Nijenhuis Operators and Abelian Extensions of Hom-δ-Jordan Lie Supertriple Systems

Abstract

Representations and cohomologies of Hom-$\delta$-Jordan Lie supertriple systems are established. As an application, Nijenhuis operators and abelian extensions of Hom-$\delta$-Jordan Lie supertriple systems are discussed. We obtain the infinitesimal deformation generated by virtue of a Nijenhuis operator. It is obtained that the sufficient and necessary condition for the equivalence of abelian extensions of Hom-$\delta$-Jordan Lie supertriple systems.

1. Introduction

In this paper, we pay our main attention to Hom-$\delta$-Jordan Lie supertriple systems over an arbitrary field. As is well-known, Lie triple systems have important applications in elementary particle theory and the theory of quantum mechanics [1]. Some results on Lie triple systems, including simple Lie triple systems over an algebraically closed field [2], the Yamaguti cohomology theory [3,4], infinitesimal deformations, abelian extensions [5], etc. were studied. Okubo reformulated the para-statistics as Lie supertriple systems and explained the relationship between Lie supertriple systems and ortho-symplectic supertriple systems [6]. In [7], the authors introduced $\delta$-Jordan Lie supertriple systems, which are a generalization of Lie supertriple systems. Some examples of quasi-classical $\delta$-Lie supertriple systems were given over a field of characteristic not two [8]. The Jordan superalgebra of F-type from Jordan Lie supertriple systems were discussed [9]. Later, Ma and Chen obtained the cohomology and deformations of $\delta$-Jordan Lie triple systems in 2017 [10]. In [11], the authors determined derivations and deformations of $\delta$-Jordan Lie supertriple systems. In 2022, we obtained 1-parameter formal deformations and abelian extensions of Lie color triple systems [12].

In recent years, structures and representations of many Hom-type algebras have been obtained [13,14,15,16,17,18,19,20,21,22,23,24]. In particular, the notion of Hom-Lie triple systems was introduced in [14]. Hom-Lie triple systems are ternary Hom–Nambu algebras whose triple product is left anti-symmetric. When the twisting maps of Hom–Lie triple systems are all equal to the identity map, one recovers Lie triple systems. In 2019, Chen and his students gave that cohomologies and 1-parameter formal deformations of Hom-$\delta$-Jordan Lie triple systems [25]. In 2022, we obtained central extensions and Nijenhuis operators of Hom-$\delta$-Jordan Lie triple systems [26].

The purpose of this paper is to consider the cohomology theory and deformations of Hom-$\delta$-Jordan Lie supertriple systems based on some work in [3,4,10,12,13,25,26,27]. The paper is organized as follows. Section 2 is devoted to some basic definitions and the cohomology theory of multiplicative Hom-$\delta$-Jordan Lie supertriple systems. In Section 3, we discuss Nijenhuis operators of Hom-$\delta$-Jordan Lie supertriple systems and obtain the infinitesimal deformation generated using a Nijenhuis operator. Section 4 is dedicated to the abelian extension theory of multiplicative Hom-$\delta$-Jordan Lie supertriple systems. We show the sufficient and necessary condition for the equivalence of abelian extensions.

Throughout this paper, $\mathbf{F}$ denotes an arbitrary field, and its characteristic is zero.

2. Preliminaries

We first recall some basic facts and definitions of Hom-Lie triple systems.

Definition 1

([14]). A Hom-Lie triple system $(T,[·,·,·],\alpha =({\alpha }_1,{\alpha }_2))$ consists of an $\mathbf{F}$-vector space T, a trilinear map $[·,·,·]:T\times T\times T\to T$, and linear maps ${\alpha }_i:T\to T$ for $i=1,2$, called twisted maps, such that for all $x,y,z,u,v\in T$,

$$[x,x,z]=0,$$

$$[x,y,z]+[y,z,x]+[z,x,y]=0,$$

$$\begin{array}{cc}\hfill [{\alpha }_1\left(u\right),{\alpha }_2\left(v\right),[x,y,z]]=& [[u,v,x],{\alpha }_1\left(y\right),{\alpha }_2\left(z\right)]+[{\alpha }_1\left(x\right),[u,v,y],{\alpha }_2\left(z\right)]\hfill \\ \hfill & +[{\alpha }_1\left(x\right),{\alpha }_2\left(y\right),[u,v,z]].\hfill \end{array}$$

If $\left|x\right|$ occurs in some expression in this paper, we always regard x as a ${\mathbb{Z}}_2$-homogeneous element and $\left|x\right|$ as the ${\mathbb{Z}}_2$-degree of x.

