On Hom-Lie superbialgebras
Mohamed Fadous, Sami Mabrouk, Abdenacer Makhlouf

TL;DR
This paper extends the theory of Hom-Lie bialgebras to the superalgebra setting, introducing new constructions, structures, and deformations specific to Hom-Lie superbialgebras.
Contribution
It generalizes Hom-Lie bialgebras to the $ ext{Z}_2$-graded case and develops new methods for constructing and analyzing Hom-Lie superbialgebras.
Findings
Introduced methods for constructing Hom-Lie superbialgebras
Defined and related Matched pairs and Manin supertriples
Studied coboundary, triangular structures, and deformations
Abstract
The purpose of this paper is to generalize to -graded case the study of Hom-Lie bialgebras which were discussed first by D. Yau, then by C. Bai and Y. Sheng. We provide different ways for constructing Hom-Lie superbialgebras. Also we define Matched pairs, Manin supertriples and discuss their relationships. Moreover, we study coboundary and triangular Hom-Lie bialgebras, as well as infinitesimal deformations of the cobracket.
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On Hom-Lie superbialgebras
Mohamed Fadous, Sami Mabrouk, Abdenacer Makhlouf
Abstract
The purpose of this paper is to generalize to -graded case the study of Hom-Lie bialgebras which were discussed first by D. Yau, then by C. Bai and Y. Sheng. We provide different ways for constructing Hom-Lie superbialgebras. Also we define Matched pairs, Manin supertriples and discuss their relationships. Moreover, we study coboundary and triangular Hom-Lie bialgebras, as well as infinitesimal deformations of the cobracket.
**Keywords :Hom-Lie superalgebra, coboundary Hom-Lie bialgebra, triangular Hom-Lie bialgebra
2010 Mathematics Subject Classification : 17B62, 17B37. **
Introduction
Lie bialgebras were introduced by V. Drinfel’d in [7, 8], they are infinitesimal versions of compatible Poisson structures on Lie groups and maybe viewed as the Lie-theoretic case of a bialgebra. He raised various problems related to quantum groups and quantization. The study of quasi-triangular quantum groups involves the solutions of the quantum Yang-Baxter equations. In the classical limit, the solutions of the classical Yang-Baxter equations provide examples of Lie bialgebras. Since then a huge research activity was dedicated to these kind of algebraic structures.
The aim of this paper is to define and study Hom-Lie superbialgebras which are Hom-type generalization of Lie superbialgebras. Hom-Lie superbialgebras are Hom-Lie superalgebras provided with a cobracket and a compatibility condition. Motivated by examples of -deformations of algebras of vector fields, J. Hartwig, D. Larsson, and S. Silvestrov introduced the notion of Hom-Lie algebra in [12], as a generalization of Lie algebras where the Jacobi condition is twisted by a Homomorphism. The graded case of Hom-Lie algebras were studied first by F. Ammar and the last author in [1], while Hom-Lie bialgebras were discussed by D. Yau, then by C. Bai and Y. Sheng. Recently L. Cai and Y. Sheng presented a slightly different approach of Hom-Lie bialgebras called purely Hom-Lie bialgebras in [5].
The paper is organized as follows, in the first section we provide the relevant definitions and some properties about Hom-Lie superbialgebras. Moreover, we give some key constructions and a classification of 3-dimensional Hom-Lie superbialgebras with 2-dimensional even part. In Section 2, We define Matched pairs and Manin supertriples, then we establish their relationships with Hom-Lie superbialgebras. We construct a Hom-Lie superalgebra structure on the direct sum of two Hom-Lie superalgebras and , such that is a -module and is a -module, also we construct a Hom-Lie superbialgebra structure on the direct sum where is the dual superspace of . Section 3 is dedicated to coboundary Hom-Lie superbialgebras and quasi-triangular Hom-Lie superbialgebras. We show how a coboundary or quasi-triangular Hom-Lie superbialgebra can be constructed from a Hom-Lie superalgebra and an -matrix. In the last section, we study perturbation of cobrackets in Hom-Lie superbialgebras, following Drinfel’d’s perturbation theory of quasi-Hopf algebras. We describe Hom-Lie superbialgebras obtained by infinitesimal deformations of the cobracket.
1 Basics and Classification of Hom-Lie superbialgebras
In this section, we introduce and study Hom-Lie superbialgebras, which are Hom-type version of Lie superbialgebras, see [7, 8]. We extend to graded case the definition of Hom-Lie bialgebra introduced in [27]. We show that the dual of a finite dimensional Hom-Lie superbialgebra is also a Hom-Lie superbialgebra (Theorem 1.18), generalizing the self-dual property of Hom-Lie bialgebras.
