([Id.sub.A] [cross product] [mu]) [omicron] [[psi].sub.X,A] = [[psi].sub.Y,A] [omicron] ([mu] [cross product] [Id.sub.A]). [T.sub.A] is a strict monoidal category, with unit object ([1.bar], [Id.sub.A]) and tensor product

(X, [[psi].sub.X,A]) [[cross product].bar](Y, [[psi].sub.Y,A]) = X [cross product] Y, [[psi].sub.X [cross product] Y,A],

with [[psi].sub.X [cross product] Y,A] := ([[psi].sub.X,A] [cross product] [Id.sub.Y])([Id.sub.X] [cross product] [[psi].sub.Y,A]). (3.2)

We leave it to the reader to introduce the monoidal category [T.sup.C] of right transfer morphisms through the coalgebra C in C.

Proposition 3.2. Let C be a monoidal category.

i) If A is an algebra in C then (C, [[psi].sub.C,A]) is a coalgebra in [T.sub.A] if and only if (A, C, [[psi].sub.C,A]) is a right entwining structure in C.

ii) If C is a coalgebra in C then (A, [[psi].sub.C,A]) is an algebra in [T.sup.C] if and only if (A, C, [[psi].sub.C,A]) is a right entwining structure in C.

Consequently, (A, C, [psi]) is a right entwining structure in C if and only if (([1.bar],[Id.sub.A]), (C, [psi]),Id) is a trivial entwining structure in [T.sub.A] or, equivalently, ((A,[psi]), ([1.bar], [Id.sub.c]), Id) is a trivial entwining structure in [T.sup.c].

Proof. Straightforward. The verification of all these details is left to the reader.

For C = [sup.k]M, the category of vector spaces, we obtain the classical definition of an entwining structure [5]. In this case, it is well known that a class of examples is given by the Doi-Koppinen structures (DK for short) over k. This can be generalized to entwining structures in braided monoidal categories.

Assume that C is a braided monoidal category, and that B is an object in C which is both an algebra and a coalgebra (but not necessarily a bialgebra) in C. A right [c.sup.[+ or -]1]-module coalgebra over B is a coalgebra C in C which is also a right B-module such that the structure map C [cross product] B [right arrow] C is a coalgebra morphism from C [[cross product].sup.[+ or -]] B to C. Similarly, a right [c.sup.[+ or -]1]-comodule algebra over C is an algebra A in C which is also a right B-comodule such that the structure morphism A [right arrow] A [cross product] B is an algebra morphism from A to A [[cross product].sub.[+ or -]] B. For example, in the c-case, C is a coalgebra and a right B-module in C, respectively A is an algebra and a right B-comodule in C, such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the following equalities hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.3)

Note that for the [c.sup.-1]-case we have to replace in the above equalities the braiding [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with its inverse [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In a similar way, we can define the notions of left [c.sup.[+ or -]1]-module coalgebra and left [c.sup.[+ or -]1]-left comodule algebra over B.

Definition 3.3. Let C be a braided monoidal category and B an object of C which has both an algebra and a coalgebra structure in C. If A is a right [c.sup.[+ or -]1]-comodule algebra over B and C is a right [c.sup.[+ or -]1]-module coalgebra over B, respectively, then we call the triple (B, A, C) a lax [c.sup.[+ or -]1]-right Doi-Koppinen (DK for short)-structure in C. If B is a bialgebra in C then we simply say that (B, A, C) is a [c.sup.[+ or -]1]-right DK-structure in C.

Proposition 3.4. Let C be a braided monoidal category, B an object of C which has both an algebra and a coalgebra structure in C, and A [member of] [C.sup.B] and C [member of] [C.sub.B]. Define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

Then the following assertions hold:

(i) If (B, A, C) is a lax c-right DK-structure in C then (A, C, [[psi].sub.+]) is a right entwining structure in C.

(ii) If (B, A, C) is a lax [c.sup.-1]-right DK-structure in C then (A, C, [[psi].sub.-]) is a right entwining structure in C.

Proof. It suffices to prove (i), as (ii) is (i) applied to [C.sup.in]. The computation

shows that (3.1.a) is satisfied. In a similar way, we have that

and this shows that (3.1.c) is satisfied. It is easy to see that (A, C, [[psi].sub.+]) satisfies (3.1.b,d), and this completes the proof.

If C is symmetric then lax DK-structures in C can be obtained from lax Yetter-Drinfeld structures (abbreviated as YD-structures) in C defined over lax Hopf algebras.

