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We give the Fundamental Theorem for Hopf modules in the category of Yetter-Drinfeld modules , where L is a quasitriangular weak Hopf algebra with a bijective antipode. We also show that H* has a right H-Hopf module structure in the Yetter-Drinfeld category. As an application we deduce the existence and uniqueness of right integral from it.

Weak Hopf algebras were introduced by G. Böhm and K. Szlachányi as a generalization of usual Hopf algebras and groupoid algebras [

Paper [

we prove the Fundamental Theorem for Hopf modules in the category of Yetter-Drinfeld modules according to the fact that the matrix R gives rise to a natural braiding for

Throughout this paper we use Sweedler’s notation for comultiplication, writing

Definition 1. A weak Hopf algebra is a vector space L with the structure of an associative unital algebra

1) The comultiplication

2) The counit satisfies the following identity

3) There is a linear map

The linear map defined in the above equations are called target and source counital maps and denoted by

For all

We will briefly recall the necessary definitions and notions on the weak Hopf algebras.

Definition 2. A quasitriangular weak Hopf algebra is a pair

for all

where

Proposition 2.1. For any quasitriangular weak Hopf algebra

Let L be a quasitriangular weak Hopf algebra with a bijective antipode

Definition 3. Let

1)

2) H is a left L-module algebra and left L-module coalgebra if H is a left L-module via

3) H is a left L-comodule algebra and left L-comodule coalgebra if H is a left L-comodule via

4) Furthermore, H is called a weak Hopf algebra in

Similar to the definition of weak Hopf algebra, we denote

Paper [

Proposition 3.1. Suppose H is a weak Hopf algebra in

Since a weak Hopf algebra H in the weak Yetter-Drinfeld categories

Definition 4. Let H be a weak Hopf algebra in

1)

2)

3)

4)

5)

We remark that

Example 3.2. H itself is a right H-Hopf module (in

when H is a weak Hopf algebra in

Applying

For

This implies that

It is clearly to prove F is a left L-colinear by the following equation

Furthermore we can obtain the Structure Theorem for right H-Hopf modules in the category of Yetter- Drinfeld modules.

Theorem 3.3. If H is a weak Hopf algebra in

1) Let

2) The map

In [

Since H is a finite-dimensional left L-comodule,

i.e.

Second,

That is

Proposition 4.1.

Proof. Now for

It implies that

Accord to

Hence

Theorem 4.2. With the notation as above, then

Proof. Now we prove that

Next we want to check

Applying the equality

It implies that

Finally we show that

From all above,

Applying Theorem 4.2 we can obtain the following result.

Corollary 4.3.

As a consequence the space of coinvariants of the finite dimensional Hopf algebra is free of rank one. This is the case for the weak Hopf algebra in the category of the Yetter-Drinfeld modules.

Theorem 5.1. If H is a finite-dimensional weak Hopf algebra in

1)

2) The map

3) There exist a right integral t in H,

a)

b)

c)

d)

4) The map

Proof. 1) Since

2) Choose

3) a) Since

b) We remark that

i.e.

c) From Theorem 3.3 we have

we can obtain

d) Applying

This means

4) For all

This implies

The author would like to thank the referee for many suggestions and comments, which have improved the overall presentations.

Research supported by the Project of Shandong Province Higher Educational Science and Technology Program

(J12LI07) and the Project of National Natural Science Foundation of China (51078225).

Yanmin Yin, (2016) Hopf Modules in the Category of Yetter-Drinfeld Modules. Applied Mathematics,07,629-637. doi: 10.4236/am.2016.77058