On Hom-Lie superbialgebras

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 $\mathbb{Z}_2$-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.

Equations

$$\xi =(1\otimes \tau )\cdot (\tau \otimes 1):x\otimes y\otimes z\mapsto (-1{)}^{|x|(|y|+|z|)}y\otimes z\otimes x,$$

$$[x,y]=-(-1{)}^{|x||y|}[y,x],$$

$$(-1{)}^{|x||z|}[\alpha (x),[y,z]]+(-1{)}^{|z||y|}[\alpha (z),[x,y]]+(-1{)}^{|y||x|}[\alpha (y),[z,x]]=0$$

$$\Delta ({\mathfrak{L}}^i)\subset \sum _{i=j+k}{\mathfrak{L}}^j\otimes {\mathfrak{L}}^{k}fori\in {\mathbb{Z}}_2,$$

$$Im\Delta \subset Im(1\otimes 1-\tau )i.e.\Delta isskew-supersymmetric,$$

$$(1\otimes 1\otimes 1+\xi +{\xi }^2)\circ (\alpha \otimes \Delta )\circ \Delta =0:\mathfrak{L}\to \mathfrak{L}\otimes \mathfrak{L}\otimes \mathfrak{L}.$$

$$a{d}_x(y_1\otimes \cdot \cdot \cdot \otimes y_n)=\sum _{i=1}^n(-1{)}^{|x|(|y_1|+|y_2|+\cdot \cdot \cdot +|y_{i-1}|)}\alpha (y_1)\otimes \cdot \cdot \cdot \otimes \alpha (y_{i-1})\otimes [x,y_i]\otimes \alpha (y_{i+1})\cdot \cdot \cdot \otimes \alpha (y_n).$$

$$a{d}_{\alpha (x)}(y_1\otimes \cdot \cdot \cdot \otimes y_n)=\sum _{i=1}^n(-1{)}^{|x|(|y_1|+|y_2|+\cdot \cdot \cdot +|y_{i-1}|)}\alpha (y_1)\otimes \cdot \cdot \cdot \otimes \alpha (y_{i-1})\otimes [\alpha (x),y_i]\otimes \alpha (y_{i+1})\cdot \cdot \cdot \otimes \alpha (y_n).$$

$$\Delta ([x,y])=a{d}_{\alpha (x)}(\Delta (y))-(-1{)}^{|x||y|}a{d}_{\alpha (y)}(\Delta (x)).$$

$$\alpha \circ f=f\circ \alpha ,f\circ [\cdot ,\cdot ]=[\cdot ,\cdot ]\circ f^{\otimes 2}\text{and }\Delta \circ f=f^{\otimes 2}\circ \Delta .$$

$$x\cdot (y_1\otimes y_2)$$

$$=[\alpha (x),y_1]\otimes \alpha (y_2)+(-1{)}^{|x||y_1|}\alpha (y_1)\otimes [\alpha (x),y_2],$$

$$({\delta }_{HL}^1\Delta )(x,y)=\Delta ([x,y])-x\cdot \Delta (y)+(-1{)}^{|x||y|}y\cdot \Delta (x)=\Delta ([x,y])-a{d}_{\alpha (x)}(\Delta (y))+(-1{)}^{|x||y|}a{d}_{\alpha (y)}(\Delta (x)).$$

$$\Delta ([x,y])=a{d}_{{\alpha }^{-1}(x)}(\Delta (y))-(-1{)}^{|x||y|}a{d}_{{\alpha }^{-1}(y)}(\Delta (x)).$$

$$\alpha (e_1)=a_1{e}_1+a_2{e}_2,\alpha (e_2)=a_3{e}_1+a_4{e}_2,\alpha (e_3)=a_5{e}_3,$$

$$[e_3,e_3]=b_1{e}_1+b_2{e}_2,[e_1,e_2]=b_3{e}_1+b_4{e}_2,[e_1,e_3]=b_5{e}_3,[e_2,e_3]=b_6{e}_3,$$

$$\Delta (e_1)=c_1(e_1\otimes e_2-e_2\otimes e_1)+c_5{e}_3\otimes e_3,\Delta (e_2)=c_2(e_1\otimes e_2-e_2\otimes e_1)+c_6{e}_3\otimes e_3,$$

$$\Delta (e_3)=c_3(e_1\otimes e_3-e_3\otimes e_1)+c_4(e_2\otimes e_3-e_3\otimes e_2),$$

$${\mathfrak{L}}_{\beta }=(\mathfrak{L},[\cdot ,\cdot {]}_{\beta }=\beta \circ [\cdot ,\cdot ],{\Delta }_{\beta }=\Delta \circ \beta ,\beta \alpha )$$

$$[x,y{]}_{\beta }=\beta ([x,y])=\beta (-(-1{)}^{|x||y|}[y,x])=-(-1{)}^{|x||y|}\beta ([y,x])=-(-1{)}^{|x||y|}[y,x{]}_{\beta }.$$

$$(-1{)}^{|x||z|}[\beta \alpha (x),[y,z{]}_{\beta }{]}_{\beta }=(-1{)}^{|x||z|}{\beta }^2[\alpha (x),[y,z]]=-(-1{)}^{|z||y|}[\beta \alpha (z),[x,y{]}_{\beta }{]}_{\beta }-(-1{)}^{|y||x|}[\beta \alpha (y),[z,x{]}_{\beta }{]}_{\beta },$$

$$(-1{)}^{|x||z|}[\beta \alpha (x),[y,z{]}_{\beta }{]}_{\beta }+(-1{)}^{|z||y|}[\beta \alpha (z),[x,y{]}_{\beta }{]}_{\beta }+(-1{)}^{|y||x|}[\beta \alpha (y),[z,x{]}_{\beta }{]}_{\beta }=0.$$

$$\tau \circ {\Delta }_{\beta }=\tau \circ \Delta \circ \beta =-\Delta \circ \beta =-{\Delta }_{\beta }.$$

$$(1\otimes 1\otimes 1+\xi +{\xi }^2)\circ (\beta \alpha \otimes {\Delta }_{\beta })\circ {\Delta }_{\beta }=({\beta }^{\otimes 3}{)}^2(1\otimes 1\otimes 1+\xi +{\xi }^2)\circ (\alpha \otimes \Delta )\circ \Delta =0.$$

$${\Delta }_{\beta }([x,y{]}_{\beta })$$

$$=({\beta }^{\otimes 2}{)}^2([\alpha (x),y_1]\otimes \alpha (y_2))+({\beta }^{\otimes 2}{)}^2((-1{)}^{|x||y_1|}\alpha (y_1)\otimes [\alpha (x),y_2])$$

$$-({\beta }^{\otimes 2}{)}^2((-1{)}^{|x||y|}[\alpha (y),x_1]\otimes \alpha (x_2))-({\beta }^{\otimes 2}{)}^2((-1{)}^{|x||y|}(-1{)}^{|y||x_1|}\alpha (x_1)\otimes [\alpha (y),x_2])$$

$$=[\beta \alpha (x),\beta (y_1){]}_{\beta }\otimes \beta \alpha (\beta (y_2))+(-1{)}^{|x||y_1|}\beta \alpha (\beta (y_1))\otimes [\beta \alpha (x),\beta (y_2){]}_{\beta }$$

$$-(-1{)}^{|x||y|}[\beta \alpha (y),\beta (x_1){]}_{\beta }\otimes \beta \alpha (\beta (x_2))-(-1{)}^{|x||y|}(-1{)}^{|y||x_1|}\beta \alpha (\beta (x_1))\otimes [\beta \alpha (y),\beta (x_2){]}_{\beta }$$

$$=[\beta \alpha (x),\beta (y_1){]}_{\beta }\otimes \beta \alpha (\beta (y_2))+(-1{)}^{|x||\beta (y_1)|}\beta \alpha (\beta (y_1))\otimes [\beta \alpha (x),\beta (y_2){]}_{\beta }$$

$$-(-1{)}^{|x||y|}[\beta \alpha (y),\beta (x_1){]}_{\beta }\otimes \beta \alpha (\beta (x_2))-(-1{)}^{|x||y|}(-1{)}^{|y||\beta (x_1)|}\beta \alpha (\beta (x_1))\otimes [\beta \alpha (y),\beta (x_2){]}_{\beta }$$

$$=a{d}_{\beta \alpha (x)}({\Delta }_{\beta }(y))-(-1{)}^{|x||y|}a{d}_{\beta \alpha (y)}({\Delta }_{\beta }(x)).$$

$${\mathfrak{L}}_{\beta }=(\mathfrak{L},[\cdot ,\cdot {]}_{\beta }=\beta \circ [\cdot ,\cdot ],{\Delta }_{\beta }=\Delta \circ \beta ,\beta )$$

$${\mathfrak{L}}_{{\alpha }^n}=(\mathfrak{L},[\cdot ,\cdot {]}_{{\alpha }^n}={\alpha }^n\circ [\cdot ,\cdot ],{\Delta }_{{\alpha }^n}=\Delta \circ {\alpha }^n,{\alpha }^{n+1})$$

$$\beta \gamma [x,y]=\gamma \alpha [x,y]=\gamma ([x,y{]}_{\alpha })=[\gamma (x),\gamma (y){]}_{\beta }=\beta [\gamma (x),\gamma (y)].$$