First, let us start by fixing some definitions and notations. Let be a -graded vector space over an arbitrary field of characteristic 0. In the sequel, we will consider only element which are -homogeneous. For , we denote by its parity, i.e., . We denote by the super-twist map of namely, for The super-cyclic map permutes the coordinates of , it is defined as
[TABLE]
for , where is the identity map on . We denote by Hom the linear dual of . For and , we often use the adjoint notation for .
For a linear map (comultiplication), we use Sweedler’s notation for . We will often omit the summation sign to make it simple. The parity of is defined as follows : since we assume homogenous, there exists , such that can be written as , are homogenous elements with .
Definition 1.1**.**
([1]). A *Hom-Lie superalgebra * is a triple consisting of a superspace , an even bilinear map and an even superspace homomorphism satisfying
[TABLE]
[TABLE]
for all homogeneous elements in .
It is multiplicative if, in addition , (i.e., , ).
Definition 1.2**.**
([13], [19]). A Hom-Lie supercoalgebra is a triple consisting of a superspace , an even superspace homomorphism and a linear map (the cobracket) such that
[TABLE]
[TABLE]
[TABLE]
If, in addition, , then is called co-multiplicative.
A Hom-Lie supercoalgebra with is exactly a Lie supercoalgebra [13].
Remark 1.3*.*
Let , then i.e. is skew-supersymmetric ([24]).
Definition 1.4**.**
(1) For an element in a Hom-Lie superalgebra and , define the adjoint map by
[TABLE]
For , .
Conversely, given , we define the map by , for .
(2) For an element in a Hom-Lie superalgebra and , define the adjoint map
by
[TABLE]
Definition 1.5**.**
A *(multiplicative) Hom-Lie superbialgebra * is a quadruple such that
is a (multiplicative) Hom-Lie superalgebra. 2. 2.
is a (co-multiplicative) Hom-Lie supercoalgebra. 3. 3.
The following compatibility condition holds for all :
[TABLE]
Definition 1.6**.**
The map is called even (resp. odd) map if (resp. for . A morphism of Hom-Lie superbialgebras is an even linear map such that
[TABLE]
An isomorphism of Hom-Lie superbialgebras is an invertible morphism of Hom-Lie superbialgebras. Two Hom-Lie superbialgebras are said to be isomorphic if there exists an isomorphism between them.
Remark 1.7*.*
A Hom-Lie superbialgebra with is exactly a Lie superbialgebra, as defined in [13, 7, 8].
Remark 1.8*.*
The compatibility condition (1.8) is, in fact, a cocycle condition in Hom-Lie superalgebra cohomology [20], just as it is the case in a Lie superbialgebra with Lie superalgebra cohomology [8]. Indeed, we can regard as an -module via the -twisted adjoint action (1.8):
[TABLE]
for and .
Then we can think of the cobracket as a 1-cochain . Here is defined as the linear super-subspace of Hom consisting of maps that commute with . Generalizing [20] to include coefficients in , the differential on is given by
[TABLE]
Therefore, (1.8) says exactly that is a 1-cocycle.
Example 1.9**.**
Classification of 2-dimensional Hom-Lie superbialgebras with 1-dimensional even part*.
Let be a 2-dimensional superspace where is generated by and is generated by . The triple is a Hom-Lie superalgebra when , and with , and , where are parameters in .
The triple is a Hom-Lie supercoalgebra if and , where .
The triple is a Hom-Lie superbialgebras if or and .*
Remark 1.10*.*
Recently Cai and Sheng introduced a different notion of Hom-Lie bialgebras called purely Hom-Lie bialgebra, see [5], which we can extend to the super case as follows. Let , where is invertible, and be two Hom-Lie superalgebras. The pair is a purely Hom-Lie superbialgebra if holds the compatibility condition
[TABLE]
Notice that this condition is different from condition (1.8). Purely Hom-Lie bialgebras in a graded case will be studied in a forthcoming paper.
Classification of 3-dimensional Hom-Lie superbialgebras with 2-dimensional even part.
Let be a 3-dimensional superspace where is generated by , and is generated by . We aim to construct Hom-Lie superbialgebras . We set for the linear map
[TABLE]
where are parameters in
The structure of the bracket is of the form
[TABLE]
where are parameters in
The structure of the cobracket is of the form
[TABLE]
where are parameters in
In the sequel, we will consider the linear map under the Jordan forms with respect to a suitable basis. We split the calculations in two cases. First, we deal with diagonal case and then with Jordan case.