Definition 3.5. Let C be a braided monoidal category. A lax Hopf algebra in C is a sextuple (B,[[m.bar].sub.B],[[[eta].bar].sub.B],[[A.bar].sub.B],[[t.bar].sub.B],[[S.bar].sub.B]) consisting in an algebra (B,[[m.bar].sub.B],[[[eta].bar].sub.B]) and a coalgebra (B,[[[DELTA].bar].sub.b],[[[epsilon].bar].sub.b]) structure on an object B of C, and an anti-algebra and anticoalgebra endomorphism [[S.bar].sub.B] of B satisfying (2.7).

It is clear that a braided Hopf algebra is a lax Hopf algebra. Let C be a braided monoidal category and B an object of C which is both an algebra and coalgebra in C. A (c, c)-bimodule coalgebra over B is a coalgebra C in C which has a B-bimodule structure such that C is both a left and right c-module coalgebra over B. Similarly, we can define (c, [c.sup.-1]), ([c.sup.-1], c) and ([c.sup.-1], [c.sup.-1])-bimodule coalgebras over B in C, respectively. Likewise, a (c, c)-bicomodule algebra in C is an algebra A of C endowed with a B-bicomodule structure such that A is simultaneously a left and right c-comodule algebra over B. In a similar manner one can define (c, [c.sup.-1]), ([c.sup.-1], c) and ([c.sup.-1], [c.sup.-1])-bicomodule algebras over B in C, respectively.

Definition 3.6. Let C be a braided monoidal category and B [member of] C such that B is both an algebra and a coalgebra in C. If C is a (c, c)-bimodule coalgebra and A is a (c,c)-bicomodule algebra over B in C, then we call the triple (B, C, A) a lax [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] YD-structure in C (the first row of the matrix refers to the bicomodule algebra structure, while the second row refers to the bimodule coalgebra structure over B). In a similar way we define lax [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] YD-structures in C. If B is a bialgebra in C then the word lax will be omitted. Also, in the particular case when C is symmetric monoidal we simply say that (B, C, A) is a (lax) YD-structure in C.

First, we show that any ([c.sup.-1], c)-bimodule coalgebra can be viewed as a c-right module coalgebra.

Lemma 3.7. Let C be a braided monoidal category, B an object of C which has both an algebra and a coalgebra structure in C, and C a ([c.sup.-1], c)-bimodule coalgebra over B in C. Then C with structure defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

is a c-right module coalgebra over [B.sup.op+] [[cross product].sup.+.sub.+] B in C. Proof. C is a right [B.sup.op+] [[cross product].sub.+] B-module since

In the equality (*), we used the definition of [[m.bar].sup.op+]. We also have that

since C [cross product] [B.sup.C]B and [c.sub.x,[1.bar]] = [c.sub.[1.bar],x] = [Id.sub.X], for any object X of C (see [19, Prop. XIII.1.2]).

Next, we prove that the structure map C [[cross product].sup.+] (B [[cross product].sup.+] B) [right arrow] C defined above is a coalgebra morphism in C. This follows from our next computation. At the very beginning we use (2.1) (a), and in (*), we use the fact that C is a [c.sup.-1]-left module coalgebra over B.

In a similar way, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here we used successively that C is a left [c.sup.-1]-module coalgebra, C is a right c-module coalgebra, and [c.sub.c,1] = [Id.sub.c].

Our next aim is to show that a bicomodule algebra over a lax Hopf algebra gives rise to a right comodule algebra.

Lemma 3.8. Let C be a braided monoidal category, B a lax Hopf algebra in C and A a ([c.sup.-1], c)-bicomodule algebra over B. Then A is a c-right comodule algebra over [B.sup.op+] [[cross product].sup.-.sub.+] B in C via the structure morphism [[[rho].bar].sub.A] : A [right arrow] A [cross product] [B.sub.op+] [R] B given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

Proof. We first check that A is a right B [[cross product].sup.+] B-comodule.

and this is exactly what we need. Note that in (*) we used the naturality of the braiding c twice.