Full text

**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 $q$-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 $(\mathfrak{g},[\cdot,\cdot],\phi)$ and $(\mathfrak{g}^{\prime},[\cdot,\cdot]^{\prime},\phi^{\prime})$, such that $\mathfrak{g}$ is a $\mathfrak{g}^{\prime}$-module and $\mathfrak{g}^{\prime}$ is a $\mathfrak{g}$-module, also we construct a Hom-Lie superbialgebra structure on the direct sum $\mathfrak{g}\oplus\mathfrak{g}^{*}$ where $\mathfrak{g}^{*}$ is the dual superspace of $\mathfrak{g}$. 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 $r$-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 $\mathfrak{L}=\mathfrak{L}_{\bar{0}}\oplus\mathfrak{L}_{\bar{1}}$ be a $\mathbb{Z}_{2}$-graded vector space over an arbitrary field $\mathbb{K}$ of characteristic 0. In the sequel, we will consider only element which are $\mathbb{Z}_{2}$-homogeneous. For $x\in\mathfrak{L}$, we denote by ${|x|}\in\mathbb{Z}_{2}$ its parity, i.e., $x\in\mathfrak{L}_{|x|}$. We denote by $\tau$ the super-twist map of $\mathfrak{L}\otimes\mathfrak{L}$ namely, $\tau(x\otimes y)=(-1)^{{|x|}{|y|}}y\otimes x$ for $x,y\in\mathfrak{L}.$ The super-cyclic map $\xi$ permutes the coordinates of $\mathfrak{L}\otimes\mathfrak{L}\otimes\mathfrak{L}$, it is defined as

[TABLE]

for $x,y,z\in\mathfrak{L}$, where $\mathbf{1}$ is the identity map on $\mathfrak{L}$. We denote by $\mathfrak{L^{\ast}}=$Hom$(\mathfrak{L},\mathbb{K})$ the linear dual of $\mathfrak{L}$. For $\phi\in\mathfrak{L^{\ast}}$ and $x\in\mathfrak{L}$, we often use the adjoint notation $\langle\phi,x\rangle$ for $\phi(x)\in\mathbb{K}$.

For a linear map $\Delta:\mathfrak{L}\rightarrow\mathfrak{L}\otimes\mathfrak{L}$ (comultiplication), we use Sweedler’s notation $\Delta(x)=\sum_{(x)}x_{1}\otimes x_{2}$ for $x\in\mathfrak{L}$. We will often omit the summation sign $\sum_{(x)}$ to make it simple. The parity $|r|$ of $r\in\mathfrak{L}^{\otimes 2}$ is defined as follows : since we assume $r$ homogenous, there exists $|r|\in\mathbb{Z}_{2}$, such that $r$ can be written as $r=\sum r_{1}\otimes r_{2}\in\mathfrak{L}^{\otimes 2}$, $r_{1},r_{2}$ are homogenous elements with $|r|=|r_{1}|+|r_{2}|$.

Definition 1.1**.**

([1]). A *Hom-Lie superalgebra * is a triple $(\mathfrak{L},[\cdot,\cdot],\alpha)$ consisting of a superspace $\mathfrak{L}$, an even bilinear map $[\cdot,\cdot]:\mathfrak{L}\times\mathfrak{L}\rightarrow\mathfrak{L}$ and an even superspace homomorphism $\alpha:\mathfrak{L}\rightarrow\mathfrak{L}$ satisfying

[TABLE]

[TABLE]

for all homogeneous elements $x,y,z$ in $\mathfrak{L}$.

It is multiplicative if, in addition $\alpha\circ[\cdot,\cdot]=[\cdot,\cdot]\circ\alpha^{\otimes 2}$, (i.e., $\alpha([x,y])=[\alpha(x),\alpha(y)]$, $\forall x,y\in\mathfrak{L}$).

Definition 1.2**.**

([13], [19]). A Hom-Lie supercoalgebra is a triple $(\mathfrak{L},\Delta,\alpha)$ consisting of a superspace $\mathfrak{L}$, an even superspace homomorphism $\alpha:\mathfrak{L}\rightarrow\mathfrak{L}$ and a linear map $\Delta:\mathfrak{L}\rightarrow\mathfrak{L}\otimes\mathfrak{L}$ (the cobracket) such that

[TABLE]

[TABLE]

[TABLE]

If, in addition, $\Delta\circ\alpha=\alpha^{\otimes 2}\circ\Delta$, then $\mathfrak{L}$ is called co-multiplicative.

A Hom-Lie supercoalgebra with $\alpha=Id$ is exactly a Lie supercoalgebra [13].

Remark 1.3*.*

Let $Im\Delta\subset Im(1\otimes 1-\tau)\subset ker(1\otimes 1+\tau)$, then $(1\otimes 1+\tau)\circ\Delta=0$   i.e.  $\Delta$ is skew-supersymmetric ([24]).

Definition 1.4**.**

(1) For an element $x$ in a Hom-Lie superalgebra $(\mathfrak{L},[\cdot,\cdot],\alpha)$ and $n\geq 2$, define the adjoint map $ad_{x}:\mathfrak{L}^{\otimes n}\rightarrow\mathfrak{L}^{\otimes n}$ by

[TABLE]

For $n=2$, $ad_{x}(y_{1}\otimes y_{2})=[x,y_{1}]\otimes\alpha(y_{2})+(-1)^{{|x|}{|y_{1}|}}\alpha(y_{1})\otimes[x,y_{2}]$.

Conversely, given $\gamma=y_{1}\otimes\cdot\cdot\cdot\otimes y_{n}$, we define the map $ad(\gamma):\mathfrak{L}\rightarrow\mathfrak{L}^{\otimes n}$ by $ad(\gamma)(x)=ad_{x}(\gamma)$, for $x\in\mathfrak{L}$.

(2) For an element $x$ in a Hom-Lie superalgebra $(\mathfrak{L},[\cdot,\cdot],\alpha)$ and $n\geq 2$, define the adjoint map

$ad_{\alpha(x)}:\mathfrak{L}^{\otimes n}\rightarrow\mathfrak{L}^{\otimes n}$ by

[TABLE]

Definition 1.5**.**

A *(multiplicative) Hom-Lie superbialgebra * is a quadruple $(\mathfrak{L},[\cdot,\cdot],\Delta,\alpha)$ such that

$(\mathfrak{L},[\cdot,\cdot],\alpha)$ is a (multiplicative) Hom-Lie superalgebra. 2. 2.

$(\mathfrak{L},\Delta,\alpha)$ is a (co-multiplicative) Hom-Lie supercoalgebra. 3. 3.

The following compatibility condition holds for all $x,y\in\mathfrak{L}$ :

[TABLE]

Definition 1.6**.**

The map $f:\mathfrak{L}\rightarrow\mathfrak{L^{\prime}}$ is called even (resp. odd) map if $f(\mathfrak{L}_{i})\subset\mathfrak{L^{\prime}}_{i}$ (resp. $f(\mathfrak{L}_{i})\subset\mathfrak{L^{\prime}}_{i+1}),$ for $i=0,1$. 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 $\alpha=Id$ 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 $\mathfrak{L}^{\otimes 2}$ as an $\mathfrak{L}$-module via the $\alpha$-twisted adjoint action (1.8):

[TABLE]

for $x\in\mathfrak{L}$ and $y_{1}\otimes y_{2}\in\mathfrak{L}^{\otimes 2}$.

Then we can think of the cobracket $\Delta:\mathfrak{L}\rightarrow\mathfrak{L}\otimes\mathfrak{L}$ as a 1-cochain $\Delta\in C^{1}(\mathfrak{L},\mathfrak{L}^{\otimes 2})$. Here $C^{1}(\mathfrak{L},\mathfrak{L}^{\otimes 2})$ is defined as the linear super-subspace of Hom$(\mathfrak{L},\mathfrak{L}^{\otimes 2})$ consisting of maps that commute with $\alpha$. Generalizing [20] to include coefficients in $\mathfrak{L}^{\otimes 2}$, the differential on $\Delta$ is given by

[TABLE]

Therefore, (1.8) says exactly that $\Delta\in C^{1}(\mathfrak{L},\mathfrak{L}^{\otimes 2})$ is a 1-cocycle.

Example 1.9**.**

Classification of 2-dimensional Hom-Lie superbialgebras with 1-dimensional even part*.