Definition 2

([28]). A Hom-Lie supertriple system $(T,[·,·,·],\alpha =({\alpha }_1,{\alpha }_2))$ consists of a ${\mathbb{Z}}_2$-graded vector space $T=T_{\overline{0}}\oplus T_{\overline{1}}$ over $\mathbf{F}$ together with a trilinear map $[·,·,·]:T\times T\times T\to T$, and even linear maps ${\alpha }_i:T\to T$ for $i=1,2$, called twisted maps, such that for all $x,y,z,u,v\in T$,

$$\begin{array}{c}\hfill [x,y,z]=-{(-1)}^{\left|x\right|\left|y\right|}[y,x,z],\end{array}$$

(1)

$$\begin{array}{c}\hfill {(-1)}^{\left|x\right|\left|z\right|}[x,y,z]+{(-1)}^{\left|y\right|\left|x\right|}[y,z,x]+{(-1)}^{\left|z\right|\left|y\right|}[z,x,y]=0,\end{array}$$

(2)

$$\begin{array}{cc}\hfill [{\alpha }_1\left(u\right),{\alpha }_2\left(v\right),[x,y,z]]=& [[u,v,x],{\alpha }_1\left(y\right),{\alpha }_2\left(z\right)]+{(-1)}^{\left|x\right|\left(\right|u|+|v\left|\right)}[{\alpha }_1\left(x\right),[u,v,y],{\alpha }_2\left(z\right)]\hfill \\ \hfill & +{(-1)}^{\left(\right|u|+|v\left|\right)\left(\right|x|+|y\left|\right)}[{\alpha }_1\left(x\right),{\alpha }_2\left(y\right),[u,v,z]].\hfill \end{array}$$

(3)

Definition 3.

A Hom-δ-Jordan Lie supertriple system $(T,[·,·,·],\alpha =({\alpha }_1,{\alpha }_2))$ consists of a ${\mathbb{Z}}_2$-graded vector space $T=T_{\overline{0}}\oplus T_{\overline{1}}$ over $\mathbf{F}$ together with a trilinear map $[·,·,·]:T\times T\times T\to T$, and even linear maps ${\alpha }_i:T\to T$ for $i=1,2$, called twisted maps, such that for all $x,y,z,u,v\in T$, $\delta =\pm 1,$

$$\begin{array}{c}\hfill [x,y,z]=-\delta {(-1)}^{\left|x\right|\left|y\right|}[y,x,z],\end{array}$$

(4)

$$\begin{array}{c}\hfill {(-1)}^{\left|x\right|\left|z\right|}[x,y,z]+{(-1)}^{\left|y\right|\left|x\right|}[y,z,x]+{(-1)}^{\left|z\right|\left|y\right|}[z,x,y]=0,\end{array}$$

(5)

$$\begin{array}{cc}\hfill [{\alpha }_1\left(u\right),{\alpha }_2\left(v\right),[x,y,z]]=& [[u,v,x],{\alpha }_1\left(y\right),{\alpha }_2\left(z\right)]+{(-1)}^{\left|x\right|\left(\right|u|+|v\left|\right)}[{\alpha }_1\left(x\right),[u,v,y],{\alpha }_2\left(z\right)]\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|u|+|v\left|\right)\left(\right|x|+|y\left|\right)}[{\alpha }_1\left(x\right),{\alpha }_2\left(y\right),[u,v,z]].\hfill \end{array}$$

(6)

Clearly, $T_{\overline{0}}$ is a Hom-$\delta$-Jordan Lie triple system, and the case of $\alpha =\mathrm{id}, \delta =1$ defines a Lie triple system, so any Hom-$\delta$-Jordan Lie supertriple system is the generalization of a Lie triple system.