**1) Diagonal case **: We consider the linear map , where with respect to a suitable basis the matrix is of the form: \alpha=\left(\begin{array}[]{lll}a_{1}\ \ 0\ \ \ 0\\ 0\ \ \ a_{4}\ \ 0\\ 0\ \ \ 0\ \ \ a_{5}\\ \end{array}\right), which corresponds to , , and the eigenvalues are pairwise non-equal.
We obtained, when the eigenvalues are nonzero, the following corresponding (multiplicative) Hom-Lie superbialgebras :
[TABLE]
[TABLE]
2) Jordan case : Now, we consider the linear map where the corresponding matrix is of the form \alpha=\left(\begin{array}[]{lll}a_{1}\ \ 1\ \ \ 0\\ 0\ \ \ a_{1}\ \ 0\\ 0\ \ \ 0\ \ \ a_{5}\\ \end{array}\right), that is , , .
We obtained the following corresponding (multiplicative) Hom-Lie superbialgebras :
[TABLE]
[TABLE]
The following result shows that a Hom-Lie superbialgebra deforms into another Hom-Lie superbialgebra along any endomorphism.
Theorem 1.11**.**
Let be a Hom-Lie superbialgebra and an even map be a Hom-Lie superbialgebra morphism. Then
[TABLE]
is also a Hom-Lie superbialgebra, which is multiplicative if is.
Proof.
It is immediate that is skew-supersymmetric (1.1) because
[TABLE]
The Hom super-Jacobi identity holds in because
[TABLE]
i.e.,
[TABLE]
is skew-supersymmetric because
[TABLE]
Likewise, the Hom-super-co-jacobi identity holds in because
[TABLE]
To check the compatibility condition (1.8) in , we compute as follows :
[TABLE]
Because , and (i.e., an even map ). We have shown that is a Hom-Lie superbialgebra. The super-multiplicativity assertion is obvious. ∎
Now we discuss two special cases of Theorem 1.11. The next result says that one can obtain multiplicative Hom-Lie superbialgebras from Lie superbialgebras and their endomorphisms. A construction result of this form for Hom-type algebras was first given in [26].
Corollary 1.12**.**
Let be a Lie superbialgebra and an even map be a Lie superbialgebra morphism. Then
[TABLE]
is a multiplicative Hom-Lie superbialgebra.
Proof.
This is the special case of Theorem 1.11. ∎
The next result says that every multiplicative Hom-Lie superbialgebra gives rise to an infinite sequence of multiplicative Hom-Lie superbialgebras.
Corollary 1.13**.**
Let be a multiplicative Hom-Lie superbialgebra. Then
[TABLE]
is also a multiplicative Hom-Lie superbialgebra for each integer .
Proof.
This is the special case of Theorem 1.11. ∎
Next we consider when Hom-Lie superbialgebra of the from , as in Corollary 1.12, are isomorphic.
Theorem 1.14**.**
*Let and be Lie superbialgebras. Let and be Lie superbialgebras morphisms with and injective. Then the following statements are equivalent :
-
The Hom-Lie superbialgebras and as in Corollary 1.12, are isomorphic.
-
There exists a Lie superbialgebra isomorphism such that *
Proof.
To show that the first statement implies the second statement, suppose that is an isomorphism of Hom-Lie superbialgebras. Then automatically.
To see that is a Lie superbialgebra isomorphism, first we check that it commutes with the Lie bracket. For any two elements and in , we have
[TABLE]
Since is injective, we conclude that i.e., is a Lie superbialgebra isomorphism.
To check that commutes with the Lie cobrackets, we compute as follows:
[TABLE]
The injectivity of now implies that commutes with the Lie cobrackets. Therefore, is a Lie superbialgebra isomorphism. The other implication is proved by a similar argument, much of which is already given above. ∎
For a Lie superbialgebra , let Aut be the group of Lie superbialgebra isomorphisms from to . In Theorem 1.14, restricting to the case with and both invertible, we obtain the following special case.
Corollary 1.15**.**
Let be a Lie superbialgebra and Aut. Then the Hom-Lie superbialgebras and , as in Corollary 1.12, are isomorphic if and only if and are conjugate in Aut.
Corollary 1.15 can be restated as follows.