Let $\mathfrak{L}=\mathfrak{L}_{\bar{0}}\oplus\mathfrak{L}_{\bar{1}}$ be a 2-dimensional superspace where $\mathfrak{L}_{\bar{0}}$ is generated by $e_{1}$ and $\mathfrak{L}_{\bar{1}}$ is generated by $e_{2}$. The triple $(\mathfrak{L},[\cdot,\cdot],\alpha)$ is a Hom-Lie superalgebra when $[e_{1},e_{1}]=0$, $[e_{1},e_{2}]=be_{2}$ and $[e_{2},e_{2}]=ce_{1}$ with $\alpha(e_{1})=a_{1}e_{1}$, $\alpha(e_{2})=a_{2}e_{2}$ and $a_{2}bc=0$, where $b,c,a_{1},a_{2}$ are parameters in $\mathbb{K}$.

The triple $(\mathfrak{L},\Delta,\alpha)$ is a Hom-Lie supercoalgebra if $\Delta(e_{1})=0$ and $\Delta(e_{2})=d(e_{1}\otimes e_{2}-e_{2}\otimes e_{1})$, where $d\in\mathbb{K}$.

The triple $(\mathfrak{L},[\cdot,\cdot],\Delta,\alpha)$ is a Hom-Lie superbialgebras if $a_{1}=1$ or $a_{1}=-1$ and $a_{2}bd=0$.*

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 $(\mathfrak{L},[\cdot,\cdot],\alpha)$, where $\alpha$ is invertible, and $(\mathfrak{L}^{*},[\cdot,\cdot],(\alpha^{-1})^{*})$ be two Hom-Lie superalgebras. The pair $(\mathfrak{L},\mathfrak{L}^{*})$ 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 $\mathfrak{L}=\mathfrak{L}_{\bar{0}}\oplus\mathfrak{L}_{\bar{1}}$ be a 3-dimensional superspace where $\mathfrak{L}_{\bar{0}}$ is generated by $e_{1}$, $e_{2}$ and $\mathfrak{L}_{\bar{1}}$ is generated by $e_{3}$. We aim to construct Hom-Lie superbialgebras . We set for the linear map $\alpha$

[TABLE]

where $a_{1},a_{2},a_{3},a_{4},a_{5}$ are parameters in $\mathbb{K}.$

The structure of the bracket $[\cdot,\cdot]$ is of the form

[TABLE]

where $b_{1},b_{2},b_{3},b_{4},b_{5},b_{6}$ are parameters in $\mathbb{K}.$

The structure of the cobracket $\Delta$ is of the form

[TABLE]

where $c_{1},c_{2},c_{3},c_{4}$ are parameters in $\mathbb{K}.$

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 $\alpha$, 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 $a_{2}=0$, $a_{3}=0$, 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 $\alpha$ 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 $a_{2}=1$, $a_{3}=0$, $a_{4}=a_{1}$.

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 $(\mathfrak{L},[\cdot,\cdot],\Delta,\alpha)$ be a Hom-Lie superbialgebra and an even map $\beta:\mathfrak{L}\rightarrow\mathfrak{L}$ be a Hom-Lie superbialgebra morphism. Then

[TABLE]

is also a Hom-Lie superbialgebra, which is multiplicative if $\mathfrak{L}$ is.

Proof.

It is immediate that $[\cdot,\cdot]_{\beta}$ is skew-supersymmetric (1.1) because

[TABLE]

The Hom super-Jacobi identity holds in $\mathfrak{L}_{\beta}$ because

[TABLE]

i.e.,

[TABLE]

$\Delta_{\beta}$ is skew-supersymmetric because

[TABLE]

Likewise, the Hom-super-co-jacobi identity holds in $\mathfrak{L}_{\beta}$ because

[TABLE]

To check the compatibility condition (1.8) in $\mathfrak{L}_{\beta}$, we compute as follows :

[TABLE]

Because $|\beta(x_{1})|=|x_{1}|$, and $|\beta(y_{1})|=|y_{1}|$ (i.e., an even map $\beta$). We have shown that $\mathfrak{L}_{\beta}$ 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 $(\mathfrak{L},[\cdot,\cdot],\Delta)$ be a Lie superbialgebra and an even map $\beta:\mathfrak{L}\rightarrow\mathfrak{L}$ be a Lie superbialgebra morphism. Then

[TABLE]

is a multiplicative Hom-Lie superbialgebra.

Proof.

This is the $\alpha=Id$ 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 $(\mathfrak{L},[\cdot,\cdot],\Delta,\alpha)$ be a multiplicative Hom-Lie superbialgebra. Then

[TABLE]

is also a multiplicative Hom-Lie superbialgebra for each integer $n\geq 0$.

Proof.

This is the $\beta=\alpha^{n}$ special case of Theorem 1.11. ∎

Next we consider when Hom-Lie superbialgebra of the from $\mathfrak{L}_{\beta}$, as in Corollary 1.12, are isomorphic.

Theorem 1.14**.**

*Let $\mathfrak{g}$ and $\mathfrak{h}$ be Lie superbialgebras. Let $\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ and $\beta:\mathfrak{h}\rightarrow\mathfrak{h}$ be Lie superbialgebras morphisms with $\beta$ and $\beta^{\otimes{2}}$ injective. Then the following statements are equivalent :

1. The Hom-Lie superbialgebras $\mathfrak{g}_{\alpha}$ and $\mathfrak{h}_{\beta}$ as in Corollary 1.12, are isomorphic.

2. There exists a Lie superbialgebra isomorphism $\gamma:\mathfrak{g}\rightarrow\mathfrak{h}$ such that $\gamma\alpha=\beta\gamma.$*

Proof.

To show that the first statement implies the second statement, suppose that $\gamma:\mathfrak{g}_{\alpha}\rightarrow\mathfrak{h}_{\beta}$ is an isomorphism of Hom-Lie superbialgebras. Then $\gamma\alpha=\beta\gamma$ automatically.

To see that $\gamma$ is a Lie superbialgebra isomorphism, first we check that it commutes with the Lie bracket. For any two elements $x$ and $y$ in $\mathfrak{g}$, we have

[TABLE]

Since $\beta$ is injective, we conclude that $\gamma[x,y]=[\gamma(x),\gamma(y)],$ i.e., $\gamma$ is a Lie superbialgebra isomorphism.

To check that $\gamma$ commutes with the Lie cobrackets, we compute as follows:

[TABLE]

The injectivity of $\beta^{\otimes{2}}$ now implies that $\gamma$ commutes with the Lie cobrackets. Therefore, $\gamma$ 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 $\mathfrak{g}$, let Aut$(\mathfrak{g})$ be the group of Lie superbialgebra isomorphisms from $\mathfrak{g}$ to $\mathfrak{g}$. In Theorem 1.14, restricting to the case $\mathfrak{g}=\mathfrak{h}$ with $\alpha$ and $\beta$ both invertible, we obtain the following special case.

Corollary 1.15**.**

Let $\mathfrak{g}$ be a Lie superbialgebra and $\alpha,\beta\in$ Aut$(\mathfrak{g})$. Then the Hom-Lie superbialgebras $\mathfrak{g}_{\alpha}$ and $\mathfrak{g}_{\beta}$, as in Corollary 1.12, are isomorphic if and only if $\alpha$ and $\beta$ are conjugate in Aut$(\mathfrak{g})$.

Corollary 1.15 can be restated as follows.

Corollary 1.16**.**

*Let $\mathfrak{g}$ be a Lie superbialgebra.Then there is a bijection between the following two sets:

1. The set of isomorphism classes of Hom-Lie superbialgebras $\mathfrak{g}_{\alpha}$ with $\alpha$ invertible.

2. The set of conjugacy classes in the group Aut$(\mathfrak{g})$.*

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*.*

$\bullet$ If $(\mathfrak{L},\Delta,\alpha)$ is a Hom-Lie supercoalgebra, then $(\mathfrak{L}^{\ast},[\cdot,\cdot],\alpha)$ is a Hom-Lie superalgebra. Here $[\cdot,\cdot]$ and $\alpha$ in $\mathfrak{L}^{\ast}$ are dual to $\Delta$ and $\alpha$, respectively, in $\mathfrak{L}$.

$\bullet$ Conversely, if $(\mathfrak{L},[\cdot,\cdot],\alpha)$ is a finite dimensional Hom-Lie superalgebra, then $(\mathfrak{L}^{\ast},\Delta,\alpha)$ is a Hom-Lie supercoalgebra, where $\Delta$ and $\alpha$ in $\mathfrak{L}^{\ast}$ are dual to $[\cdot,\cdot]$ and $\alpha$, respectively, in $\mathfrak{L}$.

Theorem 1.18**.**

Let $(\mathfrak{L},[\cdot,\cdot],\Delta,\alpha)$ be a finite dimensional (multiplicative) Hom-Lie superbialgebra. Then its linear dual $\mathfrak{L^{\ast}}=$Hom$(\mathfrak{L},\mathbb{K})$ is also a (multiplicative) Hom-Lie superbialgebra with the dual structure maps:

[TABLE]

for $x,y\in\mathfrak{L}$ and $\phi,\psi\in\mathfrak{L^{\ast}}$.

Proof.