A Hom-$\delta$-Jordan Lie supertriple system is said to be multiplicative if ${\alpha }_1={\alpha }_2=\alpha$ and $\alpha \left(\right[x,y,z\left]\right)=\left[\alpha \right(x),\alpha (y),\alpha (z\left)\right]$, and denoted by $(T,[·,·,·],\alpha )$.

A morphism $f:(T,[·,·,·],\alpha =({\alpha }_1,{\alpha }_2))\to (T^{\prime },{[·,·,·]}^{\prime },{\alpha }^{\prime }=({\alpha }_1^{\prime },{\alpha }_2^{\prime }))$ of Hom-$\delta$-Jordan Lie supertriple system is a linear map satisfying $f\left([x,y,z]\right)={[f\left(x\right),f\left(y\right),f\left(z\right)]}^{\prime }$ and $f\circ {\alpha }_i={\alpha }_i^{\prime }\circ f$ for $i=1,2$. An isomorphism is a bijective morphism.

Following the representation theory of Lie triple systems was studied by Yamaguti, we generalize the notion of the representation to Hom-$\delta$-Jordan Lie supertriple systems.

Definition 4.

Let $(T,[·,·,·],\alpha )$ be a multiplicative Hom-δ-Jordan Lie supertriple system, V a ${\mathbb{Z}}_2$-graded vector space over $\mathbf{F}$, and $A\in \mathrm{End}\left(V\right)$. V is called a $(T,[·,·,·],\alpha )$-module with respect to A if there exists a bilinear map $\theta :T\times T\to \mathrm{End}\left(V\right)$, $(a,b)\mapsto \theta (a,b)$ such that for all $a,b,c,d\in T$,

$$\begin{array}{c}\theta \left(\alpha \right(a),\alpha (b\left)\right)\circ A=A\circ \theta (a,b),\\ {(-1)}^{\left(\right|a|+|b\left|\right)\left(\right|c|+|d\left|\right)}\theta (\alpha \left(c\right),\alpha \left(d\right))\theta (a,b)-\delta {(-1)}^{\left|a\right|\left|b\right|+\left|d\right|\left(\right|a|+|c\left|\right)}\theta (\alpha \left(b\right),\alpha \left(d\right))\theta (a,c)\end{array}$$

(7)

$$\begin{array}{c}-\theta (\alpha \left(a\right),[b,c,d])\circ A+{(-1)}^{\left|a\right|\left(\right|b|+|c\left|\right)}D(\alpha \left(b\right),\alpha \left(c\right))\theta (a,d)=0,\\ \delta {(-1)}^{\left(\right|a|+|b\left|\right)\left(\right|c|+|d\left|\right)}\theta (\alpha \left(c\right),\alpha \left(d\right))D(a,b)-\delta D(\alpha \left(a\right),\alpha \left(b\right))\theta (c,d)\end{array}$$

(8)

$$\begin{array}{c}+\theta ([a,b,c],\alpha \left(d\right))\circ A+\delta {(-1)}^{\left|c\right|\left(\right|a|+|b\left|\right)}\theta (\alpha \left(c\right),[a,b,d])\circ A=0,\\ \delta {(-1)}^{\left(\right|a|+|b\left|\right)\left(\right|c|+|d\left|\right)}D(\alpha \left(c\right),\alpha \left(d\right))D(a,b)-D(\alpha \left(a\right),\alpha \left(b\right))D(c,d)\end{array}$$

(9)

$$\begin{array}{c}+\delta D([a,b,c],\alpha \left(d\right))\circ A+\delta {(-1)}^{\left|c\right|\left(\right|a|+|b\left|\right)}D(\alpha \left(c\right),[a,b,d])\circ A=0,\end{array}$$

(10)

where $D(a,b)={(-1)}^{\left|a\right|\left|b\right|}\theta (b,a)-\delta \theta (a,b)$.

Then, θ is called the representation of $(T,[·,·,·],\alpha )$ on V with respect to A. In the case $\theta =0$, V is called the trivial $(T,[·,·,·],\alpha )$-module with respect to A.

