$3$-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras

TL;DR

This paper explores the relationships between BiHom-Lie superalgebras and their induced 3-BiHom-Lie superalgebras, introducing new derivation concepts and operators, and constructing 3-BiHom-Lie superalgebras via Rota-Baxter operators.

Contribution

It introduces new derivation notions, Rota-Baxter and Nijenhuis operators for 3-BiHom-Lie superalgebras, and constructs these algebras from BiHom-Lie superalgebras.

Findings

Defined $( ext{α}^s, ext{β}^r)$-derivations and quasiderivations.

Established relations between derivations of BiHom-Lie and 3-BiHom-Lie superalgebras.

Constructed 3-BiHom-Lie superalgebras using Rota-Baxter operators.

Abstract

The purpose of this paper is to study the relationships between a BiHom-Lie superalgebras and its induced 3-BiHom-Lie superalgebras. We introduce the notion of $(\alpha^s,\beta^r)$-derivation, $(\alpha^s,\beta^r)$-quasiderivation and generalized $(\alpha^s,\beta^r)$-derivation of 3-BiHom-Lie superalgebras, and their relation with derivation of BiHom-Lie superalgebras. We introduce also the concepts of Rota-Baxter operators and Nijenhuis Operators of BiHom $3$-Lie superalgebras. We also explore the construction of $3$-BiHom-Lie superalgebras by using Rota-Baxter of BiHom-Lie superalgebras.

Equations

$$\alpha \circ \beta =\beta \circ \alpha ,$$

$$\alpha ([x,y])=[\alpha (x),\alpha (y)],\beta ([x,y])=[\beta (x),\beta (y)],$$

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

$${\circlearrowleft }_{x,y,z}(-1{)}^{|x||z|}[{\beta }^2(x),[\beta (y),\alpha (z)]]=0.$$

$$[x,y,z]$$

$$[x,y,[z,u,v]]=[[x,y,z],u,v]$$

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

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

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

$$-(-1{)}^{(|z|+|v|)(|x|+|y|)+|u||v|}[{\beta }^2(z),{\beta }^2(v),[\beta (x),\beta (y),\alpha (u)]]$$

$$+(-1{)}^{(|z|+|u|)(|x|+|y|)}[{\beta }^2(z),{\beta }^2(u),[\beta (x),\beta (y),\alpha (v)]],$$

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

$$\alpha \circ \beta =\beta \circ \alpha ,$$

$$\alpha ([x,y,z])=[\alpha (x),\alpha (y),\alpha (z)]and\beta ([x,y,z])=[\beta (x),\beta (y),\beta (z)],$$

$$[{\beta }^2(x),{\beta }^2(y),[\beta (z),\beta (t),\alpha (u){]}_{\alpha ,\beta }{]}_{\alpha ,\beta }$$

$$[x_1,x_2,x_3{]}_{\tau }=\tau (x_1)[x_2,x_3]-(-1{)}^{|x_1||x_2|}\tau (x_2)[x_1,x_3]+(-1{)}^{|x_3|(|x_1|+|x_2|)}\tau (x_3)[x_1,x_2].$$

$$\tau ([x,y])=0$$

$$\tau (\alpha (x))\beta (y)$$

$$[\beta (x_1),\beta (x_2),\alpha (x_3){]}_{\tau }$$

$$L=$$

$$=$$

$$+(-1{)}^{|y_3|(|y_1|+|y_2|)}\tau (\beta (y_3))[{\beta }^2(x_1),{\beta }^2(x_2),[\beta (y_1),\alpha (y_2)]{]}_{\tau }$$

$$=$$

$$-(-1{)}^{|y_1||y_2|}\tau (\beta (y_2))\{\tau ({\beta }^2(x_1))[{\beta }^2(x_2),[\beta (y_1),\alpha (y_3)]]$$

$$-(-1{)}^{|x_1||x_2|}\tau ({\beta }^2(x_2))[{\beta }^2(x_1),[\beta (y_1),\alpha (y_3)]]\}$$

$$+(-1{)}^{|y_3|(|y_1|+|y_2|)}\tau (\beta (y_3))\{\tau ({\beta }^2(x_1))[{\beta }^2(x_2),[\beta (y_1),\alpha (y_2)]]$$

$$-(-1{)}^{|x_1||x_2|}\tau ({\beta }^2(x_2))[{\beta }^2(x_1),[\beta (y_1),\alpha (y_2)]]\}$$

$$=$$

$$-(-1{)}^{|y_1||y_2|}\tau (\beta (y_2))\tau ({\beta }^2(x_1))[{\beta }^2(x_2),[\beta (y_1),\alpha (y_3)]]$$

$$+(-1{)}^{|x_1||x_2|+|y_1||y_2|}\tau (\beta (y_2))\tau ({\beta }^2(x_2))[{\beta }^2(x_1),[\beta (y_1),\alpha (y_3)]]$$

$$+(-1{)}^{|y_3|(|y_1|+|y_2|)}\tau (\beta (y_3))\tau ({\beta }^2(x_1))[{\beta }^2(x_2),[\beta (y_1),\alpha (y_2)]]$$