Corollary 1.16**.**
*Let be a Lie superbialgebra.Then there is a bijection between the following two sets:
-
The set of isomorphism classes of Hom-Lie superbialgebras with invertible.
-
The set of conjugacy classes in the group Aut.*
The next result shows that finite dimensional Hom-Lie superbialgebras, like Lie superbialgebras, can be dualized. A proof of this self-dual property for the special of Lie bialgebras can be found in [16].
Remark 1.17*.*
If is a Hom-Lie supercoalgebra, then is a Hom-Lie superalgebra. Here and in are dual to and , respectively, in .
Conversely, if is a finite dimensional Hom-Lie superalgebra, then is a Hom-Lie supercoalgebra, where and in are dual to and , respectively, in .
Theorem 1.18**.**
Let be a finite dimensional (multiplicative) Hom-Lie superbialgebra. Then its linear dual Hom is also a (multiplicative) Hom-Lie superbialgebra with the dual structure maps:
[TABLE]
for and .
Proof.
As we mentioned right after Remark 1.17, is a Hom-Lie superalgebra, which is true even if is not finite dimensional. Moreover, is a Hom-Lie supercoalgebra, whose validity depends on the finite dimensionality of . Thus, it remains to check the compatibility condition (1.8) between the bracket and the cobracket in , i.e.,
[TABLE]
for and .
Using Definition 1.12, the compatibility condition (1.8) in , we compute the left-hand side of (1.13) as follows:
[TABLE]
Using, in addition, the skew-supersymmetric of the bracket and the cobracket in ,
and ).
The above four terms become:
[TABLE]
This is exactly the right-hand side of (1.13). ∎
2 Matched pairs, Hom-Lie superbialgebras and Manin supertriples
In this section, we introduce the notions of matched pair of Hom-Lie superalgebras and a Manin supertriple of Hom-Lie superalgebras. First, we recall the basics about representations of a Hom-Lie superalgebras.
Definition 2.1**.**
([2, 22, 23]) Let be a Hom-Lie superalgebra and an arbitrary vector superspace. A representation of the Hom-Lie superalgebra with respect to , is an even linear map , such that where , and satisfying
[TABLE]
and
[TABLE]
for all homogeneous elements .
We denote a representation by . It is straightforward to see that is a representation, called the adjoint representation, see [4, 22].
Given a representation , define by
[TABLE]
, , This representation is called admissible representation with respect to .
Now, let and be two multiplicative Hom-Lie superalgebras. Set and . Let and be two linear maps. Define a skew-supersymmetric bracket , where is given by : , where and . We set and , for all homogeneous elements in and in . Define by
[TABLE]
and the bracket by
[TABLE]
Theorem 2.2**.**
The triple , where , , are defined above is a multiplicative Hom-Lie superalgebra if and only if and are representations of and respectively and the following conditions are satisfied
[TABLE]
[TABLE]
Proof.
Assume is a multiplicative Hom-Lie superalgebra. The multiplicativity condition writes
[TABLE]
Developing (2.7), leads to the first condition (2.1). Indeed
[TABLE]
[TABLE]
Since and are multiplicative, which implies that
[TABLE]
[TABLE]
Developing the Hom-super-Jacobi identity (1.2), that for and ,
[TABLE]
[TABLE]
leads to the second condition (2.2). Indeed,
[TABLE]
Similarly, we compute :
[TABLE]
Setting (, , and ) give
[TABLE]
[TABLE]
which implies (2.5), and (2.6).
By Eqs. (2.8) and (2.10), we deduce that is a representation of the Hom-Lie superalgebra on with respect to . By Eqs. (2.9) and (2.11), we deduce that is a representation of the Hom-Lie superalgebra on with respect to . ∎
Example 2.3** (See [2]).**
*Given a representation of a Hom-Lie superalgebra . Set and . If and , we denote .
Define a skew-supersymmetric bracket by
[TABLE]
Define by . Then is a Hom-Lie superalgebra, which we call the semi-direct product of the Hom-Lie superalgebra by .
Definition 2.4**.**
For any , define by . Its dual map is defined as
[TABLE]
for all , and , where the pairing is supersymmetric, i.e. for ,
and . More precisely, for any , we have
[TABLE]
Definition 2.5**.**
A matched pairs of Hom-Lie superalgebras, which we denote by , consists of two Hom-Lie superalgebras and , together with representations and with respect to and respectively, such that the compatibility conditions (2.5) and (2.6) are satisfied.