In particular, let $V=T$, $A=\alpha$, and $\theta (x,y)\left(z\right)={(-1)}^{\left|z\right|\left(\right|x|+|y\left|\right)}[z,x,y]$. Then, $D(x,y)\left(z\right)=\delta [x,y,z]$ and (7)–(10) hold. In this case, T is said to be the adjoint $(T,[·,·,·],\alpha )$-module and $\theta$ is called the adjoint representation of $(T,[·,·,·],\alpha )$ on itself with respect to $\alpha$.

As is the case of general algebras, we introduce the semidirect product of a multiplicative Hom-$\delta$-Jordan Lie supertriple systems and its module.

Proposition 1.

Let θ be a representation of a multiplicative Hom-δ-Jordan Lie supertriple system $(T,[·,·,·],\alpha )$ on V with respect to A. Define the operation ${[·,·,·]}_V:(T\oplus V)\times (T\oplus V)\times (T\oplus V)\to T\oplus V$ by

$${[(x,a),(y,b),(z,c)]}_V=([x,y,z],{(-1)}^{\left|x\right|\left(\right|y|+|z\left|\right)}\theta (y,z)\left(a\right)-\delta {(-1)}^{\left|y\right|\left|z\right|}\theta (x,z)\left(b\right)+\delta D(x,y)\left(c\right));$$

and define the twisted map $\alpha +A:T\oplus V\to T\oplus V$ by

$$(\alpha +A)(x,a)=\left(\alpha \right(x),A(a\left)\right).$$

Then $T\oplus V$ is a multiplicative Hom-δ-Jordan Lie supertriple system.

Proof. 

Using $D(x,y)={(-1)}^{\left|x\right|\left|y\right|}\theta (y,x)-\delta \theta (x,y)$, we obtain

$$\begin{array}{cc}\hfill & {[(x,a),(y,b),(z,c)]}_V\hfill \\ \hfill =& ([x,y,z],{(-1)}^{\left|x\right|\left(\right|y|+|z\left|\right)}\theta (y,z)\left(a\right)-\delta {(-1)}^{\left|y\right|\left|z\right|}\theta (x,z)\left(b\right)+\delta D(x,y)\left(c\right))\hfill \\ \hfill =& -\delta {(-1)}^{\left|x\right|\left|y\right|}([y,x,z],{(-1)}^{\left|y\right|\left(\right|x|+|z\left|\right)}\theta (x,z)\left(b\right)-\delta {(-1)}^{\left|x\right|\left|z\right|}\theta (y,z)\left(a\right)-{(-1)}^{\left|x\right|\left|y\right|}D(x,y)\left(c\right))\hfill \\ \hfill =& -\delta {(-1)}^{\left|x\right|\left|y\right|}([y,x,z],{(-1)}^{\left|y\right|\left(\right|x|+|z\left|\right)}\theta (x,z)\left(b\right)-\delta {(-1)}^{\left|x\right|\left|z\right|}\theta (y,z)\left(a\right)+\delta D(y,x)\left(c\right))\hfill \\ \hfill =& -\delta {(-1)}^{\left|x\right|\left|y\right|}{[(y,b),(x,a),(z,c)]}_V,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {(-1)}^{\left|x\right|\left|z\right|}{[(x,a),(y,b),(z,c)]}_V+{(-1)}^{\left|y\right|\left|x\right|}{[(y,b),(z,c),(x,a)]}_V+{(-1)}^{\left|z\right|\left|y\right|}{[(z,c),(x,a),(y,b)]}_V\hfill \\ \hfill =& ({(-1)}^{\left|x\right|\left|z\right|}[x,y,z],{(-1)}^{\left|x\right|\left|y\right|}\theta (y,z)\left(a\right)-\delta {(-1)}^{\left|z\right|\left(\right|x|+|y\left|\right)}\theta (x,z)\left(b\right)+\delta {(-1)}^{\left|x\right|\left|z\right|}D(x,y)\left(c\right))\hfill \\ \hfill & +({(-1)}^{\left|y\right|\left|x\right|}[y,z,x],{(-1)}^{\left|y\right|\left|z\right|}\theta (z,x)\left(b\right)-\delta {(-1)}^{\left|x\right|\left(\right|y|+|z\left|\right)}\theta (y,x)\left(c\right)+\delta {(-1)}^{\left|y\right|\left|x\right|}D(y,z)\left(a\right))\hfill \\ \hfill & +({(-1)}^{\left|z\right|\left|y\right|}[z,x,y],{(-1)}^{\left|z\right|\left|x\right|}\theta (x,y)\left(c\right)-\delta {(-1)}^{\left|y\right|\left(\right|x|+|z\left|\right)}\theta (z,y)\left(a\right)+\delta {(-1)}^{\left|z\right|\left|y\right|}D(z,x)\left(b\right))\hfill \\ \hfill =& (0,{(-1)}^{\left|x\right|\left|y\right|}\theta (y,z)\left(a\right)-\delta {(-1)}^{\left|y\right|\left(\right|x|+|z\left|\right)}\theta (z,y)\left(a\right)+\delta {(-1)}^{\left|y\right|\left|x\right|}D(y,z)\left(a\right)\hfill \\ \hfill & +{(-1)}^{\left|y\right|\left|z\right|}\theta (z,x)\left(b\right)-\delta {(-1)}^{\left|z\right|\left(\right|x|+|y|}\theta (x,z)\left(b\right)+\delta {(-1)}^{\left|z\right|\left|y\right|}D(z,x)\left(b\right)\hfill \\ \hfill & +{(-1)}^{\left|z\right|\left|x\right|}\theta (x,y)\left(c\right)-\delta {(-1)}^{\left|x\right|\left(\right|y|+|z\left|\right)}\theta (y,x)\left(c\right)+\delta {(-1)}^{\left|x\right|\left|z\right|}D(x,y)\left(c\right))\hfill \\ \hfill =& (0,0).\hfill \end{array}$$