We show next that [[rho].bar].sub.A] is an algebra map from A to A [[cross product].sub.+] ([B.sup.op+] [[cross product].sub.-] B). It is easy to check that [[[rho].bar].sub.A] respects the unit morphisms. We also have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the first equality we used the fact that A is a [c.sup.-1]-left comodule algebra over B; in the second equality we used the naturality of the braiding c and the fact that A is a c-right comodule algebra over B; in the third equality we used (2.8.a), the naturality of the braiding c and the definition of the multiplication of [B.sup.op+] in (2.5); in the fourth, fifth and sixth equality we used (2.3); in the seventh equality we used (2.4); in the eight equality we used (2.2); in the final equality th naturality of c was used twice. Finally, using (2.8.b), the naturality of the braiding c, [c.sub.[1.bar],a] = [Id.sub.A] and the fact that A is a B-bicomodule, we compute

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and this finishes the proof.

Applying Lemmas 3.7,3.8 and Proposition 3.4 we obtain the following result.

Proposition 3.9. Let B be a lax Hopf algebra in a symmetric monoidal category C. To any right lax YD-structure (B, C, A) in C we can associate a right lax DK-structure in C, namely ([B.sup.op] [cross product] B, C, A). Consequently, any lax YD-structure (B, C, A) over a lax Hopfalgebra B produces a right entwining structure (A, C, [psi]) in C, where [psi] can be explicitly computed using (3.4), (3.5) and (3.6).

4 Entwined modules--a module categorical approach

Let A be an algebra in a monoidal category C. To A we have associated the monoidal category [T.sub.A], called the category of transfer morphisms through A. Then it came out that there is a bijective correspondence between right entwining structures in C and coalgebras in [T.sub.A]. On the other hand, to a right entwining structure (A, C, [psi]) in C, we can associate the category of entwined modules C[([psi]).sup.C.sub.A]. The objects are right A-modules and right C-comodules in C for which the right C-comodule morphism structure is right A-linear or, equivalently, for which the right A-module morphism structure is right C-colinear. The morphisms in C[([psi]).sup.C.sub.A] are the morphisms in C which are right A-linear and right C-colinear.

Based on these observations and with the help of the notion of C-category we will see that entwined modules can be viewed as comodules over a coalgebra in the category of transfer morphisms through an algebra in C. First we generalize the classical notion of entwined module.

Definition 4.1. Let C be a monoidal category, let V be a right C-category and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a right entwining structure in C. An object M [member of] D is called a right entwined module with entwining map [psi] if M is a right module in D over A, a right comodule in V over C and the following compatibility relation holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.1)

D[([psi]).sup.C.sub.A] will be the category of right entwined modules with entwining map [psi] and right A-module and right C-colinear morphisms.

Remark 4.2. Observe that Definition 4.1 does not cover all categories of Hopf modules defined over Hopf algebras and their generalization. For example, the category of Doi-Hopf modules over a quasi-Hopf algebra H (see [7]) has not such a description. This is because in the quasi-Hopf case a right DK-structure consists of a right H-comodule algebra U (in the sense of Hausser and Nill [17]) which is associative and a right H-module coalgebra C (in the sense of [12]) which is coassociative up to the reassociator of H which is, in general, not trivial. Therefore (H, U, C) does not produce a right entwining structure in a certain monoidal category. Nevertheless, in the forthcoming paper [11] we will see that the notion of entwining structure in a monoidal category can be generalized. Then, using the point of view suggested by the result below we will define a more general category of entwined modules, unifying in this way most of the Doi-Hopf module categories known so far.

Proposition 4.3. Let C be a monoidal category, (A, C, [psi]) a right entwining structure in C and V a right C-category.

(i) If M [member of] [D.sub.A] and (X, [[psi].sub.X,A]) [member of] [T.sub.A] then M [??] X [member of] [D.sub.A] with the structure

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The associated functor from [D.sub.A] x [T.sub.A] to [D.sub.A] turns [D.sub.A] into a right [T.sub.A]-category.

(ii) The category of entwined modules D[([psi]).sup.C.sub.A] coincides with the category of right comodules in [D.sub.A] over the coalgebra (C, [psi]) in [T.sub.A].

Proof. In diagrammatic notation, we have that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

by (3.1.a). All other details are left to the reader.

Example 4.4. Let (B, A, C) be a lax c-right DK-structure in a braided monoidal category C, and let V be a right C-category. The category [D.sub.lax] [(B).sup.C.sub.A] of entwined modules in D corresponding to the right entwining structure in C defined by (3.4) is called the category of lax right Doi-Hopf modules in D over B.

Now we introduce the braided version of the category of Yetter-Drinfeld modules.