$$-(-1{)}^{|x_1||x_2|}(-1{)}^{|y_3|(|y_1|+|y_2|)}\tau (\beta (y_3))\tau ({\beta }^2(x_2))[{\beta }^2(x_1),[\beta (y_1),\alpha (y_2)]]$$

$$R=$$

$$-(-1{)}^{(|x_1|+|x_2|)(|y_1|+|y_3|)}(-1{)}^{|y_2||y_3|}[{\beta }^2(y_1),{\beta }^2(y_3),[\beta (x_1),\beta (x_2),\alpha (y_2){]}_{\tau }{]}_{\tau }$$

$$+(-1{)}^{(|x_1|+|x_2|)(|y_1|+|y_2|)}[{\beta }^2(y_1),{\beta }^2(y_2),[\beta (x_1),\beta (x_2),\alpha (y_3){]}_{\tau }{]}_{\tau }$$

$$=$$

$$-(-1{)}^{(|x_1|+|x_2|+|y_1|)(|y_2|+|y_3|)}(-1{)}^{|y_2||y_3|}\tau (\beta (x_1))\tau ({\beta }^2(y_3))[{\beta }^2(y_2),[\beta (x_2),\alpha (y_1)]]$$

$$-(-1{)}^{(|x_1|+|x_2|+|y_1|)(|y_2|+|y_3|)}(-1{)}^{|x_1||x_2|}\tau (\beta (x_2))\tau ({\beta }^2(y_2))[{\beta }^2(y_3),[\beta (x_1),\alpha (y_1)]]$$

$$+(-1{)}^{(|x_1|+|x_2|+|y_1|)(|y_2|+|y_3|)}(-1{)}^{|x_1||x_2|+|y_2||y_3|}\tau (\beta (x_2))\tau ({\beta }^2(y_3))[{\beta }^2(y_2),[\beta (x_1),\alpha (y_1)]]$$

$$+(-1{)}^{(|x_1|+|x_2|+|y_1|)(|y_2|+|y_3|)}(-1{)}^{|y_1|(|x_1|+|x_2|)}\tau (\beta (y_1))\tau ({\beta }^2(y_2))[{\beta }^2(y_3),[\beta (x_1),\alpha (x_2)]]$$

$$-(-1{)}^{(|x_1|+|x_2|+|y_1|)(|y_2|+|y_3|)}(-1{)}^{|y_1|(|x_1|+|x_2|)+|y_2||y_3|}\tau (\beta (y_1))\tau ({\beta }^2(y_3))[{\beta }^2(y_2),[\beta (x_1),\alpha (x_2)]]$$

$$-(-1{)}^{(|x_1|+|x_2|)(|y_1|+|y_3|)}(-1{)}^{|y_2||y_3|}\tau (\beta (x_1))\tau ({\beta }^2(y_1))[{\beta }^2(y_3),[\beta (x_2),\alpha (y_2)]]$$

$$+(-1{)}^{(|x_1|+|x_2|)(|y_1|+|y_3|)}(-1{)}^{|y_3|(|y_1|+|y_2|)}\tau (\beta (x_1))\tau ({\beta }^2(y_3))[{\beta }^2(y_1),[\beta (x_2),\alpha (y_2)]]$$

$$+(-1{)}^{(|x_1|+|x_2|)(|y_1|+|y_3|)}(-1{)}^{|y_2||y_3|+|x_1||x_2|}\tau (\beta (x_2))\tau ({\beta }^2(y_1))[{\beta }^2(y_3),[\beta (x_1),\alpha (y_2)]]$$

$$-(-1{)}^{(|x_1|+|x_2|)(|y_1|+|y_3|)}(-1{)}^{|y_3|(|y_1|+|y_2|)+|x_1||x_2|}\tau (\beta (x_2))\tau ({\beta }^2(y_3))[{\beta }^2(y_1),[\beta (x_1),\alpha (y_2)]]$$

$$-(-1{)}^{(|x_1|+|x_2|)(|y_1|+|y_2|+|y_3|)}(-1{)}^{|y_3||y_2|}\tau (\beta (y_2))\tau ({\beta }^2(y_1))[{\beta }^2(y_3),[\beta (x_1),\alpha (x_2)]]$$

$$+(-1{)}^{(|x_1|+|x_2|)(|y_1|+|y_2|+|y_3|)}(-1{)}^{|y_3|(|y_1|+|y_2|)}\tau (\beta (y_2))\tau ({\beta }^2(y_3))[{\beta }^2(y_1),[\beta (x_1),\alpha (x_2)]]$$

$$+(-1{)}^{(|x_1|+|x_2|)(|y_1|+|y_2|)}\tau (\beta (x_1))\tau ({\beta }^2(y_1))[{\beta }^2(y_2),[\beta (x_2),\alpha (y_3)]]$$

Full text

$3$-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras

Abdelkader Ben Hassine1,2, Sami Mabrouk3, Othmen Ncib4

(1 Department of Mathematics, Faculty of Science and Arts at Belqarn, P. O. Box 60, Sabt Al-Alaya 61985, University of Bisha, Saudi Arabia