By (8)–(10), it follows that

$$\begin{array}{cc}\hfill & {[{[(x,a),(y,b),(u,c)]}_V,(\alpha +A)(v,d),(\alpha +A)(w,e)]}_V\hfill \\ \hfill =& [([x,y,u],{(-1)}^{\left|x\right|\left(\right|y|+|u\left|\right)}\theta (y,u)\left(a\right)-\delta {(-1)}^{\left|y\right|\left|u\right|}\theta (x,u)\left(b\right)+\delta D(x,y)\left(c\right)),\hfill \\ \hfill & {(\alpha \left(v\right),A\left(d\right)),(\alpha \left(w\right),A\left(e\right))]}_V\hfill \\ \hfill =& {([[x,y,u],\alpha \left(v\right),\alpha \left(w\right)],(-1)}^{\left(\right|x|+|y|+|u\left|\right)\left(\right|v|+|w\left|\right)}{\theta (\alpha \left(v\right),\alpha \left(w\right))((-1)}^{\left|x\right|\left(\right|y|+|u\left|\right)}\theta (y,u)\left(a\right)\hfill \\ \hfill & -\delta {(-1)}^{\left|y\right|\left|u\right|}{\theta (x,u)\left(b\right)+\delta D(x,y)\left(c\right))-\delta (-1)}^{\left|v\right|\left|w\right|}\theta ([x,y,u],\alpha \left(w\right))\left(A\left(d\right)\right)\hfill \\ \hfill & +\delta D\left(\right[x,y,u],\alpha (v\left)\right)\left(A\right(e\left)\right)),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {(-1)}^{\left|u\right|\left(\right|x|+|y\left|\right)}{[(\alpha +A)(u,c),{[(x,a),(y,b),(v,d)]}_V,(\alpha +A)(w,e)]}_V\hfill \\ \hfill =& {(-1)}^{\left|u\right|\left(\right|x|+|y\left|\right)}[(\alpha \left(u\right),A\left(c\right)),([x,y,v],{(-1)}^{\left|x\right|\left(\right|y|+|v\left|\right)}\theta (y,v)\left(a\right)-\delta {(-1)}^{\left|y\right|\left|v\right|}\theta (x,v)\left(b\right)\hfill \\ \hfill & +{\delta D(x,y)\left(d\right)),(\alpha \left(w\right),A\left(e\right))]}_V\hfill \\ \hfill =& {(-1)}^{\left|u\right|\left(\right|x|+|y\left|\right)}{([\alpha \left(u\right),[x,y,v],\alpha \left(w\right)],(-1)}^{\left|u\right|\left(\right|x|+|y|+|v|+|w\left|\right)}\theta ([x,y,v],\alpha \left(w\right))\left(A\left(c\right)\right)\hfill \\ \hfill & -\delta {(-1)}^{\left(\right|x|+|y|+|v\left|\right)\left|w\right|}{\theta (\alpha \left(u\right),\alpha \left(w\right))((-1)}^{\left|x\right|\left(\right|y|+|v\left|\right)}\theta (y,v)\left(a\right)-\delta {(-1)}^{\left|y\right|\left|v\right|}\theta (x,v)\left(b\right)\hfill \\ \hfill & +\delta D(x,y)\left(d\right))+\delta D(\alpha \left(u\right),[x,y,v]\left)\right(A\left(e\right)\left)\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \delta {(-1)}^{\left(\right|x|+|y\left|\right)\left(\right|u|+|v\left|\right)}{[(\alpha +A)(u,c),(\alpha +A)(v,d),{[(x,a),(y,b),(w,e)]}_V]}_V\hfill \\ \hfill =& \delta {(-1)}^{\left(\right|x|+|y\left|\right)\left(\right|u|+|v\left|\right)}[(\alpha \left(u\right),A\left(c\right)),(\alpha \left(v\right),A\left(d\right)),([x,y,w],{(-1)}^{\left|x\right|\left(\right|y|+|w\left|\right)}\theta (y,w)\left(a\right)\hfill \\ \hfill & -\delta {(-1)}^{\left|y\right|\left|w\right|}\theta (x,w)\left(b\right)+\delta D(x,y)\left(e\right){\left)\right]}_V\hfill \\ \hfill =& \delta {(-1)}^{\left(\right|x|+|y\left|\right)\left(\right|u|+|v\left|\right)}{([\alpha \left(u\right),\alpha \left(v\right),[x,y,w]],(-1)}^{\left|u\right|\left(\right|v|+|x|+|y|+|w\left|\right)}\theta (\alpha \left(v\right),[x,y,w])\left(A\left(c\right)\right)\hfill \\ \hfill & -\delta {(-1)}^{\left|v\right|\left(\right|x|+|y|+|w\left|\right)}{\theta (\alpha \left(u\right),[x,y,w])\left(A\left(d\right)\right)+\delta D(\alpha \left(u\right),\alpha \left(v\right))((-1)}^{\left|x\right|\left(\right|y|+|w\left|\right)}\theta (y,w)\left(a\right)\hfill \\ \hfill & -\delta {(-1)}^{\left|y\right|\left|w\right|}\theta (x,w)\left(b\right)+\delta D(x,y)\left(e\right)\left)\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {[(\alpha +A)(x,a),(\alpha +A)(y,b),{[(u,c),(v,d),(w,e)]}_V]}_V\hfill \\ \hfill =& {[(\alpha \left(x\right),A\left(a\right)),(\alpha \left(y\right),A\left(b\right)),([u,v,w],(-1)}^{\left|u\right|\left(\right|v|+|w\left|\right)}\theta (v,w)\left(c\right)-\delta {(-1)}^{\left|v\right|\left|w\right|}\theta (u,w)\left(d\right)\hfill \\ \hfill & +{\delta D(u,v)\left(e\right)\left)\right]}_V\hfill \\ \hfill =& ([\alpha \left(x\right),\alpha \left(y\right),[u,v,w]],{(-1)}^{\left|x\right|\left(\right|y|+|u|+|v|+|w\left|\right)}\theta (\alpha \left(y\right),[u,v,w])\left(A\left(a\right)\right)\hfill \\ \hfill & -\delta {(-1)}^{\left|y\right|\left(\right|u|+|v|+|w\left|\right)}{\theta (\alpha \left(x\right),[u,v,w])\left(A\left(b\right)\right)+\delta D(\alpha \left(x\right),\alpha \left(y\right))((-1)}^{\left|u\right|\left(\right|v|+|w\left|\right)}\theta (v,w)\left(c\right)\hfill \\ \hfill & -\delta {(-1)}^{\left|v\right|\left|w\right|}\theta (u,w)\left(d\right)+\delta D(u,v)\left(e\right)\left)\right).\hfill \end{array}$$

The calculation above shows that (1)–(3) hold.