Definition 4.5. Let (B, A, C) be a lax [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] YD-structure in a braided monoidal category C, and let D be a right C-category. An object M of D is called a lax right Yetter-Drinfeld module if M [member of] [D.sub.A], M [member of] [D.sub.C] and the following compatibility relation between these structures holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.2)

Then Y[D.sub.lax] [(B).sup.C.sub.A] will be the notation for the category of lax right Yetter-Drinfeld modules in V over B, and right A-linear and right C-colinear morphisms.

Theorem 4.6 is the braided version of the main result in [15], and tells us that Y[D.sub.lax][(B).sup.C.sub.A] is a category of entwined modules, at least if we work over a symmetric monoidal category C.

Theorem 4.6. Let D be a right C-category with C symmetric monoidal. If B is a lax Hopf algebra in C and (B, A, C) is a lax YD-structure in C then Y[D.sub.lax] [(B).sup.C.sub.A] = [D.sub.lax] [([B.sup.op] [cross product] B).sup.C.sub.A].

Proof. By Example 4.4 and Proposition 3.9 an object M of [D.sub.lax] [([B.sup.op] [cross product] B).sup.C.sub.A] is a right module in D over A and a right comodule in D over C such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.3)

Therefore it suffices to show that (4.2) and (4.3) are equivalent. The following computation shows that (4.2) implies (4.3). Observe that (*) follows from the naturality of the braiding c and the fact that [c.sub.[1.bar],X] = [c.sub.X,[1.bar]] = [Id.sub.X], and that the last equality follows from the unit and counit axioms.

Conversely, assume that (4.3) holds. We then compute:

and this ends the proof. In the last equality we also used the fact that [c.sub.[1.bar],X] = [c.sub.X,[1.bar]] = [Id.sub.X], for all X [member of] C.

5 Monoidal entwining structures defined by weak Hopf algebra actions and coactions

The aim of this Section is to present examples of entwining structures in monoidal categories obtained from actions and coactions of a weak bialgebra. Then we relate our results to some existing results on Doi-Hopf modules and Yetter-Drinfeld modules over a weak Hopf algebra.

Let k be a field. Recall from [4] that a weak bialgebra is a k-module H together with a k-algebra structure (H, m, u) and a k-coalgebra structure (H, [DELTA], [epsilon]) such that [DELTA] is multiplicative and the following relations hold

[1.sup.1] [cross product] [1.sub.2] [cross product] [1.sup.3] = [1.sup.1] [cross product] [1.sup.2] [1.sup.1'] [cross product] [1.sup.2'] = [1.sub.1] [cross product] [1.sub.1'] [1.sub.2] [cross product] [1.sub.2'], (5.1)

[epsilon](ghl) = [epsilon]([gh.sub.1])[epsilon]([h.sub.2]1) = [epsilon]([gh.sub.2])[epsilon]([h.sub.1]l), [for all] g,h, l [member of] H. (5.2)

[1.sub.1'] [cross product] [1.sub.2] is a second copy of [DELTA](1), 1 is the unit of H and [DELTA](h) = [h.sub.1] [cross product] [h.sub.2], h [member of] H. It is known that the category of right H-(co)representations is monoidal; this can be explained easily using the following arguments, as presented in [3].

The endomorphisms [[epsilon].sub.s], [[epsilon].sub.t] : H [right arrow] H, [[epsilon].sub.s] (h) = [epsilon](h[1.sub.2])[1.sub.1], [[epsilon].sub.t](h) = [epsilon]([1.sub.1] h)[1.sub.2] are idempotent. Their images

[H.sub.t] := {h [member of] H | [DELTA](h) = [1.sup.1] h [cross product] [1.sub.2]} and [H.sub.s] := {h [member of] H | [DELTA](h) = [1.sub.1] [cross product] h[1.sub.2]},

called the target and source subspace of H, are subalgebras of H.

If H a weak k-bialgebra then so is H[o.sup.p], and it can be easily seen that

[[epsilon].sup.opf.sub.s] (h) = [epsilon]([1.sub.2]h)[1.sub.1] := [[bar.[epsilon]].sub.s] (h) and [[epsilon].sup.op.sub.t] (h) = [epsilon](h[1.sub.1])[l.sub.2] := [[bar.[epsilon]].sub.f] (h),

for all h [member of] H. Note that the map [[epsilon].sub.t] restricts to an anti-algebra isomorphism from [H.sub.s] to [H.sub.t] with inverse [[bar.[epsilon]].sub.S], while [[epsilon].sub.s] restricts to an anti-algebra isomorphism from [H.sub.t] to [H.sub.s] with inverse [[bar.[epsilon]].sub.t].