2 Faculty of Sciences, University of Sfax, BP 1171, 3000 Sfax, Tunisia

3,4 Faculty of Sciences, University of Gafsa, BP 2100, Gafsa, Tunisia )

Abstract

The purpose of this paper is to study the relationships between a BiHom-Lie superalgebras and its induced 3-BiHom-Lie superalgebras. We introduce the notion of $(\alpha^{s},\beta^{r})$-derivation, $(\alpha^{s},\beta^{r})$-quasiderivation and generalized $(\alpha^{s},\beta^{r})$-derivation of 3-BiHom-Lie superalgebras, and their relation with derivation of BiHom-Lie superalgebras. We introduce also the concepts of Rota-Baxter operators and Nijenhuis Operators of BiHom $3$-Lie superalgebras. We also explore the construction of $3$-BiHom-Lie superalgebras by using Rota-Baxter of BiHom-Lie superalgebras.

Keywords: BiHom-Lie superalgebras, 3-BiHom-Lie superalgebras, $(\alpha^{s},\beta^{r})$-derivation, Rota-Baxter operators, Nijenhuis Operators.

Keywords: MSC(2010): 17A40; 17B70

000 Corresponding author(S.Mabrouk): mabrouksami00@yahoo.fr

Introduction

BiHom-type generalizations of $n$-ary Nambu-Lie algebras, called $n$-ary BiHom-Nambu-Lie algebras, were introduced by Kitouni, Makhlouf, and Silvestrov in [12]. Each $n$-ary BiHom-Nambu-Lie algebra has $(n-1)$-linear twisting maps, which appear in a twisted generalization of the $n$-ary Nambu identity called the $n$-ary BiHom-Nambu identity. If the twisting maps are all equal to the identity, one recovers an $n$-ary Nambu-Lie algebra. The twisting maps provide a substantial amount of freedom in manipulating Nambu-Lie algebras. In recent years, Rota-Baxter (associative) algebras, originated from the work of G. Baxter [4] in probability and populated by the work of Cartier and Rota [7, 17], have also been studied in connection with many areas of mathematics and physics, including combinatorics, number theory, operators and quantum field theory [16]. Furthermore, Rota-Baxter operators on a Lie algebra are an operator form of the classical Yang-Baxter equations and contribute to the study of integrable systems [5]. Further Rota-Baxter $3$-Lie algebras are closely related to pre-Lie algebras [6]. Rota-Baxter of multiplicative $3$-ary Hom-Nambu-Lie algebras were introduced by Sun and Chen, in [14].

Deformations of $n$-Lie algebras have been studied from several aspects. See [1, 9] for more details. In particular, a notion of a Nijenhuis operator on a $3$-Lie algebra was introduced in [15] in the study of the $1$-order deformations of a $3$-Lie algebra. But there are some quite strong conditions in this definition of a Nijenhuis operator. In the case of Lie algebras, one could obtain fruitful results by considering one-parameter infinitesimal deformations, i.e. $1$-order deformations. However, for $n$-Lie algebras, we believe that one should consider $(n-1)$-order deformations to obtain similar results. In [9], for $3$-Lie algebras, the author had already considered $2$-order deformations. For the case of Hom-Lie superalgebras, the authors in [13] give the notion of Hom-Nijenhuis operator.

Thus it is time to study $3$-BiHom-Lie superalgebras, Rota-Baxter algebras and Nijenhuis operator together to get a suitable definition of Rota-Baxter of $3$-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras. Similarly, we give the relationship between Nijenhuis operator of $3$-BiHom-Lie superalgebras and BiHom-Lie superalgebras.

This paper is organized as follows: In Section $1$, we recall the concepts of BiHom-Lie superalgebras and introduce the notion of $3$-BiHom-Lie superalgebras. The construction of $3$-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras are established in Section $2$. In section $3$, we give the definition of $(\alpha^{s},\beta^{r})$-derivation and $(\alpha^{s},\beta^{r})$-quasiderivation of $3$-BiHom-Lie superalgebras. In section $4$, we give the definition of Rota-Baxter of $3$-BiHom-Lie superalgebras and the realizations of Rota-Baxter of $3$-BiHom-Lie superalgebras from Rota-Baxter BiHom-Lie superalgebras. The Section $5$ is dedicated to study the second order deformation of $3$-BiHom-Lie superalgebras, and introduce the notion of Nijenhuis operator on $3$-BiHom-Lie superalgebras, which could generate a trivial deformation. In the other part of this section we give some properties and results of Nijenhuis operators.

1 Definitions and Notations

In this section, we review basics definition of BiHom-Lie superalgebras, $3$-Lie superalgebras and generalize the notion of $3$-BiHom-Lie algebras to the super case.

Let $V=V_{\overline{0}}\oplus V_{\overline{1}}$ be a $\mathbb{Z}_{2}$-graded vector space. If $v\in V$ is a homogenous element, then its degree will be denoted by $|v|$, where $|v|\in\mathbb{Z}_{2}$ and $\mathbb{Z}_{2}=\{\overline{0},\overline{1}\}$. Let $End(V)$ be the $\mathbb{Z}_{2}$-graded vector space of endomorphisms of a $\mathbb{Z}_{2}$-graded vector space $V=V_{\overline{0}}\oplus V_{\overline{1}}$, we denote by $\mathcal{H}(V)$ the set of homogenous elements of $V$. The composition of two endomorphisms $a\circ b$ determines the structure of superalgebra in $End(V)$, and the graded binary commutator $[a,b]=a\circ b-(-1)^{|a||b|}b\circ a$ induces the structure of Lie superalgebras in $End(V)$.

