Received 1 August 2014; revised 2 September 2014; accepted 13 September 2014

ABSTRACT

In this paper, the fundamental theorem of Yetter-Drinfeld Hopf module is proved. As applications, the freedom of tensor and twisted tensor of two Yetter-Drinfeld Hopf algebras is given. Let A be a Yetter-Drinfeld Hopf algebra. It is proved that the category of A-bimodule is equivalent to the category of -twisted module.

Keywords:

Hopf Algebra, Hopf Module, Yetter-Drinfeld Module, Yetter-Drinfeld Hopf Algebra

1. Introduction

Let be a field and an algebra. A left -module is a -vector space together with a -linear map such that and. The category of left -module is denoted by. Dually, let be a coalgebra. A left -comodule is a -vector space together with a -linear map such that

The category of left -comodule is denoted by. For more about modules and comodules, see [1] -[3] .

Assume that is a Hopf algebra with antipode, a left Yetter-Drinfeld module over is a -vector space which is both a left -module and left -comodule and satisfies the compatibility condition

for all. The category of left Yetter-Drinfeld module is denoted by. Yetter-Drinfeld modules category constitutes a monomidal category, see [4] . The category is pre-braided; the pre-braiding is given by

The map is a braiding in precisely when Hopf algebra has a bijective antipode with inverse of. In this case, the inverse of is

Let be a Hopf algebra and the category of left Yetter-Drinfeld module over. We call a Hopf algebra in or Yetter-Drinfeld Hopf algebra if is a -algebra and a -coalgebra, and the following conditions (a1)-(a6) hold for,

(a1) is a left -module algebra, i.e.,

(a2) is a left -comodule algebra, i.e.,

(a3) is a left -module coalgebra, i.e.,

(a4) is a left -comodule coalgebra, i.e.,

(a5) are algebra maps in, i.e.,

(a6) There exists a -linear map in such that

One easily get that is both -linear and -colinear. In general, Yetter-Drinfeld Hopf algebras are not ordinary Hopf algebras because the bialgebra axiom asserts that they obey (a5). However, it may happen that Yetter-Drinfeld Hopf algebras are ordinary Hopf algebras when the pre-braiding is trivial, for details see [5] .

Yetter-Drinfeld Hopf algebras are generalizations of Hopf algebras. Some important properties of Hopf algebras can be applied to Yetter-Drinfeld Hopf algebra. For example: Doi gave the trace formular of Yetter-Drin- feld Hopf algebras in [6] and studied Hopf module in [7] ; Chen and Zhang constructed Four-dimensional Yetter-Drinfeld module algebras in [8] ; Zhu and Chen studied Yetter-Drinfeld modules over the Hopf-Ore Extension of Group algebra of Dihedral group in [9] ; Alonso Álvarez, Fernández Vilaboa, González Rodríguez and Soneira Calvoar considered Yetter-Drinfeld modules over a weak braided Hopf algebra in [10] , and so on.

Hopf module fundamental theorem plays an important role in Hopf algebras. This theory can be generalized to Yetter-Drinfel Hopf algebras.

Theorem 1.1. Let be a Yetter-Drinfeld Hopf algebra, be a Yetter-Drinfeld Hopf module, then as left Yetter-Drinfeld Hopf module.

Note that Theorem 1.1 was appeared in [7] , but we give a different proof with Doi’s here.

Let be a Yetter-Drinfeld Hopf algebra. Define the multiplication of as

,

then is an algebra. But it is not a Yetter-Drinfeld Hopf algebra if is not the trivial twist T. As applications of Yetter-Drinfeld Hopf module fundamental theory, we have the freedom of the tensor of Yetter-Drinfeld Hopf algebras and twisted tensor of Yetter-Drinfeld Hopf algebras.

Theorem 1.2. Let be a Yetter-Drinfeld Hopf algebra, then and are free over.

We also proved the category of Yetter-Drinfeld. -bimodule is equivalent to the category of - module.