Note that $\alpha +A$ is a linear map and, using (7), we have

$$\begin{array}{cc}\hfill & (\alpha +A){[(x,a),(y,b),(z,c)]}_V\hfill \\ \hfill =& (\alpha +A)([x,y,z],{(-1)}^{\left|x\right|\left(\right|y|+|z\left|\right)}\theta (y,z)\left(a\right)-\delta {(-1)}^{\left|y\right|\left|z\right|}\theta (x,z)\left(b\right)+\delta D(x,y)\left(c\right))\hfill \\ \hfill =& (\alpha \left([x,y,z]\right),A\circ ({(-1)}^{\left|x\right|\left(\right|y|+|z\left|\right)}\theta (y,z)\left(a\right)-\delta {(-1)}^{\left|y\right|\left|z\right|}\theta (x,z)\left(b\right)+\delta D(x,y)\left(c\right)))\hfill \\ \hfill =& ([\alpha \left(x\right),\alpha \left(y\right),\alpha \left(z\right)],{(-1)}^{\left|x\right|\left(\right|y|+|z\left|\right)}\theta (\alpha \left(y\right),\alpha \left(z\right))A\left(a\right)-\delta {(-1)}^{\left|y\right|\left|z\right|}\theta (\alpha \left(x\right),\alpha \left(z\right))A\left(b\right)\hfill \\ \hfill & +\delta D\left(\alpha \right(x),\alpha (y\left)\right)A\left(c\right))\hfill \\ \hfill =& {[(\alpha \left(x\right),A\left(a\right)),(\alpha \left(y\right),A\left(b\right)),(\alpha \left(z\right),A\left(c\right))]}_V\hfill \\ \hfill =& {[(\alpha +A)(x,a),(\alpha +A)(y,b),(\alpha +A)(z,c)]}_V.\hfill \end{array}$$

Thus, $(T\oplus V,{[·,·,·]}_V,\alpha +A)$ is a multiplicative Hom-$\delta$-Jordan Lie supertriple system. □

Let $\theta$ be a representation of $(T,[·,·,·],\alpha )$ on V with respect to A. If an n-linear map $f:\underset{n\mathrm{times}}{\underbrace{T\times \cdots \times T}}\to V$ satisfies

$$A\left(f(x_1,\cdots ,x_n)\right)=f(\alpha \left(x_1\right),\cdots ,\alpha \left(x_n\right)),$$

$$f(x_1,\cdots ,x,y,x_{n-2})=-{(-1)}^{\left|x\right|\left|y\right|}f(x_1,\cdots ,y,x,x_{n-2}),$$

$$\begin{array}{cc}\hfill & {(-1)}^{\left|x\right|\left|z\right|}f(x_1,\cdots ,x_{n-3},x,y,z)+{(-1)}^{\left|y\right|\left|x\right|}f(x_1,\cdots ,x_{n-3},y,z,x)\hfill \\ \hfill & +{(-1)}^{\left|z\right|\left|y\right|}f(x_1,\cdots ,x_{n-3},z,x,y)=0,\hfill \end{array}$$

then f is called an n-Hom-cochain on T. Denote by $C_{\alpha ,A}^n(T,V)$ the set of all n-Hom-cochains, $\forall n\ge 1$.

Definition 5.

For $n=1,2,3,4$, the coboundary operator $d_{hom}^n:C_{\alpha ,A}^n(T,V)\to C_{\alpha ,A}^{n+2}(T,V)$ is defined as follows.