Definition 1.1**.**

[18]** A BiHom-Lie superalgebra is a triple $(\mathfrak{g},[\cdot,\cdot],\alpha,\beta)$ consisting of a $\mathbb{Z}_{2}$-graded vector space $\mathfrak{g}=\mathfrak{g}_{\overline{0}}\oplus\mathfrak{g}_{\overline{1}}$, an even bilinear map $[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\longrightarrow\mathfrak{g}$ and a two even homomorphisms $\alpha,\beta:\mathfrak{g}\rightarrow\mathfrak{g}$ satisfying the following identities:

[TABLE]

*where $x,y$ and $z$ are homogeneous elements in $\mathfrak{g}$. The condition (1.4) is called BiHom-super-Jacobi identity.

If the conditions (1.1) and (1.2) are not satisfying, then BiHom-Lie superalgebra is called nonmultiplicative BiHom-Lie superalgebra.*

Definition 1.2**.**

[8]** A $\mathbb{Z}_{2}$-graded vector space $\mathfrak{g}=\mathfrak{g}_{\overline{0}}\oplus\mathfrak{g}_{\overline{1}}$ is said to be a $3$-Lie superalgebra, if it is endowed with a trilinear map (bracket) $[\cdot,\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$. If it satisfies the following conditions:

[TABLE]

where $x,y,z,u,v\in\mathfrak{g}$ are homogeneous elements.

Definition 1.3**.**

A nonmultiplicative $3$-BiHom-Lie superalgebra is a quadruple $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$ consisting of a $\mathbb{Z}_{2}$-graded vector space $\mathfrak{g}=\mathfrak{g}_{\overline{0}}\oplus\mathfrak{g}_{\overline{1}}$, an even trilinear map (bracket) $[\cdot,\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ and two even endomorphisms $\alpha,\beta:\mathfrak{g}\rightarrow\mathfrak{g}$. If it satisfies the following conditions:

[TABLE]

where $x,y,z,u,v\in\mathfrak{g}$ are homogeneous elements. The condition (1.5) is called the $3$-BiHom-super-Jacobi identity.

Remark 1.4**.**

The identity (1.5) is equivalent to

[TABLE]

where $\gamma=(-1)^{(|u|+|v|)(|x|+|y|)+|z||u|}.$

Definition 1.5**.**

A $3$-BiHom-Lie superalgebra is a nonmultiplicative $3$-BiHom-Lie superalgebra $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$ such that

[TABLE]

for all $x,y,z\in\mathcal{H}(\mathfrak{g}).$

Remark 1.6**.**

If   $\alpha=\beta=Id$ we recover the $3$-Lie superalgebra.

Proposition 1.7**.**

*Let $(\mathfrak{g},[\cdot,\cdot,\cdot])$ be a $3$-Lie superalgebra and $\alpha,\beta:\mathfrak{g}\rightarrow\mathfrak{g}$ are two commuting morphisms on $\mathfrak{g}$, then $(\mathfrak{g},[\cdot,\cdot,\cdot]_{\alpha,\beta},\alpha,\beta)$ is a $3$-BiHom-Lie superalgebra,

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

Proof. .

It is easy to see that, $\forall x,y,z\in\mathcal{H(\mathfrak{g})}$ we are :

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

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

Let $x,y,z,t,u\in\mathcal{H(\mathfrak{g})}$,we have

[TABLE]

then the $3$-BiHom-super-Jacobi identity it satisfies. ∎

2 $3$-BiHom-Lie Superalgebras Induced by BiHom-Lie

Superalgebras

Now we generalize the result given in [11] to super case. Given a nonmultplicative BiHom-Lie superalgebra $(\mathfrak{g},[\cdot,\cdot],\alpha,\beta)$ and a linear form $\tau:\mathfrak{g}\rightarrow K$. For any $x_{1},x_{2},x_{3}\in\mathcal{H}(\mathfrak{g})$, we define the 3-ary bracket by

[TABLE]

Theorem 2.1**.**

*Let $(\mathfrak{g},[\cdot,\cdot],\alpha,\beta)$ be a nonmultiplicative BiHom-Lie superalgebra. If the conditions *

[TABLE]

*are satisfied for any $x,y\in\mathfrak{g}$, then $(\mathfrak{g},[\cdot,\cdot,\cdot]_{\tau},\alpha,\beta)$ is a nonmultiplicative 3-BiHom-Lie superalgebra.

We say that $(\mathfrak{g},[\cdot,\cdot,\cdot]_{\tau},\alpha,\beta)$ induced by $(\mathfrak{g},[\cdot,\cdot],\alpha,\beta)$, it is denoted by $\mathfrak{g}_{\tau}$.*

Proof. .