Now, if M is a right H-module then M becomes an [H.sub.s]-bimodule via r x m x r' = m[[bar.[epsilon]].sub.t] (r)r' = mr'[[bar.[epsilon]].sub.t] (r), for all m [member of] M and r,r' [member of] [H.sub.s]. Furthermore, [H.sub.s] is a right H-module via r [??] h = [[epsilon].sub.s] (rh), for all r [member of] [H.sub.s] and h [member of] H. Then ([M.sub.H], [cross product][H.sub.s], [H.sub.s]) is monoidal in such a way that the forgetful functor from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] turns into a monoidal functor.

We call a coalgebra C in [M.sub.H] a right H-module coalgebra. The next result says that this notion is equivalent to the notion of weak H-module coalgebra, as introduced in [2].

Proposition 5.1. If H is a weak k-bialgebra then C is a right H-module coalgebra if and only if C has a right H-module structure and a coalgebra structure (C, [[DELTA].sub.C], [[epsilon].sub.C]) in [sup.k]M that are compatible in the following sense,

[[DELTA].sub.C] (c x h) = [c.sub.1] x [h.sub.1] [cross product] [c.sub.2] x [h.sub.2], and (5.3) [[epsilon].sub.C] (c x [[epsilon].sub.t] (h)) = [[epsilon].sub.C] (c x h), [for all] c [member of] C, h [member of] H. (5.4)

Proof First let (C,[[DELTA].bar],[[epsilon].bar]) be a coalgebra in [M.sub.H]. We have a well-defined map [[DELTA].sub.C] : C [right arrow] C [cross product] C given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for all c [member of] C. Moreover, (C, [[DELTA].sub.q], [[epsilon].sub.c] := [epsilon][[epsilon].bar]) is a K- coalgebra, a right H-module and the conditions in the statement are satisfied.

Conversely, if (C, [DELTA]C, [[epsilon].sub.C]) is a k-coalgebra and a right H-module obeying (5.3) and (5.4) then C becomes a right H-module coalgebra via the comultiplication [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the counit [[epsilon].bar] : C [right arrow] [H.sub.s] given by [e.bar](c) = [[epsilon].sub.c] (c x [1.sub.2])[1.sub.1], for all c [member of] C.

The monoidal structure on the category Mh is defined in a similar manner. In this case the key observation is the fact that H is an Hs-coring. Indeed, it is well-known that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

together with [??] := [[epsilon].sub.s] defines a coalgebra structure on H within the monoidal category [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This fact allows us to define a right corepresentation over a weak bialgebra:

Definition 5.2. A right comodule over a weak k-bialgebra H consists in a right comodule over the [H.sub.s]-coring H. Otherwise stated, a right comodule over H is a right [H.sub.s]-module M together with a right [H.sub.s]-module map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that, via the canonical identifications in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A morphism f : M [right arrow] N between two right comodules over H is a right [H.sub.s]-module map satisfying [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The category of right comodules over H and right H-comodule maps is denoted by [M.sup.H].

Although a right comodule M over H is not necessarily a left [H.sub.s]-module, one can easily see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a left [H.sub.s]-module by defining [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all r [member of] [H.sub.s], m [member of] M and h [member of] H. Observe that this action is well-defined because of the [H.sub.s]-bimodule structure of H. Using this observation and specializing [1, Proposition 1.1] for a weak bialgebra we obtain the following result.

Proposition 5.3. Let H be a weak bialgebra and M a right comodule over H via the structure morphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all m [member of] M. Then there is a unique left [H.sub.s]-module structure on M making [[rho].sub.M] a left Hs-module morphism. Namely, r x m = [m.sub.(0)] x [[epsilon].sub.s] ([rm.sub.(1)]),for all r [member of] [H.sub.s] and m [member of] M. Furthermore, with this additional structure,

(1) M becomes an [H.sub.s]-bimodule;

(2) [[rho].sup.M] becomes an [H.sub.s]-bimodule map;

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(4) any morphism in [M.sup.H] becomes an [H.sub.s]-bimodule map.

We are now able to describe the monoidal structure of [M.sup.H], when H is a weak k-bialgebra.