As we mentioned right after Remark 1.17, $(\mathfrak{L}^{\ast},[\cdot,\cdot],\alpha)$ is a Hom-Lie superalgebra, which is true even if $\mathfrak{L}$ is not finite dimensional. Moreover, $(\mathfrak{L}^{\ast},\Delta,\alpha)$ is a Hom-Lie supercoalgebra, whose validity depends on the finite dimensionality of $\mathfrak{L}$. Thus, it remains to check the compatibility condition (1.8) between the bracket and the cobracket in $\mathfrak{L}^{\ast}$, i.e.,

[TABLE]

for $x,y\in\mathfrak{L}$ and $\phi,\psi\in\mathfrak{L^{\ast}}$.

Using Definition 1.12, the compatibility condition (1.8) in $\mathfrak{L}$, 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 $\mathfrak{L}^{\ast}$,

$([\phi_{1},\alpha(\psi)]=-(-1)^{{|\phi_{1}|}{|\psi|}}[\alpha(\psi),\phi_{1}]$ and $\phi_{1}\otimes\phi_{2}=-(-1)^{{|\phi_{1}|}{|\phi_{2}|}}\phi_{2}\otimes\phi_{1}$).

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 $(\mathfrak{L},[\cdot,\cdot],\alpha)$ be a Hom-Lie superalgebra and $M=M_{\bar{0}}\oplus M_{\bar{1}}$ an arbitrary vector superspace. A representation of the Hom-Lie superalgebra with respect to $A\in\mathfrak{gl}(M)$, is an even linear map $\rho:\mathfrak{L}\rightarrow End(M)$, such that $\rho(\mathfrak{L_{i}})(M_{j})\subset M_{i+j}$ where $i,j\in\mathbb{Z}_{2}$, and satisfying

[TABLE]

and

[TABLE]

for all homogeneous elements $x,y\in\mathfrak{L}$.

We denote a representation by $(M,\rho,A)$. It is straightforward to see that $(\mathfrak{L},ad,\alpha)$ is a representation, called the adjoint representation, see [4, 22].

Given a representation $(M,\rho,A)$, define $\rho^{\ast}:\mathfrak{L}\rightarrow End(M^{\ast})$ by

[TABLE]

$\forall$ $x\in\mathfrak{L}$, $\xi\in M^{\ast}$, $\nu\in M.$ This representation is called admissible representation with respect to $(M,\rho,A)$.

Now, let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\alpha_{\mathfrak{g}})$ and $(\mathfrak{g^{\prime}},[\cdot,\cdot]_{\mathfrak{g^{\prime}}},\alpha_{\mathfrak{g^{\prime}}})$ be two multiplicative Hom-Lie superalgebras. Set $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ and $\mathfrak{g^{\prime}}=\mathfrak{g^{\prime}}_{\bar{0}}\oplus\mathfrak{g^{\prime}}_{\bar{1}}$. Let $\rho:\mathfrak{g}\rightarrow\mathfrak{gl}(\mathfrak{g^{\prime}})$ and $\rho^{\prime}:\mathfrak{g^{\prime}}\rightarrow\mathfrak{gl}(\mathfrak{g})$ be two linear maps. Define a skew-supersymmetric bracket $[\cdot,\cdot]_{\widetilde{G}}:\widetilde{G}\times\widetilde{G}\rightarrow\widetilde{G}$, where $\widetilde{G}$ is given by : $\widetilde{G}=\widetilde{G}_{\bar{0}}\oplus\widetilde{G}_{\bar{1}}$, where $\widetilde{G}_{\bar{0}}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g^{\prime}}_{\bar{0}}$ and $\widetilde{G}_{\bar{1}}=\mathfrak{g}_{\bar{1}}\oplus\mathfrak{g^{\prime}}_{\bar{1}}$. We set $|(x,x^{\prime})|=|x|=|x^{\prime}|$ and $|(y,y^{\prime})|=|y|=|y^{\prime}|$, for all homogeneous elements $x,y$ in $\mathfrak{g}$ and $x^{\prime},y^{\prime}$ in $\mathfrak{g^{\prime}}$. Define $\widetilde{\alpha}:\widetilde{G}\rightarrow\widetilde{G}$ by

[TABLE]

and the bracket $\widetilde{G}$ by

[TABLE]

Theorem 2.2**.**

The triple $(\widetilde{G}=\mathfrak{g}\oplus\mathfrak{g^{\prime}},[\cdot,\cdot]_{\widetilde{G}},\widetilde{\alpha})$, where $\widetilde{G}$, $[\cdot,\cdot]_{\widetilde{G}}$, $\widetilde{\alpha}$ are defined above is a multiplicative Hom-Lie superalgebra if and only if $\rho$ and $\rho^{\prime}$ are representations of $\mathfrak{g}$ and $\mathfrak{g^{\prime}}$ respectively and the following conditions are satisfied

[TABLE]

[TABLE]

Proof.

Assume $(\widetilde{G}=\mathfrak{g}\oplus\mathfrak{g^{\prime}},[\cdot,\cdot]_{\widetilde{G}},\widetilde{\alpha})$ 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 $\alpha_{\mathfrak{g}}$ and $\alpha_{\mathfrak{g^{\prime}}}$ are multiplicative, which implies that

[TABLE]

[TABLE]

Developing the Hom-super-Jacobi identity (1.2), that for $x,y,z\in\mathfrak{g}$ and $x^{\prime},y^{\prime},z^{\prime}\in\mathfrak{g^{\prime}}$,

[TABLE]

[TABLE]

leads to the second condition (2.2). Indeed,

[TABLE]

Similarly, we compute :

[TABLE]

Setting ($|x|=|x^{\prime}|$, $|y|=|y^{\prime}|$, and $|z|=|z^{\prime}|$ ) give

[TABLE]

[TABLE]

which implies (2.5), and (2.6).

By Eqs. (2.8) and (2.10), we deduce that $\rho$ is a representation of the Hom-Lie superalgebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\alpha_{\mathfrak{g}})$ on $\mathfrak{g^{\prime}}$ with respect to $\alpha_{\mathfrak{g^{\prime}}}$. By Eqs. (2.9) and (2.11), we deduce that $\rho^{\prime}$ is a representation of the Hom-Lie superalgebra $(\mathfrak{g^{\prime}},[\cdot,\cdot]_{\mathfrak{g^{\prime}}},\alpha_{\mathfrak{g^{\prime}}})$ on $\mathfrak{g}$ with respect to $\alpha_{\mathfrak{g}}$. ∎

Example 2.3** (See [2]).**

*Given a representation $(V,[.,.]_{V},\beta)$ of a Hom-Lie superalgebra $(G,[\cdot,\cdot],\alpha)$. Set $\widetilde{G}=G\oplus V$ and $\widetilde{G_{k}}=G_{k}\oplus V_{k}$. If $x\in G_{i}$ and $v\in V_{i}$ $(i\in\mathbb{Z}_{2})$, we denote $|(x,v)|=|x|$.

Define a skew-supersymmetric bracket $[\cdot,\cdot]_{\widetilde{G}}:\wedge^{2}(G\oplus V)\rightarrow G\oplus V$ by

[TABLE]

Define $\widetilde{\alpha}:G\oplus V\rightarrow G\oplus V$ by $\widetilde{\alpha}(x,v)=(\alpha(x),\beta(v))$. Then $(G\oplus V,[-,-]_{\widetilde{G}},\widetilde{\alpha})$ is a Hom-Lie superalgebra, which we call the semi-direct product of the Hom-Lie superalgebra $(G,[\cdot,\cdot],\alpha)$ by $V$.

Definition 2.4**.**

For any $x\in\mathfrak{g}$, define $ad_{x}:\wedge^{n}\mathfrak{g}\rightarrow\wedge^{n}\mathfrak{g}$ by $ad_{x}P=[x,P]_{\mathfrak{g}}$. Its dual map $ad^{\ast}:\wedge^{n}\mathfrak{g}^{\ast}\rightarrow\wedge^{n}\mathfrak{g}^{\ast}$ is defined as

[TABLE]

for all $x,y\in\mathfrak{g}$, and $\xi\in\mathfrak{g}^{\ast}$, where the pairing $\langle\ ,\ \rangle$ is supersymmetric, i.e. $\langle x,\xi\rangle=(-1)^{{|x|}{|\xi|}}\langle\xi,x\rangle,$ for $x\in\mathfrak{g}$,

and $\xi\in\mathfrak{g}^{\ast}$. More precisely, for any $\xi_{1},\cdot\cdot\cdot,\xi_{n}\in\mathfrak{g}^{\ast}$, we have

[TABLE]

Definition 2.5**.**

A matched pairs of Hom-Lie superalgebras, which we denote by $(\mathfrak{g},\mathfrak{g^{\prime}},\rho,\rho^{\prime})$, consists of two Hom-Lie superalgebras $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\alpha_{\mathfrak{g}})$ and $(\mathfrak{g^{\prime}},[\cdot,\cdot]_{\mathfrak{g^{\prime}}},\alpha_{\mathfrak{g^{\prime}}})$, together with representations $\rho:\mathfrak{g}\rightarrow\mathfrak{gl}(\mathfrak{g^{\prime}})$ and $\rho^{\prime}:\mathfrak{g^{\prime}}\rightarrow\mathfrak{gl}(\mathfrak{g})$ with respect to $\alpha_{\mathfrak{g^{\prime}}}$ and $\alpha_{\mathfrak{g}}$ respectively, such that the compatibility conditions (2.5) and (2.6) are satisfied.