For any $x_{1},x_{2},x_{3}\in\mathcal{H}(\mathfrak{g})$, we have

[TABLE]

Similarly, one gets $[\beta(x_{1}),\beta(x_{2}),\alpha(x_{3})]_{\tau}=-(-1)^{|x_{2}||x_{3}|}[\beta(x_{1}),\beta(x_{3}),\alpha(x_{2})]_{\tau}$.

We next show that $[\cdot,\cdot,\cdot]_{\tau}$ is satisfies the $3$-BiHom-super-Jacobi identity.

Let $L$ be its left-hand side, and $R$ its right-hand side of the identity (1.5). Using the definition of $[\cdot,\cdot,\cdot]_{\tau}$, the equations (2.2) and (2.3), we have

[TABLE]

And

[TABLE]

By the identity of BiHom-super-Jacobi and the equation (2.2) we get

[TABLE]

Where

[TABLE]

If $x_{1}$ or $y_{1}$ is odd, then $\tau(\beta(y_{1}))\tau(\beta^{2}(x_{1}))=0$ which gives $\tau(\beta(y_{1}))\tau(\beta^{2}(x_{1}))\Omega_{1}=0$.

If $x_{1}$ and $y_{1}$ are even, then:

[TABLE]

since the identity of BiHom-super-Jacobi and equation (1.3). So $\tau(\beta(y_{1}))\tau(\beta^{2}(x_{1}))\Omega_{1}=0$.

For the same reason we show that each $\Omega_{i}$, where $1\leq i\leq 6$, multiplied by their coefficient is zero. From where $L-R=0$. The theorem is proved. ∎

3 Derivations and $(\alpha^{s},\beta^{r})$-derivations of $3$-BiHom-Lie superalgebras

In this section, we introduce the notion of $(\alpha^{s},\beta^{r})$-derivations of $3$-BiHom-Lie superalgebras generalize the notion of $(\alpha^{s},\beta^{r})$-derivations of $3$-BiHom-Lie algebras introduced in [2] and we give some results.

Definition 3.1**.**

Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$ be a $3$-BiHom-Lie superalgebra. A linear map $\mathfrak{D}:\mathfrak{g}\rightarrow\mathfrak{g}$ is a derivation if it satisfait for all $x,y,z\in\mathcal{H}(\mathfrak{g})$ :

[TABLE]

and it is called an $(\alpha^{s},\beta^{r})$-derivation of $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$, if it satisfies :

[TABLE]

Let ${\rm Der}_{(\alpha^{s},\beta^{r})}(\mathfrak{g})$ be the set of $(\alpha^{s},\beta^{r})$-derivations of $\mathfrak{g}$ and set

[TABLE]

We show that ${\rm Der}(\mathfrak{g})$ is equipped with a Lie superalgebra structure. In fact, for $\mathfrak{D}\in{\rm Der}_{(\alpha^{s},\beta^{r})}(\mathfrak{g})$ and $\mathfrak{D}^{\prime}\in{\rm Der}_{(\alpha^{s^{\prime}},\beta^{r^{\prime}})}(\mathfrak{g})$, we have $[\mathfrak{D},\mathfrak{D}^{\prime}]\in{\rm Der}_{(\alpha^{s+s^{\prime}},\beta^{r+r^{\prime}})}(\mathfrak{g})$ , where $[\mathfrak{D},\mathfrak{D}^{\prime}]$ is the standard supercommutator defined by $[\mathfrak{D},\mathfrak{D}^{\prime}]=\mathfrak{D}\mathfrak{D}^{\prime}-(-1)^{|\mathfrak{D}||\mathfrak{D}^{\prime}|}\mathfrak{D}^{\prime}\mathfrak{D}.$

Definition 3.2**.**

An endomorphism $\mathfrak{D}$ of a $3$-BiHom-Lie superalgebra $\mathfrak{g}$ is called $(\alpha^{s},\beta^{r})$-quasiderivation if there exists an endomorphism $\mathfrak{D}^{\prime}$ of $\mathfrak{g}$ such that

[TABLE]

[TABLE]

for any $x,y,z\in\mathfrak{g}.$

Then we define

[TABLE]

Proposition 3.3**.**

Let $(\mathfrak{g},[\cdot,\cdot],\alpha,\beta)$ be a nonmultiplicative BiHom-Lie superalgebra. Let $\mathfrak{D}:\mathfrak{g}\rightarrow\mathfrak{g}$ be an $(\alpha^{s},\beta^{r})$-derivation of $(\mathfrak{g},[\cdot,\cdot],\alpha,\beta)$. If the following identity holds, for all $x,y,z\in\mathcal{H}(\mathfrak{g}),$

[TABLE]

then $\mathfrak{D}$ is an $(\alpha^{s},\beta^{r})$-derivation of the induced nonmultiplicative $3$-BiHom-Lie superalgebra $\mathfrak{g}_{\tau}$.

Proof. .

