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).