In the following, we concentrate on the case that $\mathfrak{g^{\prime}}$ is $\mathfrak{g}^{\ast}$, the dual space of $\mathfrak{g}$, and $\alpha_{\mathfrak{g^{\prime}}}=\alpha^{\ast}_{\mathfrak{g}}$, $\rho=ad^{\ast}$, $\rho^{\prime}=\mathfrak{ad}^{\ast}$, where $\mathfrak{ad}^{\ast}$ is the dual map of $\mathfrak{ad}$. Notice that $ad$ and $\mathfrak{ad}$ are the adjoint representations associated to Hom-Lie superalgebras $\mathfrak{g}$ and $\mathfrak{g}^{\ast}$ respectively. Let $x,y,z$ be elements in $\mathfrak{g}$ and $\xi,\eta$ elements in $\mathfrak{g}^{\ast}$.

For a Hom-Lie superalgebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\alpha_{\mathfrak{g}})$ (resp. $(\mathfrak{g}^{\ast},[\cdot,\cdot]_{\mathfrak{g}^{\ast}},\alpha_{\mathfrak{g}^{\ast}})$), let $\Delta^{\ast}:\mathfrak{g}^{\ast}\rightarrow\wedge^{2}\mathfrak{g}^{\ast}$ (resp. $\Delta:\mathfrak{g}\rightarrow\wedge^{2}\mathfrak{g}$) be the dual map of $[\cdot,\cdot]_{\mathfrak{g}}:\wedge^{2}\mathfrak{g}\rightarrow\mathfrak{g}$ (resp. $[\cdot,\cdot]_{\mathfrak{g}^{\ast}}:\wedge^{2}\mathfrak{g}^{\ast}\rightarrow\mathfrak{g}^{\ast}$), i.e.

[TABLE]

A Hom-Lie superalgebra $(\mathfrak{L},[\cdot,\cdot],\alpha)$ is called admissible if its adjoint representation is admissible, that is $[(Id-\alpha^{2})(x),\alpha(y)]=0$. In particular, if a Hom-Lie superalgebra $(\mathfrak{L},[\cdot,\cdot],\alpha)$ is * involutive*, that is $\alpha^{2}=Id$, then it is admissible.

Proposition 2.6**.**

A pair of admissible Hom-Lie superalgebras $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\alpha_{\mathfrak{g}})$ and $(\mathfrak{g}^{\ast},[\cdot,\cdot]_{\mathfrak{g}^{\ast}},\alpha_{\mathfrak{g}}^{\ast})$ determines a Hom-Lie superbialgebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\Delta,\alpha_{\mathfrak{g}})$, where $\Delta$ is the dual operation of $[\cdot,\cdot]_{\mathfrak{g}^{\ast}}$ if

[TABLE]

[TABLE]

Remark 2.7*.*

Following Bai and Sheng, we may denote this Hom-Lie superbialgebra by $(\mathfrak{g},\mathfrak{g}^{\ast})$.

Theorem 2.8**.**

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\alpha_{\mathfrak{g}})$ and $(\mathfrak{g}^{\ast},[\cdot,\cdot]_{\mathfrak{g}^{\ast}},\alpha_{\mathfrak{g}}^{\ast})$ be a pair of admissible Hom-Lie superalgebra. Then $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\Delta,\alpha_{\mathfrak{g}})$, where $\Delta$ is the dual of $[\cdot,\cdot]_{\mathfrak{g}^{\ast}}$, is a Hom-Lie superbialgebra if and only if $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\alpha_{\mathfrak{g}})$ and $(\mathfrak{g}^{\ast},[\cdot,\cdot]_{\mathfrak{g}^{\ast}},\alpha_{\mathfrak{g}}^{\ast})$ is a matched pairs of Hom-Lie superalgebras, i.e. $(\mathfrak{g}\oplus\mathfrak{g}^{\ast},[\cdot,\cdot]_{\widetilde{G}},\alpha_{\mathfrak{g}}\oplus\alpha_{\mathfrak{g}}^{\ast})$ is a multiplicative Hom-Lie superalgebra, where $[\cdot,\cdot]_{\widetilde{G}}$ is given by Eq. (2.4), in which $\rho=ad^{\ast}$ and $\rho^{\prime}=\mathfrak{ad}^{\ast}.$

Proof.

By Theorem 2.2, two admissible Hom-Lie superalgebras $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\alpha_{\mathfrak{g}})$ and $(\mathfrak{g}^{\ast},[\cdot,\cdot]_{\mathfrak{g}^{\ast}},\alpha_{\mathfrak{g}}^{\ast})$ form a matched pair of Hom-Lie superalgebras if and only if

[TABLE]

$+(-1)^{{{|\xi|}{|\eta|}}+{{|\eta|}{|z|}}}ad^{\ast}_{\mathfrak{ad}^{\ast}_{\eta}(z)}(\alpha_{\mathfrak{g}}^{\ast}(\xi))-(-1)^{{|\xi|}{|z|}}ad^{\ast}_{\mathfrak{ad}^{\ast}_{\xi}(z)}(\alpha_{\mathfrak{g}}^{\ast}(\eta)).$

[TABLE]

$+(-1)^{{{|x|}{|y|}}+{{|y|}{|\xi|}}}\mathfrak{ad}^{\ast}_{ad^{\ast}_{y}(\xi)}(\alpha_{\mathfrak{g}}(x))-(-1)^{{|x|}{|\xi|}}\mathfrak{ad}^{\ast}_{ad^{\ast}_{x}(\xi)}(\alpha_{\mathfrak{g}}(y)).$

Since $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\alpha_{\mathfrak{g}})$ and $(\mathfrak{g}^{\ast},[\cdot,\cdot]_{\mathfrak{g}^{\ast}},\alpha_{\mathfrak{g}}^{\ast})$ are admissible Hom-Lie superalgebras i.e. ($\alpha_{\mathfrak{g}}^{2}=Id$, and $(\alpha_{\mathfrak{g}}^{\ast})^{2}=Id$), 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 $V$ be a superspace and $\langle\ ,\ \rangle:V^{\ast}\times V\rightarrow\mathbb{K}$ be the canonical pairing. Then we identify $V$ with $V^{\ast}$ by the pairing $\langle x,\xi\rangle=(-1)^{{|x|}{|\xi|}}\langle\xi,x\rangle$, $x\in V$ and $\xi\in V^{\ast}$. On the other hand, we shall say that a bilinear form $(|):V\times V\rightarrow\mathbb{K}$ is supersymmetric if $(\upsilon|\omega)=(-1)^{{|\upsilon|}{|\omega|}}(\omega|\upsilon)$.

Definition 2.9**.**

A Manin supertriple of Hom-Lie superalgebras is a triple of Hom-Lie superalgebras $(\mathfrak{M},\mathfrak{g},\mathfrak{g^{\prime}})$ together with a nondegenerate supersymmetric bilinear form $S$ on $\mathfrak{M}$ such that

$S$ is invariant, i.e. for any $x,y,z\in\mathfrak{M}$, we have

[TABLE]

[TABLE] 2. 2.

$\mathfrak{g}$ and $\mathfrak{g^{\prime}}$ are isotropic Hom-Lie sub-superalgebra of $\mathfrak{M}$, such that $\mathfrak{M}=\mathfrak{g}\oplus\mathfrak{g^{\prime}}$ as vector superspace.

Proposition 2.10**.**

Let $(\mathfrak{g},\mathfrak{g}^{\ast})$ be a Hom-Lie superbialgebra in the sense of Proposition 2.6. Then $(\mathfrak{g}\oplus\mathfrak{g}^{\ast},\mathfrak{g},\mathfrak{g}^{\ast})$ is a Manin supertriple of Hom-Lie superalgebras.

Proof.

Let $(\mathfrak{g},\mathfrak{g}^{\ast})$ be a Hom-Lie superbialgebra in the sense of Proposition 2.6, i.e. $\mathfrak{g}$ and $\mathfrak{g}^{\ast}$ are admissible Hom-Lie superalgebras such that Eqs. (2.14), (2.15) are satisfied. By Theorem 2.8, we know that $(\mathfrak{g}\oplus\mathfrak{g}^{\ast},[\cdot,\cdot]_{\mathfrak{g}\oplus\mathfrak{g}^{\ast}},\alpha_{\mathfrak{g}}\oplus\alpha_{\mathfrak{g}}^{\ast})$ is a Hom-Lie superalgebra, where $[-,-]_{\mathfrak{g}\oplus\mathfrak{g}^{\ast}}$ is given by

[TABLE]

for all homogeneous elements $x,y$ and $\xi,\eta$ in $\mathfrak{g}$ and $\mathfrak{g}^{\ast}$ respectively. From the construction above we have $|x|=|\xi|$ and $|y|=|\eta|$.