In the following, we concentrate on the case that is , the dual space of , and , , , where is the dual map of . Notice that and are the adjoint representations associated to Hom-Lie superalgebras and respectively. Let be elements in and elements in .
For a Hom-Lie superalgebra (resp. ), let (resp. ) be the dual map of (resp. ), i.e.
[TABLE]
A Hom-Lie superalgebra is called admissible if its adjoint representation is admissible, that is . In particular, if a Hom-Lie superalgebra is * involutive*, that is , then it is admissible.
Proposition 2.6**.**
A pair of admissible Hom-Lie superalgebras and determines a Hom-Lie superbialgebra , where is the dual operation of if
[TABLE]
[TABLE]
Remark 2.7*.*
Following Bai and Sheng, we may denote this Hom-Lie superbialgebra by .
Theorem 2.8**.**
Let and be a pair of admissible Hom-Lie superalgebra. Then , where is the dual of , is a Hom-Lie superbialgebra if and only if and is a matched pairs of Hom-Lie superalgebras, i.e. is a multiplicative Hom-Lie superalgebra, where is given by Eq. (2.4), in which and
Proof.
By Theorem 2.2, two admissible Hom-Lie superalgebras and form a matched pair of Hom-Lie superalgebras if and only if
[TABLE]
[TABLE]
Since and are admissible Hom-Lie superalgebras i.e. (, and ), see Lemma 2.9 in [23]. By Eq. (2.17), we get
[TABLE]
[TABLE]
which implies that
[TABLE]
which is exactly Eq. (2.14). Similarly, one deduces that Eq. (2.16) is equivalent to Eq. (2.15). ∎
Let be a superspace and be the canonical pairing. Then we identify with by the pairing , and . On the other hand, we shall say that a bilinear form is supersymmetric if .
Definition 2.9**.**
A Manin supertriple of Hom-Lie superalgebras is a triple of Hom-Lie superalgebras together with a nondegenerate supersymmetric bilinear form on such that
is invariant, i.e. for any , we have
[TABLE]
[TABLE] 2. 2.
and are isotropic Hom-Lie sub-superalgebra of , such that as vector superspace.
Proposition 2.10**.**
Let be a Hom-Lie superbialgebra in the sense of Proposition 2.6. Then is a Manin supertriple of Hom-Lie superalgebras.
Proof.
Let be a Hom-Lie superbialgebra in the sense of Proposition 2.6, i.e. and are admissible Hom-Lie superalgebras such that Eqs. (2.14), (2.15) are satisfied. By Theorem 2.8, we know that is a Hom-Lie superalgebra, where is given by
[TABLE]
for all homogeneous elements and in and respectively. From the construction above we have and .
Furthermore, there is an obvious supersymmetric bilinear form on :
[TABLE]
It’s straightforward, using the supersymmetry and , with and , that Eqs. (2.18) and (2.19) are satisfied, i.e. the bilinear form defined by Eq. (2.21) is invariant. ∎
Conversely, if is a Manin supertriple of Hom-Lie superalgebras with the invariant bilinear from given by Eq. (2.21), then for any and , we have the natural scalar product on defined by
[TABLE]
Due to the invariance of , we have
[TABLE]
which implies that :
[TABLE]
that is, the Hom-Lie bracket on is given by Eq. (2.20). Therefore, is a matched pair of Hom-Lie superalgebras and hence is a Hom-Lie superbialgebra. Note that we deduce naturally that both and are admissible Hom-Lie superalgebras.
Summarizing the above study, Theorem 2.8 and Proposition 2.10, we have the following conclusion.
Theorem 2.11**.**
Let and be two admissible Hom-Lie superalgebras. Then the following conditions are equivalent.
* is a Hom-Lie superbialgebra in the sense of Proposition 2.6.* 2. 2.
* is a matched pair of Hom-Lie superalgebras.* 3. 3.
* is a Manin supertriple of Hom-Lie superalgebras with the invariant bilinear from (2.21).*
3 Coboundary and quasi-triangular Hom-Lie superbialgebras
In this section, we define and study coboundary Hom-Lie superbialgebras and quasi-triangular Hom-Lie superbialgebras. Then we show how a coboundary or a quasi-triangular Hom-Lie superbialgebra can be constructed from a Hom-Lie superalgebra and an -matrix.
Definition 3.1**.**
A (multiplicative) coboundary Hom-Lie superbialgebra consists of a (multiplicative) Hom-Lie superbialgebra and an element such that and
[TABLE]
for all .