We have to prove that

[TABLE]

By applying $\mathfrak{D}$ to each side of equation (2.1), we get

[TABLE]

while,

[TABLE]

Using the fact that

[TABLE]

we can rewrite the right hand side as

[TABLE]

We use the property $\tau|_{\mathfrak{g}_{1}}\equiv 0$ we find the following equality

[TABLE]

We then obtain the result by assuming that the following identity holds, $\forall x,y,z\in\mathcal{H}(\mathfrak{g})$,

[TABLE]

This ends the proof. ∎

Proposition 3.4**.**

Let $(\mathfrak{g},[\cdot,\cdot],\alpha,\beta)$ be a nonmultiplicative BiHom-Lie superalgebra. Let $\mathfrak{D}$ be an $(\alpha^{s},\beta^{r})$-quasiderivation of $(\mathfrak{g},[\cdot,\cdot],\alpha,\beta)$. If the following identity holds, for all $x,y,z\in\mathcal{H}(\mathfrak{g}),$

[TABLE]

then $\mathfrak{D}$ is an $(\alpha^{s},\beta^{r})$-super-quasiderivation of the induced ternary BiHom-Lie superalgebra $\mathfrak{g}_{\tau}$.

4 Rota-Baxter of $3$-BiHom-Lie superalgebras

In this section, we give the definition of Rota-Baxter of $3$-BiHom-Lie superalgebras and the realizations of Rota-Baxter of $3$-BiHom-Lie superalgebras from Rota-Baxter of BiHom-Lie superalgebras.

Definition 4.1**.**

Let $(\mathfrak{g},[\cdot,\cdot],\alpha,\beta)$ be a BiHom-Lie superalgebra and $\lambda\in\mathbb{K}$. If an even linear map $R:\mathfrak{g}\rightarrow\mathfrak{g}$ satisfies

[TABLE]

for all $x,y\in\mathfrak{g}$, then $R$ is called a Rota-Baxter operator of weight $\lambda$ on $(\mathfrak{g},[.,.],\alpha,\beta)$ .

Definition 4.2**.**

Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$ be a $3$-BiHom-Lie superalgebra and $\lambda\in\mathbb{K}$. If an even linear map $R:\mathfrak{g}\rightarrow\mathfrak{g}$ satisfies

[TABLE]

for all $x,y,z\in\mathfrak{g}$, then $R$ is called a Rota-Baxter operator of weight $\lambda$ on $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$ .

Proposition 4.3**.**

Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$ be a $3$-BiHom-Lie superalgebra over $\mathbb{K}$. An invertible linear mapping $R:\mathfrak{g}\rightarrow\mathfrak{g}$ is a Rota-Baxter operator of weight [math] on $\mathfrak{g}$ if and only if $R^{-1}$ is an even derivation on $\mathfrak{g}$.

Proof. .

$R$ is an even invertible Rota-Baxter operator of weight [math] on $\mathfrak{g}$, if and only if:

$[R(x),R(y),R(z)]=R([R(x),R(y),z]+[R(x),R(y),z]+[x,R(y),R(z)]),\;\forall x,y,z\in\mathfrak{g}$.

Suppose that $X=R(x),\;Y=R(y)$ and $Z=R(z)$, we have:

$[X,Y,Z]=R([X,Y,R^{-1}(Z)]+[X,R^{-1}(Y),Z]+[R^{-1}(X),Y,Z])$, which gives

$R^{-1}([X,Y,Z])=[X,Y,R^{-1}(Z)]+[X,R^{-1}(Y),Z]+[R^{-1}(X),Y,Z]$. Thus $R^{-1}$ is an even derivation on $\mathfrak{g}$. ∎

Proposition 4.4**.**

Let $R$ be a Rota-Baxter of weight $\lambda$ on a nonmultiplicative BiHom-Lie superalgebra $(\mathfrak{g},[\cdot,\cdot],\alpha,\beta)$ and $\tau\in\mathfrak{g}^{*}$ satisfying to the two conditions (2.2) and (2.3). Then $R$ is a Rota-Baxter operator of weight $\lambda$ on the induced nonmultiplicative $3$-BiHom-Lie superalgebra $\mathfrak{g}_{\tau}$ if and only if $R$ satisfies

[TABLE]

Proof. .

Let $x,y,z\in\mathcal{H}(\mathfrak{g})$, we have

[TABLE]

where

[TABLE]

thus, $R$ is a Rota-Baxter operator on $(\mathfrak{g},[.,.,.],\alpha,\beta)$ if and only if $A=0$. ∎

Let $R$ be a Rota-Baxter operator of weight $\lambda$ on a $3$-BiHom-Lie superalgebra $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$, we define a ternary operation on $\mathfrak{g}$ by

[TABLE]

for all $x_{1},x_{2},x_{3}\in\mathcal{H}(\mathfrak{g})$, where \widehat{R}_{I}(x_{i})=\left\{\begin{array}[]{ll}x_{i}&\;i\;\in I\hbox{ ;}\\ R(x_{i})&\;i\;\not\in I\hbox{.}\end{array}\right.

Then we have the following result.

Theorem 4.5**.**

Let $R$ be a Rota-Baxter operator of weight $\lambda$ on a $3$-BiHom-Lie superalgebra $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$. Then $(\mathfrak{g},[\cdot,\cdot,\cdot]_{R},\alpha,\beta)$ is a $3$-BiHom-Lie superalgebra and $R$ be a Rota-Baxter operator of weight $\lambda$ on $(\mathfrak{g},[\cdot,\cdot,\cdot]_{R},\alpha,\beta)$.