Furthermore, there is an obvious supersymmetric bilinear form on $\mathfrak{g}\oplus\mathfrak{g}^{\ast}$:

[TABLE]

It’s straightforward, using the supersymmetry and $\langle ad_{x}(y),\xi\rangle=-(-1)^{{|x|}{|y|}}\langle y,ad^{\ast}_{x}(\xi)\rangle$, with $|x|=|\xi|$ and $|y|=|\eta|$, that Eqs. (2.18) and (2.19) are satisfied, i.e. the bilinear form defined by Eq. (2.21) is invariant. ∎

Conversely, if $(\mathfrak{g}\oplus{\mathfrak{g}}^{\ast},\mathfrak{g},{\mathfrak{g}}^{\ast})$ is a Manin supertriple of Hom-Lie superalgebras with the invariant bilinear from $S$ given by Eq. (2.21), then for any $x,y\in\mathfrak{g}$ and $\xi,\eta\in{\mathfrak{g}}^{\ast}$, we have the natural scalar product on $\mathfrak{g}\oplus{\mathfrak{g}}^{\ast}$ defined by

[TABLE]

Due to the invariance of $S$, we have

[TABLE]

$S([x,\xi]_{\mathfrak{g}\oplus{\mathfrak{g}}^{\ast}},\eta)=S(x,[\xi,\eta]_{\mathfrak{g}\oplus{\mathfrak{g}}^{\ast}})=S(x,[\xi,\eta]_{{\mathfrak{g}}^{\ast}})=\langle x,\mathfrak{ad}_{\xi}(\eta)\rangle=-(-1)^{{|x|}{|\xi|}}\langle\mathfrak{ad}^{\ast}_{\xi}(x),\eta\rangle,$

which implies that :

[TABLE]

that is, the Hom-Lie bracket on $\mathfrak{g}\oplus{\mathfrak{g}}^{\ast}$ is given by Eq. (2.20). Therefore, $(\mathfrak{g},{\mathfrak{g}}^{\ast},ad^{\ast},\mathfrak{ad}^{\ast})$ is a matched pair of Hom-Lie superalgebras and hence $(\mathfrak{g},{\mathfrak{g}}^{\ast})$ is a Hom-Lie superbialgebra. Note that we deduce naturally that both $\mathfrak{g}$ and $\mathfrak{g}^{\ast}$ 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 $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\alpha_{\mathfrak{g}})$ and $(\mathfrak{g}^{\ast},[\cdot,\cdot]_{\mathfrak{g}^{\ast}},\alpha_{\mathfrak{g}}^{\ast})$ be two admissible Hom-Lie superalgebras. Then the following conditions are equivalent.

$(\mathfrak{g},{\mathfrak{g}}^{\ast})$* is a Hom-Lie superbialgebra in the sense of Proposition 2.6.* 2. 2.

$(\mathfrak{g},{\mathfrak{g}}^{\ast},ad^{\ast},\mathfrak{ad}^{\ast})$* is a matched pair of Hom-Lie superalgebras.* 3. 3.

$(\mathfrak{g}\oplus{\mathfrak{g}}^{\ast},\mathfrak{g},{\mathfrak{g}}^{\ast})$* 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 $r$-matrix.

Definition 3.1**.**

A (multiplicative) coboundary Hom-Lie superbialgebra $(\mathfrak{L},[\cdot,\cdot],\Delta,\alpha,r)$ consists of a (multiplicative) Hom-Lie superbialgebra $(\mathfrak{L},[\cdot,\cdot],\Delta,\alpha)$ and an element $r=\sum r_{1}\otimes r_{2}\in\mathfrak{L}^{\otimes 2}$ such that $\alpha^{\otimes 2}(r)=r$ and

[TABLE]

for all $x\in\mathfrak{L}$.

Remark 3.2*.*

[TABLE]

for $x\in\mathfrak{L}$, where the parity $|r|$ of $r$ is defined as follows : since we assume $r$ is homogenous, there exists $|r|\in\mathbb{Z}_{2}$, such that $r$ can be written as $r=\sum r_{1}\otimes r_{2}\in\mathfrak{L}^{\otimes 2}$, $r_{1},r_{2}$ are homogenous elements with $|r|=|r_{1}|+|r_{2}|$. (Note that equation (3.2) and (1.3) show that we have $|r|=\bar{0}$, namely $|r_{1}|=|r_{2}|$). So we get (3.1).

Definition 3.3**.**

The classical Yang-Baxter equation (CYBE):

[TABLE]

where $r_{ij}$ are defined by

[TABLE]

and considered as elements in ${U}(\mathfrak{L})$, the universal enveloping algebra of a Lie superalgebra $\mathfrak{L}$. Elements (3.3) belongs to $\mathfrak{L}\otimes\mathfrak{L}\otimes\mathfrak{L}$.

Definition 3.4**.**

The classical Hom-Yang-Baxter equation (CHYBE) in a Hom-Lie superalgebra $(\mathfrak{L},[\cdot,\cdot],\alpha)$ is

[TABLE]

for $r\in\mathfrak{L}\otimes\mathfrak{L}$. The three brackets in (3.4) are defined as

[TABLE]

where $r=\sum r_{1}\otimes r_{2}$ and $r^{\prime}=\sum r^{\prime}_{1}\otimes r^{\prime}_{2}$ $\in\mathfrak{L}\otimes\mathfrak{L}$. If $\alpha=Id$, 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 $r$ is a solution of the CHYBE (3.4). In these cases, we also write $\Delta$ as $ad(r)$.

Remark 3.6*.*

Note that we do not require $r$ to be skew-supersymmetric in a coboundary Hom-Lie superbialgebra, whereas in [8] $r$ 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 $\mathfrak{L}$ says that the cobracket $\Delta$ is a 1-cocycle in $C^{1}(\mathfrak{L},\mathfrak{L}^{\otimes 2})$, where $\mathfrak{L}$ acts on $\mathfrak{L}^{\otimes 2}$ via the $\alpha$-twisted adjoint action (1.9). The simplest 1-cocycles are the 1-coboundaries, i.e, images of $\delta^{0}_{HL}$. We can define the Hom-Lie 0-cochains and 0 th differential as follows, extending the definition in [20]. Set $C^{0}(\mathfrak{L},\mathfrak{L}^{\otimes 2})$ as the subspace of $\mathfrak{L}^{\otimes 2}$ consisting of elements that are fixed by $\alpha^{\otimes 2}$. Then we define the differential

[TABLE]

by setting $\delta^{0}_{HL}(r)=ad(r),$ as in (1.6). It is not hard to check that, for $r\in C^{0}(\mathfrak{L},\mathfrak{L}^{\otimes 2})$, we have $\delta^{1}_{HL}(\delta^{0}_{HL}(r))=0,$ where $\delta^{1}_{HL}$ is defined in (1.10). In fact, what this condition says is that

[TABLE]

for all $x,y\in\mathfrak{L}$. We will prove (3.6) in Lemma 3.11 below. Thus, such an element $\delta^{0}_{HL}(r)=ad(r)$ is a 1-coboundary, and hence a 1-cocycle. This fact makes $ad(r)$ (with $\alpha^{\otimes 2}(r)=r$) 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 $(\mathfrak{L},[\cdot,\cdot],\Delta=ad(r),\alpha,r)$ be a coboundary Hom-Lie superbialgebra and an even map $\beta:\mathfrak{L}\rightarrow\mathfrak{L}$ be a morphism such that $\beta^{\otimes 2}(r)=r$. Then $\mathfrak{L}_{\beta}=(\mathfrak{L},[\cdot,\cdot]_{\beta}=\beta\circ[\cdot,\cdot],\Delta_{\beta}=\Delta\circ\beta,\beta\alpha,r)$ is also a coboundary Hom-Lie superbialgebra, which is multiplicative if $\mathfrak{L}$ is. Moreover, if $\mathfrak{L}$ is quasi-triangular, then so is $\mathfrak{L}_{\beta}$.

Proof.

By Theorem 1.11 we know that $\mathfrak{L}_{\beta}$ is a Hom-Lie superbialgebra, multiplicative if $\mathfrak{L}$ is. To check that $\mathfrak{L}_{\beta}$ is coboundary, first note that $(\beta\alpha)^{\otimes 2}(r)=\beta^{\otimes 2}\alpha^{\otimes 2}(r)=r.$

To check the condition (3.1) in $\mathfrak{L}_{\beta}$, we compute as follows:

[TABLE]

The last expression above is $ad_{x}(r)$ in $\mathfrak{L}_{\beta}$, which shows that $\mathfrak{L}_{\beta}$ is coboundary.