Remark 3.2*.*
[TABLE]
for , where the parity of is defined as follows : since we assume is homogenous, there exists , such that can be written as , are homogenous elements with . (Note that equation (3.2) and (1.3) show that we have , namely ). So we get (3.1).
Definition 3.3**.**
The classical Yang-Baxter equation (CYBE):
[TABLE]
where are defined by
[TABLE]
and considered as elements in , the universal enveloping algebra of a Lie superalgebra . Elements (3.3) belongs to .
Definition 3.4**.**
The classical Hom-Yang-Baxter equation (CHYBE) in a Hom-Lie superalgebra is
[TABLE]
for . The three brackets in (3.4) are defined as
[TABLE]
where and . If , then the (CHYBE) reduces to the (CYBE).
Definition 3.5**.**
A (multiplicative) quasi-triangular Hom-Lie superbialgebra is a (multiplicative) coboundary Hom-Lie superbialgebra in which is a solution of the CHYBE (3.4). In these cases, we also write as .
Remark 3.6*.*
Note that we do not require to be skew-supersymmetric in a coboundary Hom-Lie superbialgebra, whereas in [8] is assumed to be skew-supersymmetric in a coboundary Lie bialgebra. We follow the convention in [16] and [27].
Remark 3.7*.*
Condition (3.1) is a natural because from Remark 1.8 the compatibility condition (1.8) in Hom-Lie superbialgebra says that the cobracket is a 1-cocycle in , where acts on via the -twisted adjoint action (1.9). The simplest 1-cocycles are the 1-coboundaries, i.e, images of . We can define the Hom-Lie 0-cochains and 0 th differential as follows, extending the definition in [20]. Set as the subspace of consisting of elements that are fixed by . Then we define the differential
[TABLE]
by setting as in (1.6). It is not hard to check that, for , we have where is defined in (1.10). In fact, what this condition says is that
[TABLE]
for all . We will prove (3.6) in Lemma 3.11 below. Thus, such an element is a 1-coboundary, and hence a 1-cocycle. This fact makes (with ) a natural candidate for the cobracket in a Hom-Lie superbialgebra and also justifies the name coboundary Hom-Lie superbialgebra.
The following result is the analogue of Theorem 1.11 for coboundary or quasi-triangular Hom-Lie superbialgebras. It says that coboundary or quasi-triangular Hom-Lie superbialgebras deform into other coboundary or quasi-triangular Hom-Lie superbialgebras via suitable endomorphisms.
Theorem 3.8**.**
Let be a coboundary Hom-Lie superbialgebra and an even map be a morphism such that . Then is also a coboundary Hom-Lie superbialgebra, which is multiplicative if is. Moreover, if is quasi-triangular, then so is .
Proof.
By Theorem 1.11 we know that is a Hom-Lie superbialgebra, multiplicative if is. To check that is coboundary, first note that
To check the condition (3.1) in , we compute as follows:
[TABLE]
The last expression above is in , which shows that is coboundary.
Finally, suppose in addition that is quasi-triangular, i.e., is a solution of the CHYBE in . Using the notation in (3.4) we have:
[TABLE]
where the last expression is defined in . This shows that is a solution of the CHYBE in , so is quasi-triangular.
The following result is the analogue of Corollary 1.12 for coboundary or quasi-triangular Hom-Lie superbialgebras. It says that these objects can be obtained by twisting coboundary or quasi-triangular Lie superbialgebras via suitable endomorphisms. ∎
Corollary 3.9**.**
Let be a coboundary Lie superbialgebra and an even map be a Lie superalgebra morphism such that . Then is a multiplicative coboundary Hom-Lie superbialgebra. If, in addition, is a quasi-triangular Lie superbialgebra, then is a multiplicative quasi-triangular Hom-Lie superbialgebra.
Proof.
This is the special case of Theorem 3.8, provided that we can show that
. We compute as follows:
[TABLE]
∎
The next result says that every multiplicative coboundary or quasi-triangular Hom-Lie superbialgebra gives rise to an infinite sequence of multiplicative coboundary or quasi-triangular Hom-Lie superbialgebra. It is similar to Corollary 1.13.
Corollary 3.10**.**
Let be a multiplicative coboundary (resp. quasi-triangular) Hom-Lie superbialgebra. Then is also a multiplicative coboundary (resp. quasi-triangular) Hom-Lie superbialgebra for each integer .
Proof.
This is the special case of Theorem 3.8. ∎
In the following result, we describe some sufficient condition under which a Hom-Lie superalgebra becomes a coboundary Hom-Lie superbialgebra.
