From Braided Infinitesimal Bialgebras to Braided Lie Bialgebras ()
1. Introduction
An infinitesimal bialgebra is a triple
, where
is an associative algebra (possibly without unit),
is a coassociative coalgebra (possibly without counit) such that
Infinitesimal bialgebras were introduced by Joni and Rota in [2] , called infini- tesimal coalgebra there, in the context of the calculus of divided differences [3] . In combinatorics, they were further studied in [4] [5] [6] . Aguiar established the basic theory of infinitesimal bialgebras in [7] [8] by investigating several examples and the notions of antipode, Drinfeld double and the associative Yang- Baxter equation keeping close to ordinary Hopf algebras. In [9] , Yau introduced the notion of Hom-infinitesimal bialgebras and extended Aguiar’s main results in [7] [8] to Hom-infinitesimal bialgebras.
One of the motivations of studying infinitesimal bialgebras is that they are closely related to Drinfeld’s Lie bialgebras (see [10] ). The cobracket
in a Lie bialgebra is a 1-cocycle in Chevalley-Eilenberg cohomology, which is a 1-cocycle in Hochschild cohomology (i.e., a derivation) in a infinitesimal bialgebra. So the compatible condition in a infinitesimal bialgebra can be seen as an associative analog of the cocycle condition in a Lie bialgebra.
Motivated by [1] , in which we considered infinitesimal Hopf algebras in the Yetter-Drinfeld categories, called braided infinitesimal Hopf algebras, the natural idea is whether we can obtain braided Lie bialgebras (called generalized H-Lie bialgebras in [11] [12] ) from braided infinitesimal Hopf algebras. This becomes our motivation of writing this paper.
To give a positive answer to the question above, we organize this paper as follows.
In Section 1, we recall some basic definitions about Yetter-Drinfeld modules and braided Lie bialgerbas. In Section 2, we introduce the notion of the balanceator of a braided infinitesimal bialgerba and show that a braided infinitesimal bialgerba gives rise to a braided Lie bialgerba if and only if the balanceator is symmetric (see Theorem 2.3).
2. Preliminaries
In this paper, k always denotes a fixed field, often omitted from the notation. We use Sweedler’s ( [13] ) notation for the comultiplication:
, for all
. Let H be a Hopf algebra. We denote the category of left H-modules by
. Similarly, we have the category
of left H-comodules. For a left H- comodules
, we also use Sweedler’s notation:
for all
.
A left-left Yetter-Drinfeld module M is both a left H-module and a left H- comodule satisfying the compatibility condition
(2.1)
for all
and
. The equation (1.1) is equivalent to
(2.2)
By [14] [15] , the left-left Yetter-Drinfeld category
is a braided monoi- dal category whose objects are Yetter-Drinfeld modules, morphisms are both left H-linear and H-colinear maps, and its braiding
is given by
for all
and
.
Let
be an object in
, the braiding
is called symmetric on
if the following condition holds:
(2.3)
which is equivalent to the following condition:
(2.4)
for any
In the category
, we call an (co)algebra simply if it is both a left H- module (co)algebra and a left H-comodule (co)algebra. For more details about (co)module-(co)algebras, the reader can refer to [16] [17] .
A braided Lie algebra ( [11] ) in
, called generalized H-Lie algebra there, is an object L in
together with a bracket operation
, which is a morphism in
satisfying
(1) H-anti-commutativity:
(2) H-Jacobi identity:
for all
, where
denotes
and
the braiding for L.
Let A be an associative algebra in
. Assume that the braiding is symmetric on A. Define
Then
is a braided Lie algebra (see [11] ).
A braided Lie coalgebra ( [12] )
is an object in
together with a linear map
(called the cobracket), which is also a morphism in
subject to the following conditions:
(1) H-anti-cocommutativity:
(2) H-coJacobi identity:
where
denotes the braiding for L.