Finally, suppose in addition that $\mathfrak{L}$ is quasi-triangular, i.e., $r$ is a solution of the CHYBE in $\mathfrak{L}$. Using the notation in (3.4) we have:

[TABLE]

where the last expression is defined in $\mathfrak{L}_{\beta}$. This shows that $r$ is a solution of the CHYBE in $\mathfrak{L}_{\beta}$, so $\mathfrak{L}_{\beta}$ 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 $(\mathfrak{L},[\cdot,\cdot],\Delta,r)$ be a coboundary Lie superbialgebra and an even map $\beta:\mathfrak{L}\rightarrow\mathfrak{L}$ be a Lie superalgebra morphism such that $\beta^{\otimes 2}(r)=r$. Then $\mathfrak{L}_{\beta}=(\mathfrak{L},[\cdot,\cdot]_{\beta}=\beta\circ[\cdot,\cdot],\Delta_{\beta}=\Delta\circ\beta,\beta,r)$ is a multiplicative coboundary Hom-Lie superbialgebra. If, in addition, $\mathfrak{L}$ is a quasi-triangular Lie superbialgebra, then $\mathfrak{L}_{\beta}$ is a multiplicative quasi-triangular Hom-Lie superbialgebra.

Proof.

This is the $\alpha=Id$ special case of Theorem 3.8, provided that we can show that

$\Delta\circ\beta=\beta^{\otimes 2}\circ\Delta$. 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 $(\mathfrak{L},[\cdot,\cdot],\Delta,\alpha,r)$ be a multiplicative coboundary (resp. quasi-triangular) Hom-Lie superbialgebra. Then $\mathfrak{L}_{\alpha^{n}}=(\mathfrak{L},[\cdot,\cdot]_{\alpha^{n}}=\alpha^{n}\circ[\cdot,\cdot],\Delta_{\alpha^{n}}=\Delta\circ\alpha^{n},\alpha^{n+1},r)$ is also a multiplicative coboundary (resp. quasi-triangular) Hom-Lie superbialgebra for each integer $n\geq 0$.

Proof.

This is the $\beta=\alpha^{n}$ 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 $r=\sum r_{1}\otimes r_{2}$, we write $r_{21}$ for $\tau(r)=\sum(-1)^{{|r_{1}|}{|r_{2}|}}r_{2}\otimes r_{1}.$

Lemma 3.11**.**

Let $(\mathfrak{L},[\cdot,\cdot],\alpha)$ be a multiplicative Hom-Lie superalgebra and $r\in\mathfrak{L}^{\otimes 2}$ be an element such that $\alpha^{\otimes 2}(r)=r$, (1.3) and (3.2), ((3.2) and (1.3),  i.e., $|r|=\bar{0})$. Then $\Delta=ad(r):\mathfrak{L}\rightarrow\mathfrak{L}^{\otimes 2}$ satisfies (1.8), i.e., $ad_{[x,y]}(r)=ad_{\alpha(x)}(ad_{y}(r))-(-1)^{{|x|}{|y|}}ad_{\alpha(y)}(ad_{x}(r))$ for $x,y\in\mathfrak{L}$.

Proof.

We will use $\alpha^{\otimes 2}(r)=r$, the skew-supersymmetry (1.1) and the Hom-Jacobi identity of (1.2) and $\alpha([x,y])=[\alpha(x),\alpha(y)]$, (multiplicative) in the computation below. For $x,y\in\mathfrak{L}$, we have:

[TABLE]

∎

Theorem 3.12**.**

Let $(\mathfrak{L},[\cdot,\cdot],\alpha)$ be a multiplicative Hom-Lie superalgebra and $r\in\mathfrak{L}^{\otimes 2}$ be an element such that $\alpha^{\otimes 2}(r)=r,\ \ r_{21}=-r,$  (1.3) and (3.2), ((3.2) and (1.3),  i.e., $|r|=\bar{0})$, and

[TABLE]

*for all $x\in\mathfrak{L}$, where $[[r,r]]^{\alpha}$ is defined in (3.4). Define $\Delta:\mathfrak{L}\rightarrow\mathfrak{L}^{\otimes 2}$ as $\Delta(x)=ad_{x}(r)$ as in (3.1).

Then $(\mathfrak{L},[\cdot,\cdot],\Delta,\alpha,r)$ is a multiplicative coboundary Hom-Lie superbialgebra.*

Proof.

We will show the following statements:

1. $\Delta=ad(r)$ commutes with $\alpha$.

2. $\Delta$ is skew-supersymmetric.

3. The compatibility condition (1.8) holds.

4. The condition (3.7) is equivalent Hom-super-coJacobi identity of $\Delta$ (1.5).

Write $r=\sum r_{1}\otimes r_{2}$, $r^{\prime}=\sum r^{\prime}_{1}\otimes r^{\prime}_{2}$ and $\tau(r)=\sum(-1)^{{|r_{1}|}{|r_{2}|}}r_{2}\otimes r_{1}$, $\tau(r^{\prime})=\sum(-1)^{{|r^{\prime}_{1}|}{|r^{\prime}_{2}|}}r^{\prime}_{2}\otimes r^{\prime}_{1}$. To show that $\Delta=ad(r)$ commutes with $\alpha$, pick an element $x\in\mathfrak{L}$, the summation sign will often be omitted in computation to simplify the typography. Using the definition $\Delta=ad(r)$, $\alpha([x,r_{1}])=[\alpha(x),\alpha(r_{1})]$, $\alpha([x,r_{2}])=[\alpha(x),\alpha(r_{2})]$ and the assumption $\alpha^{\otimes 2}(r)=r$ we have

[TABLE]

This shows that $\Delta$ commutes with $\alpha$.

Now we show that $\Delta=ad(r)$ is skew-supersymmetric. We have

[TABLE]

Then $\Delta(x)+\tau(\Delta(x))=ad_{x}(r+r_{21})=ad_{x}(0)=0,$ 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 $\Delta=ad(r)$ is equivalent to (3.7). Let us unwrap the Hom-super-coJacobi identity. Fix an element $x\in\mathfrak{L}$, and let $r^{\prime}=\sum r^{\prime}_{1}\otimes r^{\prime}_{2}$ be another copy of $r$. Then we write

[TABLE]

Note

[TABLE]

we get $\omega=A_{1}+B_{1}+C_{1}+D_{1}.$

With these notations, the Hom-super-coJacobi identity of $\Delta=ad(r)$ (applied to $x$) becomes

[TABLE]

Therefore, to prove the equivalence between the Hom-super-coJacobi identity of $\Delta$ 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 $[[r,r]]^{\alpha}$, 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 $r=r^{\prime}$ and $r_{12}=-r$, we have

[TABLE]

In the fourth equality we used $\alpha^{\otimes 2}(r)=r$, $\alpha^{\otimes 2}(r^{\prime})=r^{\prime}$. In the equality we used the skew-supersymmetry of $[-,-]$ and $\alpha([r_{1},r^{\prime}_{1}])=[\alpha(r_{1}),\alpha(r^{\prime}_{1})]$, we know $(-1)^{{|x|}{|r_{1}|}}(-1)^{{|x|}{|r^{\prime}_{1}|}}=1$, 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 $C_{3}$ and $D_{2}$:

[TABLE]

Using the definition (1.6) of $ad_{x}$, 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 $\Delta=ad(r)$ is equivalent to $\alpha^{\otimes 3}(ad_{x}([[r,r]]^{\alpha}))=0$ ∎

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 $(\mathfrak{L},[\cdot,\cdot],\alpha)$ be a multiplicative Hom-Lie superalgebra and $r\in\mathfrak{L}^{\otimes 2}$ be an element such that $\alpha^{\otimes 2}(r)=r,\ \ r_{21}=-r,$  (1.3) and (3.2), ((3.2) and (1.3),  i.e., $|r|=\bar{0})$, and $[[r,r]]^{\alpha}=0.$ Then $(\mathfrak{L},[\cdot,\cdot],ad(r),\alpha,r)$ is a multiplicative quasi-triangular Hom-Lie superbialgebra.

Theorem 3.15**.**

*Let $(\mathfrak{L},[\cdot,\cdot],\Delta,\alpha,r)$ be a coboundary Hom-Lie superbialgebra. Then the following statements are equivalent,

(1) $\mathfrak{L}$ is a quasi-triangular Hom-Lie superbialgebra, i.e., $[[r,r]]^{\alpha}=0$ (3.4).

(2) The equality $(\alpha\otimes\Delta)(r)=-[r_{12},r_{13}]$ holds, where the bracket is defined in (3.5).

(3) The equality $(\Delta\otimes\alpha)(r)=[r_{13},r_{23}]$ holds, where the bracket is defined in (3.5).*

Proof.