In what follows, for an element , we write for
Lemma 3.11**.**
Let be a multiplicative Hom-Lie superalgebra and be an element such that , (1.3) and (3.2), ((3.2) and (1.3), i.e., . Then satisfies (1.8), i.e., for .
Proof.
We will use , the skew-supersymmetry (1.1) and the Hom-Jacobi identity of (1.2) and , (multiplicative) in the computation below. For , we have:
[TABLE]
∎
Theorem 3.12**.**
Let be a multiplicative Hom-Lie superalgebra and be an element such that (1.3) and (3.2), ((3.2) and (1.3), i.e., , and
[TABLE]
*for all , where is defined in (3.4). Define as as in (3.1).
Then is a multiplicative coboundary Hom-Lie superbialgebra.*
Proof.
We will show the following statements:
-
commutes with .
-
is skew-supersymmetric.
-
The compatibility condition (1.8) holds.
-
The condition (3.7) is equivalent Hom-super-coJacobi identity of (1.5).
Write , and , . To show that commutes with , pick an element , the summation sign will often be omitted in computation to simplify the typography. Using the definition , , and the assumption we have
[TABLE]
This shows that commutes with .
Now we show that is skew-supersymmetric. We have
[TABLE]
Then since
[TABLE]
We will prove that the compatibility condition (1.8) holds in Lemma 3.11.
Finally, we show that the Hom-super-coJacobi identity (1.5) of is equivalent to (3.7). Let us unwrap the Hom-super-coJacobi identity. Fix an element , and let be another copy of . Then we write
[TABLE]
Note
[TABLE]
we get
With these notations, the Hom-super-coJacobi identity of (applied to ) becomes
[TABLE]
Therefore, to prove the equivalence between the Hom-super-coJacobi identity of and (3.7), it suffices to show
[TABLE]
which we will prove in Lemma 3.13 below.
The proof of Theorem 3.12 will be completed once we prove the Lemma below. ∎
Lemma 3.13**.**
The condition (3.9) holds.
Proof.
It suffices to show the following three equalities:
[TABLE]
[TABLE]
[TABLE]
where the three bracket, which add up to , are defined in (3.5). The proofs for the three equalities are very similar, so we will only give the proof of (3.10).
Since and , we have
[TABLE]
In the fourth equality we used , . In the equality we used the skew-supersymmetry of and , we know , just to check the calculations. Similar computation give
[TABLE]
[TABLE]
[TABLE]
Using, in addition, the skew-supersymmetry (1.1) and the Hom-Jacobi identity (1.2) of , we add and :
[TABLE]
Using the definition (1.6) of , we now conclude that:
[TABLE]
This proves (3.10).
The equalities (3.11) and (3.12) are proved by very similar computations.
Therefore, the equality (3.9) holds. Together with (3.8) we have shown that the the Hom-super-coJacobi identity of is equivalent to ∎
The following result is an immediate consequence of Theorem (3.12). It gives sufficient conditions under which a Hom-Lie superalgebra becomes a quasi-triangular Hom-Lie superbialgebra.
Corollary 3.14**.**
Let be a multiplicative Hom-Lie superalgebra and be an element such that (1.3) and (3.2), ((3.2) and (1.3), i.e., , and Then is a multiplicative quasi-triangular Hom-Lie superbialgebra.
Theorem 3.15**.**
*Let be a coboundary Hom-Lie superbialgebra. Then the following statements are equivalent,
(1) is a quasi-triangular Hom-Lie superbialgebra, i.e., (3.4).
(2) The equality holds, where the bracket is defined in (3.5).
(3) The equality holds, where the bracket is defined in (3.5).*
Proof.
The equivalence between three statements clearly follows from the equalities. Let be another copy of . Since (3.1), and . By calculation we will find results
[TABLE]
This shows the equivalence between statements (1) and (2). Likewise, we have
[TABLE]
This shows the equivalence between statements (1) and (3). ∎
4 Cobracket perturbation in Hom-Lie superbialgebras
The purpose of this section is to study perturbation of cobrackets in Hom-Lie superbialgebras, following Drinfel’d’s perturbation theory of quasi-Hopf algebras ([6], [9], [10], [11]).
We address the following question : ” If is a Hom-Lie superbialgebra (Definition 1.5) and , under what conditions does the perturbed cobracket give another Hom-Lie superbialgebra ?”