Proof. .

It is clear that $[\cdot,\cdot,\cdot]_{R}$ is super-skewsymetric.

Let $x_{1},x_{2},x_{3},x_{4},x_{5}\in\mathcal{H}(\mathfrak{g})$. Denote $y_{1}=\beta^{2}(x_{1}),\;y_{2}=\beta^{2}(x_{2})$ and $y_{3}=[\beta(x_{3}),\beta(x_{4}),\alpha(x_{5})]_{R}$, we have

$[\beta^{2}(x_{1}),\beta^{2}(x_{2}),[\beta(x_{3}),\beta(x_{4}),\alpha(x_{5})]_{R}]_{R}$

[TABLE]

Since $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$ is a $3$-BiHom-Lie superalgebra, $R\circ\alpha=\alpha\circ R$ and $R\circ\beta=\beta\circ R$, for any given $\emptyset\neq I\subseteq[5]$, we have

$[\widehat{R}_{I}(\beta^{2}(x_{1})),\widehat{R}_{I}(\beta^{2}(x_{2})),[\widehat{R}_{I}(\beta(x_{3}),\widehat{R}_{I}(\beta(x_{4})),\widehat{R}_{I}(\alpha(x_{5}))]]$

[TABLE]

Thus from the above sum, we conclude that $(\mathfrak{g},[~{},~{},~{}]_{R},\alpha,\beta)$ is a BiHom 3-Lie superalgebra.

It remains to show that $R$ is a Rota-Baxter operator of weight $\lambda$ on $(\mathfrak{g},[~{},~{},~{}]_{R},\alpha,\beta)$.

[TABLE]

which gives the requested result. The theorem is proved. ∎

Proposition 4.6**.**

Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$ be a $3$-BiHom-Lie superalgebra and $R$ be a Rota-Baxter operator of weight $\lambda$ on $\mathfrak{g}$ such that $R^{2}=R$, then $(\mathfrak{g},[\cdot,\cdot,\cdot]_{R},\alpha\circ R,\beta\circ R)$ is a noncommutative $3$-BiHom-Lie superalgebra.

5 Nijenhuis Operators on BiHom $3$-Lie superalgebras

In this section, we study the second order deformation of $3$-BiHom-Lie superalgebras, and introduce the notion of Nijenhuis operator on $3$-BiHom-Lie superalgebras, which could generate a trivial deformation. In the other part of this section we give some properties and results of Nijenhuis operators.

5.1 Second-order deformation of $3$-BiHom-Lie superalgebras

Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$ be a $3$-BiHom-Lie superalgebras and $\omega_{i}:\mathfrak{g}\times\mathfrak{g}\times\mathfrak{g}\longrightarrow\mathfrak{g},\;i=1,2$ be a super-skewsymetric multilinear maps. Consider a $\lambda$-parametrized family of $3$-linear operations:

[TABLE]

where $\lambda\in\mathbb{K}$. If all $[\cdot,\cdot,\cdot]_{\lambda}$ are $3$-BiHom-Lie superalgebra structures, we say that $\omega_{1},\;\omega_{2}$ generate an second-order $1$-parameter deformation of the $3$-BiHom-Lie superalgebra $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$.

Proposition 5.1**.**

With the above notations, $\omega_{1},\;\omega_{2}$ generate a second-order $1$-parameter deformation of the $3$-BiHom-Lie superalgebra $(\mathfrak{g},\omega_{0}=[\cdot,\cdot,\cdot],\alpha,\beta)$ if and only if for all $i,j=1,2$ and $l=1,\cdots,4$ the following conditions are satisfied :

[TABLE]

where $\omega_{i}\circ_{\alpha,\beta}\omega_{j}:\wedge^{2}\mathfrak{g}\otimes\wedge^{2}\mathfrak{g}\wedge\mathfrak{g}\longrightarrow\mathfrak{g}$ is defined by

[TABLE]

where $\omega_{j}(X,.)\ast_{\alpha,\beta}Y\in\wedge^{2}\mathfrak{g}$ is given by

[TABLE]

Proof. .

$(\mathfrak{g},\omega_{\lambda},\alpha,\beta)$ are $3$-BiHom-Lie superalgebra structures if and only if

[TABLE]

[TABLE]

By (5.4), we have

[TABLE]

Expending the equations in (5.5) and collecting coefficients of $\lambda^{l}$, we see that (5.5) is equivalent to the system of equation

[TABLE]

Thus,we have

[TABLE]

∎

Corollary 5.2**.**

If $\omega_{1},\omega_{2}$ generate a second-order $1$-parameter deformation of the $3$-BiHom-Lie superalgebra $(\mathfrak{g},\omega_{0}=[\cdot,\cdot,\cdot],\alpha,\beta)$, then $\omega_{1}$ is $2$-cocycle of the $3$-BiHom-Lie superalgebra $(\mathfrak{g},\omega_{0}=[\cdot,\cdot,\cdot],\alpha,\beta)$ with the coefficients in the adjoint representation

Proof. .