For X, Y [member of] [M.sup.H] we have seen that X, Y are [H.sub.s]-bimodules, and so we can define their tensor product as being [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the tensor product in the monoidal category of [H.sub.s]-bimodules. If we endow [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with the right H- coaction [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then this coaction is well-defined and determines on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a right comodule structure over H. Furthermore, in this way we have a monoidal category [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where the unit object [H.sub.s] is a right comodule over H via the trivial coaction [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This monoidal structure is designed in such a way that the forgetful functor from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] becomes a monoidal functor.

An algebra in the monoidal category [M.sup.H] will be called a right H-comodule algebra. As the reader might expect, this notion is equivalent to the notion of right weak comodule algebra over H, in the sense of [2]. The next result is the (improved) right version of [6, Proposition 3.9].

Proposition 5.4. Let H be a weak k-bialgebra. Then to give a right H-comodule algebra is equivalent to give a k-algebra A with unit 1 such that A is a right H-comodule in [sup.k]M, the comodule structure morphism [rho] : A [right arrow] A [cross product] H is multiplicative and [rho](1) [member of] A [cross product] [H.sub.t].

Proof. We sketch the proof, leaving further detail to the reader.

If A is a right comodule algebra over H then H is a k-algebra with multiplication

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and unit 1, where (A, [[m.bar].sub.A],1) stands for the algebra structure of A in [M.sup.H]. In addition, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the right coaction of the [H.sub.s]-coring H on A then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is well-defined and satisfies the requirements in the statement.

Conversely, let A be a k-algebra and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a multiplicative map that endows A with a right H-comodule structure in [sup.k]M such that [[rho].sup.A] (1) [member of] A [cross product] [H.sub.t]. Then A is a right [H.sub.s]-module via a x r = [epsilon][([a.sub.<1>]r).sub.a<0>], for a [member of] A and r [member of] [H.sub.s], and a right H-comodule via the structure morphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let H be a weak k-bialgebra, A a right H-comodule algebra H and C a right H-module coalgebra. We call (H, A, C) a right Doi-Koppinen (DK for short) structure over H.

Proposition 5.5. If(H, A, C) is a right DK structure over a weak k-bialgebra H then (A, C, [psi]) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

is a right entwining structure in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. This follows after we specialize [9, Proposition 5.13] to a weak bialgebra.

It is immediate that the bifunctor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] defines a right [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]-category structure on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If we consider the associated category of entwined modules [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as in Definition 4.1 then by the comments made after the proof of [9, Prop. 5.13], see also [6, Theorem 3.11 & Prop. 4.1], we obtain that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is isomorphic to the category of weak Doi-Hopf modules M[(H).sub.C.sub.A] in the sense of [2] and to the category of Doi-Koppinen modules over (H, A, C) in the sense of [6].

An alternative approach to Doi-Hopf modules over a weak bialgebra is the following.

Proposition 5.6. Let H be a weak k-bialgebra and A a right H-comodule algebra. Then [M.sub.A], the category of right A-modules in [sup.k]M is a right [M.sub.H]-category via the functor [??] : [M.sub.A] x [M.sub.H] [right arrow] [M.sub.A] defined as follows. If M [member of] [M.sub.A] and X [member of] [M.sub.H] then M [??] X := [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where M is a right [H.sub.s]-module via m * r = m x (1 x r), for all m [member of] M and r [member of] [H.sub.s], and where X inherits the left [H.sub.s]-module of H, i.e., r x x = x x [[bar.[epsilon]].sub.t] (r), for all r [member of] [H.sub.s] and x [member of] X. M [??] X [member of] [M.sub.A] with the right A-action, m [member of] M, x [member of] X and a [member of] A,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined by the right unit constraint r of [M.sub.H]. Furthermore, if C is a coalgebra in [M.sub.H], that is, a right H-module coalgebra, then a right comodule over C in [M.sub.A] is precisely a right Doi-Hopfmodule over H in the sense of [61

Proof. We only prove that o is well-defined. To this end we compute

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for all m [member of] M, x [member of] X and r [member of] [H.sub.s]. Note that we used in the fourth equality the fact that the multiplication of A is [H.sub.s]-balanced.

It follows now that M [??] X is a right A-module and that for any f : M [right arrow] M' in [M.sub.A] and g : X [right arrow] X in [M.sub.H] the morphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], as required.