Dually, let
be a coassociative coalgebra in
. Assume that the braiding on C is symmetric. Define
by
Then
is a braided Lie coalgebras in
(see [12] ).
A braided Lie bialgebra ( [18] ) is
in
, where
is a braided Lie algebra, and
is a braided Lie coalgebra, such that the compatibility condition holds:
where
denotes the braiding for L.
3. Main Results
In this section, we will study the relation between braided infinitesimal bialge- bras and braided Lie bialgebras as a generalization of Aguiar’s result in [8] .
Let
be a braided e-bialgebra in
. For any
, define an action of A on
by
Then the action ® is a morphism in
. In fact, for any
and
, we have
So ® is left H-linear. To show the left H-colinearity of the action ®, we compute
and
as desired.
Definition 2.1. Let
be a braided infinitesimal bialgebra and
the braiding of A. The map
defined by
(3.1)
is called the balanceator of A. The balanceator B is called symmetric if
. The braided infinitesimal bialgebra A is called balanced if
on A.
Condition (2.1) can be written as follows:
Obviously,
Lemma 2.2. Let
be a braided infinitesimal bialgebra and
. Assume that the braiding
on A is symmetric. Then the following equations hold:
(1)
(2)
(3)
Proof. (1) Since the braiding
on A is symmetric, for all
, we have
, then
that is,
So (1) holds.
(2) To show the Equation (2.2), we need the following computation:
The last equality holds since
is symmetric on A. Hence (2) holds.
(3) Finally, we check the Equation (2.3) as follows:
The last equality holds since
is symmetric on A. Hence (3) holds as required. ,
Therem 2.3. Let
be a braided infinitesimal bialgebra. Assume that the braiding
on A is symmetric. Then
is a braided Lie bialgebra if and only if
.
Proof. Since
is an associative algebra and
is a coassociative coalgebra in
,
is a braided Lie algebra and
is a braided Lie coalgebra. Therefore it remains to check the compatible condition:
for all
. In fact, on the one hand, we have
On the other hand, we have
According to Lemma 2.2, we have
Therefore,
as desired. We complete the proof. ,
Corollary 2.4. Let
be a braided infinitesimal bialgebra. Assume that the braiding
on A is symmetric and the balanceator
. Then
is a braided Lie bialgebra.
Proof. Straightforward from Theorem 2.3. ,
Example 2.5. Let q be an 2th root of unit of k and G the cyclic group of order 2 generated by g,
be the group algebra in the usual way. We consider the algebra
. By [8] ,
is a infinitesimal bialgebra equipped with the comultiplication:
Define the left-H-module action and the left-H-comodule coaction of A by
It is not hard to check that the multiplication and the comultiplicaition are both H-linear and H-colinear, therefore
is a braided infinitesimal bialgebra. Since
and
it is clear that
if and only if
. If
, it is not hard to check that the balanceator is symmetric on
. By Theorem 2.3,
is a braided Lie bialgebra.
Example 2.6. Let q be a 4th root of unit of k. Consider the Hopf algebra
, where G is a cyclic group of order 4 generated by g. Recall from [1] that
is a braided infinitesimal bialgebra in
equipped with the comultiplication:
and the H-module action, the H-comodule coaction:
Since
we claim that the balanceator is not symmetric. By Theorem 2.3,
is not a braided Lie bialgebra, where m is the multiplication of A.
Let
. It is clear that
is both H-stable and
H-costable, hence
is also a braided infinitesimal bialgebra contained in A. One can check easily that the balanceator
on
. By Corollary 2.4,
is a braided Lie bialgebra.
Acknowledgements
The paper is partially supported by the China Postdoctoral Science Foundation (No. 2015M571725), the Key University Science Research Project of Anhui Province (Nos. KJ2015A294 and KJ2016A545), the outstanding top-notch talent cultivation project of Anhui Province (No. gxfx2017123) and the NSF of Chuzhou University (No. 2015qd01).