Theorem 1.3. Let be a Yetter-Drinfeld Hopf algebra. Then the category of and are equivalent.

2. The Freedom of Yetter-Drinfeld Hopf Algebras

In this section, we require is a Hopf algebra and is a Yetter-Drinfeld module over. Moreover, we need is a Yetter-Drinfeld Hopf algebra. Next, we will give the definition of Yetter-Drinfeld Hopf module, also see [7] .

Definition 2.1. Let be a Yetter-Drinfeld Hopf algebra. The Yetter-Drinfeld Hopf module over is defined by the following

1) is a left -module and left -comodule with comodule map,

2) is a -module map, i.e., , where.

Note that is a left -comodule with, and is a left -comodule with

. The Yetter-Drinfeld Hopf module category over is denoted by.

Define is the set of coinvariant elelments of. Next conclusion is similar to the fundamental theorem of Hopf algebra, we call it as the fundamental theorem of Yetter-Drinfeld Hopf module.

Theorem 2.2. Let be a Yetter-Drinfeld Hopf algebra, be a Yetter-Drinfeld Hopf module. Then as left Yetter-Drinfeld Hopf module.

Proof: We define by and by

.

First, we show that is well-defined, i.e.,. In fact, we have

So. Thus is well-defined.

We will show that is the inverse of. Indeed, if we have

Hence. Conversely, if, then

which show that too. It remains to show that is a morphism of -module and -comodule. The first assertion is clear, since

Next, we show that is a -comodule morphism, i.e.. Indeed, we have

This complete the proof.

Proposition 2.3. We have is a Yetter-Drinfeld Hopf module over.

Proof: is an -module by the trivial module action:. In fact, for

, we have and. The A-co-

module structure of is defined by. It is easy to check is an Yetter-Drinfeld Hopf module over, we omit it.

Theorem 2.4. Let be a Yetter-Drinfeld Hopf algebra, then is free over.

Proof: Apply to the fundamental theorem of Yetter-Drinfeld Hopf algebra, then and become and

Next, we show that. In fact, we have

and

Hence, we have.

Moreover, is an -module map. Since and

. Furthermore, is also an -comodule map by the following. Take

, then. We have

and

In a word, , so is free over. This completes the proof.

3. Twist Yetter-Drinfeld Hopf Module

Let be a Yetter-Drinfeld Hopf algebra over Hopf. Define the multiplication of as follows:

(1)

Lemma 3.1. Let be a Yetter-Drinfeld Hopf algebra, then is an algebra with multiplication (1).

Proof: We only need to check the associativity of

And is the unit element of. Thus is an algebra.

Remark 3.2. is a Yetter-Drinfeld Hopf algebra if and only if. See reference [5] for the details.

Denote the -bimodule category by, and -module category by.

Theorem 3.3. Let be a Yetter-Drinfeld Hopf algebra. Then the category of and are

equivalent.

Proof: we are going to construct the functor as follows. Let be an -bimodule.

We denote the two-side action on by “·”. Define as -space with the left action given by

We claim that the action is well-defined, i.e.. In fact, we have

and

.

By comparing the above two identities, we have and -module.

Moreover, we have the functor given as follows: Let be a left -module,

define to be as -space, and its -bimodule structure given by and

. Note that denotes the inverse of and denotes the inverse of

. Clearly, is a left -module. Note that

Hence, is also a right -module. We have

and

therefore, is a -bimodule. It is easy to check that the functors and are inverse to each other. This completes the proof.

Let be a Yetter-Drinfeld Hopf algebra, then is a right -module by

.

Recall that if is a vector space, then is a free -module with the action.

Theorem 3.4. The right -module defined above is free over.

Proof: Let denote the underlying space of. Thus become a right free module. Define a map

. It is obvious that is a bijection with inverse

. We claim that is a right -module morphism, then we are done.

In fact, we have

Note that the right -module structure on is, so. Thus we have proved that. This completes the proof.

Acknowledgements

Supported by the National Nature Science Foundation of China (Grant No. 11271239).

References