Now let C be a coalgebra in [M.sub.H]. Then a right comodule over C in [M.sub.A] is a k-vector space M equipped with the following structure:

--M is a right A-module in [sup.k]M inheriting the right [H.sub.s]-module structure from the right A-action, that is, m * r = m x (1 x r), for all r [member of] [H.sub.s] and m [member of] M;

--C coacts on M to the right in the sense that there exists [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that the following relations hold,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for all m [member of] M and a [member of] A, where we denoted [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all m [member of] M. But this is nothing else than a right (H, A, C) Hopf-module in the sense of [6].

We will now show that particular examples of DK structures over a weak bialgebra H can be constructed from YD structures over H, at least if H has an antipode. This means that there is a k-linear map S : H [right arrow] H such that

S([h.sub.1])[h.sup.2] = [[epsilon].sub.s](h), [h.sub.1] S([h.sub.2]) = [[epsilon].sub.t] (h) and S([h.sub.1])[h.sub.2]S([h.sub.3]) = S(h),

for all h [member of] H. To this end we need the notions of bicomodule algebra and bimodule coalgebra over a weak bialgebra. A left comodule algebra over a weak bialgebra H is an algebra in the monoidal category [sup.H]M of left H-corepresentations. But this time we have to deal with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] rather than [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore we have to move to the category of vector spaces in order to unify the left and right version. Proceeding as in Proposition 5.4, we can show that giving a left Hcomodule algebra A is equivalent to giving a k-algebra with unit 1 and a left H-comodule structure on A in [sup.k]M such that the comodule morphism structure A [member of] a [??] [[lambda].sub.A] (a) = [a.sub.[-1]] [cross product] [a.sub.[0]] [member of] H [cross product] A is multiplicative and satisfies [[lambda].sub.A] (1) [member of] [H.sub.s] [cross product] A. Thus by an H-bicomodule algebra we mean a k-algebra A with unit 1 which is an H-bicomodule via some morphisms [[lambda].sub.A] : A [right arrow] H [cross product] A and [[rho].sub.A] : A [right arrow] A [cross product] H that are multiplicative and such that [[lambda].sub.A] (1) [member of] [H.sub.s] [cross product] A and [[rho].sub.A] (1) [member of] A [cross product] [H.sub.t].

Likewise, by an H-bimodule coalgebra C we mean a k-coalgebra C that is an H-bimodule, and for which [DELTA] is an H-bilinear morphism and the counit satisfies [epsilon](h x c) = [epsilon]([[epsilon].sub.s](h) x c) and [epsilon](c x h) = [epsilon](c x [[epsilon].sub.t] (h)), for all c [member of] C and h [member of] H.

If H is a weak bialgebra, A an H-bicomodule algebra and C an H-bimodule coalgebra then we call the triple (H, A, C) a right weak YD structure over H.

Proposition 5.7. Let H be a weak Hopf algebra and (H, A, C) a right weak YD structure over H. Then A with A [??] a [??] [rho](a) := [a.sub.(0)] [cross product] (S([a.sub.(-1)]) [cross product] [a.sub.(1)]) [member of] A [cross product] ([H.sup.op] [cross product] H) is a right weak [H.sup.op] [cross product] H-comodule algebra and C with the action given by c x (h' [cross product] h) = h' x c x h, for all c [member of] C and h, h' [member of] H, is a right weak [H.sup.op] [cross product] H-module coalgebra. Consequently, a right weak (H, A, C) YD structure defines a right weak DK structure ([H.sup.op] [cross product] H, A, C), and therefore a right DK structure ([H.sup.op] [cross product] H, A, C).

Proof. It is well-known that the antipode of a weak Hopf algebra is an anti-algebra and an anti-coalgebra endomorphism of H. From here it is immediate that p defines a right [H.sup.op] [cross product] H-comodule structure on A, and that [rho] is multiplicative. Since S[([H.sub.s]) [subset or equal to] [H.sub.t], ([H.sup.op] [cross product] H).sub.t] = [H.sub.t] [cross product] [H.sub.t] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

it follows that [rho](1) [member of] [([H.sup.op] [cross product] H).sub.t]. Hence A is a right weak [H.sup.op] [cross product] H-comodule algebra.

It is easy to verify that C is a right [H.sup.op] [cross product] H-module, and that [DELTA] is right [H.sup.op] [cross product] H-linear. We have also that Ct of [H.sup.op] [cross product] H is [[bar.[epsilon]].sub.t] [cross product] [[epsilon].sub.t], and since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we conclude that C is a right weak [H.sup.op] [cross product] H-module coalgebra.