For $l=1$, the condition $(24)$ gives the following equality

[TABLE]

which is equivalent to that $\omega_{1}$ is a $2$-cocycle. ∎

Corollary 5.3**.**

If $\omega_{1},\omega_{2}$ generate a second-ordre $1$-parameter deformation of the $3$-BiHom-Lie superalgebra $(\mathfrak{g},\omega_{0},\alpha,\beta)$, then $(\mathfrak{g},\omega_{2},\alpha,\beta)$ is a $3$-BiHom-Lie superalgebra.

Proof. .

By $(23)$, let $i=2$, we deduce that

[TABLE]

and by $(24)$, let $l=4$, we deduce that

[TABLE]

which equivalent to that $(\mathfrak{g},\omega_{2},\alpha,\beta)$ is a $3$-BiHom-Lie superalgebra. ∎

Definition 5.4**.**

A deformation is said to be trivial if there exists an even linear map $N:\mathfrak{g}\longrightarrow\mathfrak{g}$ such that for all $\lambda,\;T_{\lambda}=id+\lambda N$ satisfies

[TABLE]

[TABLE]

Eq. (5.6) equals to

[TABLE]

The left hand side of Eq. (5.7) equals to

[TABLE]

The right hand side of Eq. (5.7) equals to

[TABLE]

Therefore, by Eq. (5.7), we have

[TABLE]

Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$ be a $3$-BiHom-Lie superalgebra, and $N:\mathfrak{g}\longrightarrow\mathfrak{g}$ a linear map. Define a $3$-ary bracket $[\cdot,\cdot,\cdot]^{1}_{N}:\wedge^{3}\mathfrak{g}\longrightarrow\mathfrak{g}$ by

[TABLE]

Then we define $3$-ary bracket $[\cdot,\cdot,\cdot]^{2}_{N}:\wedge^{3}\mathfrak{g}\longrightarrow\mathfrak{g}$, via induction by

[TABLE]

Definition 5.5**.**

Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$ be a $3$-BiHom-Lie superalgebra. An even linear map $N:\mathfrak{g}\rightarrow\mathfrak{g}$ is called a Nijenhuis operator if

[TABLE]

$\forall x_{1},x_{2},x_{3}\in\mathfrak{g}$, where \widetilde{N}(x_{i})=\left\{\begin{array}[]{ll}x_{i}&\;i\;\in I\hbox{ ;}\\ N(x_{i})&\;i\;\not\in I\hbox{.}\end{array}\right.

5.2 Some properties of Nijenhuis operators

Let $(\mathfrak{g},[\cdot,\cdot],\alpha,\beta)$ be a BiHom-Lie superalgebra, a Nijenhuis operator of $\mathfrak{g}$ is a linear map $N:\mathfrak{g}\to\mathfrak{g}$ compatible with $\alpha$ and $\beta$ $($ i.e $N\circ\alpha=\alpha\circ N$ and $N\circ\beta=\beta\circ N$ $)$ defined by

[TABLE]

Proposition 5.6**.**

Let $N$ be a Nijenhuis operator on a BiHom-Lie superalgebra $(\mathfrak{g},[\cdot,\cdot],\alpha,\beta)$ and $\tau\in\wedge\mathfrak{g}^{*}$ satisfying to the two conditions (2.2) and (2.3). Then $N$ is a Nijenhuis operator on the induced $3$-BiHom-Lie superalgebra $\mathfrak{g}_{\tau}$.

Proof. .

Let $x_{1},x_{2},x_{3}\in\mathcal{H}(\mathfrak{g})$. In the first hand, we have

[TABLE]

Since $N$ is a Nijenhuis operator on a BiHom-Lie superalgebra $(\mathfrak{g},[\cdot,\cdot],\alpha,\beta)$.

In the other hand, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We suppose that:

[TABLE]

It is easy to see that

[TABLE]

which gives that $N$ is a Nijenhuis operator on $(\mathfrak{g},[\cdot,\cdot,\cdot]_{\tau},\alpha,\beta)$. ∎

Proposition 5.7**.**

Let $N$ be a Nijenhuis operator on a $3$-BiHom-Lie superalgebras $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$ and $R$ be a Rota-Baxter operator on a $3$-BiHom-Lie superalgebras $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$ such that $R\circ N=N\circ R$ then $N$ be a Nijenhuis operator on a $3$-BiHom-Lie superalgebras $(\mathfrak{g},[\cdot,\cdot,\cdot]_{R},\alpha,\beta)$.

Proof. .

For all $x_{1},x_{2},x_{3}\in\mathcal{H}(\mathfrak{g})$, we have

[TABLE]

then $N$ is a Nijenhuis operator on a $3$-BiHom-Lie superalgebras $(\mathfrak{g},[\cdot,\cdot,\cdot]_{R},\alpha,\beta)$. ∎

Proposition 5.8**.**

Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha,\beta)$ be a $3$-Bihom-Lie superalgebra. If an even endomorphism $N$ is a derivation, then $N$ is a Nijenhuis operator if and only if $N$ is a Rota-Baxter operator of weight [math] on $\mathfrak{g}$.