The equivalence between three statements clearly follows from the equalities. Let $r^{\prime}=\sum r^{\prime}_{1}\otimes r^{\prime}_{2}$ be another copy of $r$. Since $\Delta=ad(r)$ (3.1), $r=r^{\prime}$ and $r_{1}\otimes r_{2}=-(-1)^{{|r_{1}|}{|r_{2}|}}r_{2}\otimes r_{1}$. 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 $(\mathfrak{L},[\cdot,\cdot],\Delta,\alpha)$ is a Hom-Lie superbialgebra (Definition 1.5) and $t\in\mathfrak{L}^{\otimes 2}$, under what conditions does the perturbed cobracket $\Delta_{t}=\Delta+ad(t)$ give another Hom-Lie superbialgebra $(\mathfrak{L},[\cdot,\cdot],\Delta_{t},\alpha)$ ?”

Define the perturbed cobracket $\Delta_{t}=\Delta+ad(t)$. For $x\in\mathfrak{L}$ and $t=\sum t_{1}\otimes t_{2}\in\mathfrak{L}^{\otimes 2}$ and also recall the adjoint map $ad_{x}:\mathfrak{L}^{\otimes n}\rightarrow\mathfrak{L}^{\otimes n}$ (1.6) we have :

[TABLE]

This is a natural question because $\Delta$ is a 1-cocycle (Remark 1.8), $ad(t)$ (1.6) is a 1-coboundary when $\alpha^{\otimes 2}(t)=t$ (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 $(\mathfrak{L},\Delta_{t},\alpha)$ must be a Hom-Lie supercoalgebra (Definition 1.2).

In the following result, we give some sufficient conditions under which the perturbed cobracket $\Delta_{t}$ 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 $t=\sum t_{1}\otimes t_{2}\in\mathfrak{L}^{\otimes 2}$, the symbol $t_{21}$ denotes $\tau(t)=\sum(-1)^{{|t_{1}|}{|t_{2}|}}t_{2}\otimes t_{1}$. If $\varphi(x,y)$ is an expression in the elements $x$ and $y$, we set

[TABLE]

For example, the compatibility condition $\Delta([x,y])=ad_{\alpha(x)}(\Delta(y))-(-1)^{{|x|}{|y|}}ad_{\alpha(y)}(\Delta(x))$ (1.8)

is equivalent to

[TABLE]

Moreover $[[x,y],\alpha(z)]=|[\alpha(x),[y,z]]|$, where $|[\alpha(x),[y,z]]|=[\alpha(x),[y,z]]-(-1)^{{|x|}{|y|}}[\alpha(y),[x,z]]$.

By calculation the Hom-super-Jacobi identity (1.2) is equivalent to $[[x,y],\alpha(z)]=|[\alpha(x),[y,z]]|$.

Note that we have

[TABLE]

It is also noted to simplify writing $(1\otimes 1\otimes 1+\xi+\xi^{2})=\circlearrowleft$.

Theorem 4.1**.**

Let $(\mathfrak{L},[\cdot,\cdot],\Delta,\alpha)$ be a multiplicative Hom-Lie superbialgebra and $t\in\mathfrak{L}^{\otimes 2}$ be an element such that $\alpha^{\otimes 2}(t)=t,\ \ t_{21}=-t,$  (1.3) and (3.2), ((3.2) and (1.3),  i.e., $|t|=\bar{0})$ and

[TABLE]

for all $x\in\mathfrak{L}$. Then $\mathfrak{L}_{t}=(\mathfrak{L},[\cdot,\cdot],\Delta_{t}=\Delta+ad(t),\alpha)$ is multiplicative Hom-Lie superbialgebra.

Proof.

To show that $\mathfrak{L}_{t}$ is a multiplicative Hom-Lie superbialgebra, we need to prove the following conditions:

It is clear that $(\mathfrak{L},[\cdot,\cdot],\alpha)$ is a multiplicative Hom-Lie superalgebra.

It remains to show that $(\mathfrak{L},\Delta_{t},\alpha)$ co-multiplicative Hom-Lie supercoalgebra, and the compatibility condition (1.8) holds for $\Delta_{t}$ and $[\cdot,\cdot]$.

Precisely we need to prove four things:

$\bullet$ $\alpha^{\otimes 2}\circ\Delta_{t}=\Delta_{t}\circ\alpha$, equality is true because:

[TABLE]

Using $\alpha^{\otimes 2}(t)=t$, we have

[TABLE]

$\bullet$ $\Delta_{t}$ is skew-supersymmetric because:

[TABLE]

Then : $\Delta_{t}(x)+\tau(\Delta_{t}(x))=ad_{x}\textbf{(}t+(-1)^{{|t_{1}|}{|t_{2}|}}t_{2}\otimes t_{1}\textbf{)}=ad_{x}(t+t_{21})=ad_{x}(0)=0.$

$\bullet$ Now, we need to show the compatibility condition (1.8) holds for $\Delta_{t}$ and $[\cdot,\cdot]$:

[TABLE]

which is equivalent to

[TABLE]

Since $\Delta_{t}=\Delta+ad(t)$, (4.3) is equivalent to

[TABLE]

Moreover, since $\Delta([x,y])=ad_{\alpha(x)}(\Delta(y))-(-1)^{{|x|}{|y|}}ad_{\alpha(y)}(\Delta(x))=|ad_{\alpha(x)}(\Delta(y))|$, because $\mathfrak{L}$ is a Hom-Lie superbialgebra, (4.3) is equivalent to, $ad_{[x,y]}(t)=|ad_{\alpha(x)}(ad_{y}(t))|,$ which holds by Lemma 3.11.

$\bullet$ Finally, we must show the Hom-super-coJacobi identity of $\Delta_{t}$, which states

[TABLE]

for all $x\in\mathfrak{L}$. Using the definition $\Delta_{t}=\Delta+ad(t)$. We can rewrite (4.4) as

[TABLE]

We already know that $\circlearrowleft(\alpha\otimes\Delta)(\Delta(x))=0$, which is the Hom-super-coJacobi identity of $\Delta$.

Moreover, in (3.8) and (3.9) (in the proof of Theorem 3.12 with $t$ instead of $r$), we already showed that

[TABLE]

In view of (4.5) and (4.6), the Hom-super-coJacobi identity of $\Delta_{t}$ (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 $\Delta(x)=\sum x_{1}\otimes x_{2}$ and $t=\sum t_{1}\otimes t_{2}\in\mathfrak{L}^{\otimes 2}$. Then the left-hand side of (4.8) is:

[TABLE]

Write $\Delta(t_{2})=\sum t^{\prime}_{2}\otimes t^{\prime\prime}_{2}$. Recall from (1.8) that: $\Delta([x,t_{2}])=ad_{\alpha(x)}(\Delta(t_{2}))-(-1)^{{|x|}{|t_{2}|}}ad_{\alpha(t_{2})}(\Delta(x))$, because $\mathfrak{L}$ is a Hom-Lie superbialgebra. We can continue the above computation as follows :

[TABLE]

It follows the skew-supersymmetry of $\Delta$ applied to $x$ (i.e, $\sum x_{1}\otimes x_{2}=-\sum(-1)^{{|x_{1}|}{|x_{2}|}}x_{2}\otimes x_{1}$), $|t|=\bar{0}$, $t_{21}=-t$, and $\alpha^{\otimes 2}(t)=t$, We find that

$\bullet$ $\alpha^{2}(t_{1})\otimes[\alpha(t_{2}),x_{1}]\otimes\alpha(x_{2})=(-1)^{{|x_{1}|}{|x_{2}|}}\alpha(x_{2})\otimes\alpha^{2}(t_{1})\otimes[\alpha(t_{2}),x_{1}].$

$\bullet$ $(-1)^{{|x_{1}|}{|t_{2}|}}\alpha^{2}(t_{1})\otimes\alpha(x_{1})\otimes[\alpha(t_{2}),x_{2}]=(-1)^{{|t_{1}|}{|t_{2}|}}(-1)^{{|x_{2}|}{|t_{1}|}}\alpha(x_{1})\otimes[\alpha(t_{2}),x_{2}]\otimes\alpha^{2}(t_{1}).$ That the first two terms and the last two terms above cancel out. Using the commutation of $\alpha$ with $[\cdot,\cdot]$ and $\Delta$ and $\alpha^{\otimes 2}(t)=t$, 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 $(\mathfrak{L},[\cdot,\cdot],\Delta,\alpha)$ be a multiplicative Hom-Lie superbialgebra and $t\in\mathfrak{L}^{\otimes 2}$ be an element such that $\alpha^{\otimes 2}(t)=t,\ \ t_{21}=-t,$  (1.3) and (3.2), ((3.2) and (1.3),  i.e., $|t|=\bar{0})$ and $[[t,t]]^{\alpha}+\circlearrowleft(\alpha\otimes\Delta)(t)=0$ for all $x\in\mathfrak{L}$. Then $\mathfrak{L}_{t}=(\mathfrak{L},[\cdot,\cdot],\Delta_{t}=\Delta+ad(t),\alpha)$ is multiplicative Hom-Lie superbialgebra.