Definition 5.8. Let H be a weak Hopf algebra and (H, A, C) a right weak YD structure over H. The category of entwined modules corresponding to the right weak DK structure ([H.sup.op] [cross product] H, A, C) will be denoted by YD[(H).sup.C.sub.A] and called the category of right (A, C)-Yetter-Drinfeld modules over H.

For the sake of simplicity, we will describe the weak version of YD[(H).sup.C.sub.A], see the comments made after the proof of Proposition 5.4. A right weak (A, C)-YetterDrinfeld module over H is a k-vector space M that is at the same time a right A-module and a right C-comodule such that

[(m x a).sub.{0}] [cross product] [(m x a).sub.{1}] = [m.sub.{0}] x [a.sub.(0)] [cross product] S([a.sub.(-1)]) x [m.sub.{1}] x [a.sub.(1)], [for all] m [member of] M, a [member of] A, (5.5)

where M [??] m [??] [m.sub.{0}] [cross product] [m.sub.{1}] [member of] M [cross product] C is the right coaction of C on M.

Proposition 5.9. For a vector space M that is at the same time a right A-module and a right C-comodule, (5.5) is equivalent to the following two relations:

[(m x [a.sub.[0]]).sub.{0}] [cross product] [a.sub.[-1]] x [(m x [a.sub.[0]]).sub.{1}] = [m.sub.{0}] x [a.sup.(0)] [cross product] [m.sub.{1}] x [a.sub.<1>], [for all] m [member of] M, a [member of] A;(5.6)

[m.sub.{0}] [cross product] [m.sub.{1}] = [m.sub.{0}] x [1.sub.<0>] [cross product] [m.sub.{1}] x [1.sub.<1>], [for all] m [member of] M. (5.7)

Proof. In the definition of a weak left H-comodule algebra A the condition [[lambda].sub.A] (1) [member of] [H.sub.s] [cross product] A is equivalent to ([Id.sub.H] [cross product] [[lambda].sub.A])[[lambda].sub.A] (1) = [1.sub.1] [cross product] [1.sup.2] [cross product] [1.sub.[0]] and to ([Id.sub.H] [cross product] [[lambda].sub.A])[[lambda].sub.A] (1) = [1.sub.1] [cross product] [1.sub.2] [1.sub.[-1]] [cross product] [1.sub.[0]]. We then have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for all a [cross product] A. In a similar way, [a.sub.[0]] [cross product] [[epsilon].sub.s] ([a.sub.[-1]]) = [l.sub.[0]] a [cross product] [1.sub.[-1]], for a [member of] A.

Now assume that (5.5) holds. Using the identity S [omicron] [[bar.[epsilon]].sub.s] = [[epsilon].sub.t], we compute that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for all m [member of] M and a [member of] A, as desired With the help of this equality and of the fact that [[lambda].sub.A] (1) [member of] [H.sub.s] [cross product] A we deduce that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and this finishes the proof of the direct implication. The converse can be proved as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 5.10. Theorem 4.6 was our source of inspiration for the definition of Yetter-Drinfeld modules over a weak Hopf algebra. Moving backwards, by Proposition 5.9 it makes sense to define weak right Yetter-Drinfeld modules over a weak bialgebra: all we have to do is to replace (5.5) with (5.6) and (5.7). If we do this then in the case when H is a weak Hopf algebra we can identify the category of right weak Yetter-Drinfeld modules with a category of weak right Doi-Hopf modules. This identification can be regarded as the weak Hopf algebra version of Theorem 4.6 and at the same time as a generalization of [16, Corollary 3.3].

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Faculty of Mathematics and Informatics, University of Bucharest, Str. Academiei 14, RO-010014 Bucharest 1, Romania

email:daniel.bulacu@fmi.unibuc.ro

Department of Algebra and Analysis

Universidad de Almeria

E-04071 Almeria, Spain

email:btorreci@ual.es

Daniel Bulacu * Blas Torrecillas

* The first author was supported by the strategic grant POSDRU/89/1.5/S/58852, Project "Postdoctoral program for training scientific researchers" cofinanced by the European Social Fund within the Sectorial Operational Program Human Resources Development 2007-2013. The second author was partially supported by FQM 3128 from Junta Andalucia MTM2011-27090 from MCI. The first author thanks the Universidad de Almeria for its warm hospitality. The authors also thank Bodo Pareigis for sharing his "diagrams" program.

Received by the editors in March 2013.

Communicated by S. Caenepeel.

2010 Mathematics Subject Classification : Primary 16W30; Secondary 18D10; 16S34.
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