Define the perturbed cobracket . For and and also recall the adjoint map (1.6) we have :
[TABLE]
This is a natural question because is a 1-cocycle (Remark 1.8), (1.6) is a 1-coboundary when (Remark 3.7), and perturbation of cocycles by coboundaries is a natural concept in homological algebra. Of course, we have more to worry about than just the cocycle condition (1.8) because must be a Hom-Lie supercoalgebra (Definition 1.2).
In the following result, we give some sufficient conditions under which the perturbed cobracket gives another Hom-Lie superbialgebra. This is a generalization of [16], which deals with cobracket perturbation in Lie superbialgebras.
A result about cobracket perturbation in a quasi-triangular Hom-Lie superbialgebra, is given after the following result. We also briefly discuss triangular Hom-Lie superbialgebra, which is the Hom-Type version of Drinfel’d’s triangular Lie bialgebra [8].
Let us recall some notations first. For , the symbol denotes . If is an expression in the elements and , we set
[TABLE]
For example, the compatibility condition (1.8)
is equivalent to
[TABLE]
Moreover , where .
By calculation the Hom-super-Jacobi identity (1.2) is equivalent to .
Note that we have
[TABLE]
It is also noted to simplify writing .
Theorem 4.1**.**
Let be a multiplicative Hom-Lie superbialgebra and be an element such that (1.3) and (3.2), ((3.2) and (1.3), i.e., and
[TABLE]
for all . Then is multiplicative Hom-Lie superbialgebra.
Proof.
To show that is a multiplicative Hom-Lie superbialgebra, we need to prove the following conditions:
It is clear that is a multiplicative Hom-Lie superalgebra.
It remains to show that co-multiplicative Hom-Lie supercoalgebra, and the compatibility condition (1.8) holds for and .
Precisely we need to prove four things:
, equality is true because:
[TABLE]
Using , we have
[TABLE]
is skew-supersymmetric because:
[TABLE]
Then :
Now, we need to show the compatibility condition (1.8) holds for and :
[TABLE]
which is equivalent to
[TABLE]
Since , (4.3) is equivalent to
[TABLE]
Moreover, since , because is a Hom-Lie superbialgebra, (4.3) is equivalent to, which holds by Lemma 3.11.
Finally, we must show the Hom-super-coJacobi identity of , which states
[TABLE]
for all . Using the definition . We can rewrite (4.4) as
[TABLE]
We already know that , which is the Hom-super-coJacobi identity of .
Moreover, in (3.8) and (3.9) (in the proof of Theorem 3.12 with instead of ), we already showed that
[TABLE]
In view of (4.5) and (4.6), the Hom-super-coJacobi identity of (4.4) is equivalent to
[TABLE]
Using the assumption (4.1), the condition (4.7) is equivalent to
[TABLE]
We will prove (4.8) in Lemma 4.2 below.
The proof of Theorem 4.1 will be complete once we prove Lemma 4.2. ∎
Lemma 4.2**.**
The condition (4.8) holds.
Proof.
Write and . Then the left-hand side of (4.8) is:
[TABLE]
Write . Recall from (1.8) that: , because is a Hom-Lie superbialgebra. We can continue the above computation as follows :
[TABLE]
It follows the skew-supersymmetry of applied to (i.e, ), , , and , We find that
That the first two terms and the last two terms above cancel out. Using the commutation of with and and , the above computation continues as follows:
[TABLE]
This proves (4.8). ∎
The following result is a special case of the previous theorem.
Corollary 4.3**.**
Let be a multiplicative Hom-Lie superbialgebra and be an element such that (1.3) and (3.2), ((3.2) and (1.3), i.e., and for all . Then is multiplicative Hom-Lie superbialgebra.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[2] Ammar F., Makhlouf A. and Saadaoui N., Cohomology of Hom-Lie superalgebras and q-deformed Witt superalgebra , Czechoslovak Mathematical Journal, Vol. 63, No. 3, (2013) 721-761.
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- 4[4] Benayadi S. and Makhlouf A., Hom-Lie algebras with symmetric invariant non Degenerate bilinear forms, Journal of Geometry and Physics 76 (2014) 38-60.
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- 6[6] Drinfel’d V.G., Constant quasiclassical solution of the Yang-Baxter quantum equation , Sov. Math. Dokl., 28 (1983), 667-671.
- 7[7] Drinfel’d V.G., Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations , Sov., Dokl. Akad.Nauk SSSR 268 (1983), no. 2, 285-287. English translation: Soviet Math. Dokl. 27 (1983), no.1, 68-71.
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