• If $f\in C_{\alpha }^1(T,V),$ then

  $$\begin{array}{cc}\hfill & d_{hom}^{1}f(x_1,x_2,x_3)\hfill \\ \hfill =& {(-1)}^{\left(\right|f|+|x_1\left|\right)\left(\right|x_2|+|x_3\left|\right)}\theta (x_2,x_3)f\left(x_1\right)-\delta {(-1)}^{|x_2\left|\right|x_3|+|f\left|\right(|x_1|+|x_3\left|\right)}\hfill \\ \hfill & \theta (x_1,x_3)f\left(x_2\right)+\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_2\left|\right)}D(x_1,x_2)f\left(x_3\right)-f\left([x_1,x_2,x_3]\right).\hfill \end{array}$$
• If $f\in C_{\alpha }^2(T,V),$ then

  $$\begin{array}{cc}\hfill & d_{hom}^{2}f(y,x_1,x_2,x_3)\hfill \\ \hfill =& {(-1)}^{\left(\right|f|+|y|+|x_1\left|\right)\left(\right|x_2|+|x_3\left|\right)}\theta (\alpha \left(x_2\right),\alpha \left(x_3\right))f(y,x_1)-\delta {(-1)}^{|x_2\left|\right|x_3|+(\left|f\right|+|y_1\left|\right)\left(\right|x_1|+|x_3\left|\right)}\hfill \\ \hfill & \theta (\alpha \left(x_1\right),\alpha \left(x_3\right))f(y,x_2)+\delta {(-1)}^{\left(\right|f|+\left|y\right|\left)\right(|x_1|+|x_2\left|\right)}D(\alpha \left(x_1\right),\alpha \left(x_2\right))f(y,x_3)\hfill \\ \hfill & -f(\alpha \left(y\right),[x_1,x_2,x_3]).\hfill \end{array}$$
• If $f\in C_{\alpha }^3(T,V),$ then

  $$\begin{array}{cc}\hfill & d_{hom}^{3}f(x_1,x_2,x_3,x_4,x_5)\hfill \\ \hfill =& {(-1)}^{\left(\right|f|+|x_1|+|x_2|+|x_3\left|\right)\left(\right|x_4|+|x_5\left|\right)}\theta (\alpha \left(x_4\right),\alpha \left(x_5\right))f(x_1,x_2,x_3)\hfill \\ \hfill & -\delta {(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_5\left|\right)+|x_4\left|\right|x_5|}\theta (\alpha \left(x_3\right),\alpha \left(x_5\right))f(x_1,x_2,x_4)\hfill \\ \hfill & -\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_2\left|\right)}D(\alpha \left(x_1\right),\alpha \left(x_2\right))f(x_3,x_4,x_5)\hfill \\ \hfill & +{(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)}D(\alpha \left(x_3\right),\alpha \left(x_4\right))f(x_1,x_2,x_5)\hfill \\ \hfill & +f([x_1,x_2,x_3],\alpha \left(x_4\right),\alpha \left(x_5\right))+{(-1)}^{|x_3\left|\right(|x_1|+|x_2\left|\right)}f(\alpha \left(x_3\right),[x_1,x_2,x_4],\alpha \left(x_5\right))\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)}f(\alpha \left(x_3\right),\alpha \left(x_4\right),[x_1,x_2,x_5])-f(\alpha \left(x_1\right),\alpha \left(x_2\right),[x_3,x_4,x_5]).\hfill \end{array}$$
• If $f\in C_{\alpha }^4(T,V),$ then

  $$\begin{array}{cc}\hfill & d_{hom}^{4}f(y,x_1,x_2,x_3,x_4,x_5)\hfill \\ \hfill =& {(-1)}^{\left(\right|f|+|y|+|x_1|+|x_2|+|x_3\left|\right)\left(\right|x_4|+|x_5\left|\right)}\theta ({\alpha }^2\left(x_4\right),{\alpha }^2\left(x_5\right))f(y,x_1,x_2,x_3)\hfill \\ \hfill & -\delta {(-1)}^{\left(\right|f|+|y|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_5\left|\right)+|x_4\left|\right|x_5|}\theta ({\alpha }^2\left(x_3\right),{\alpha }^2\left(x_5\right))f(y,x_1,x_2,x_4)\hfill \\ \hfill & -\delta {(-1)}^{\left(\right|f|+\left|y\right|\left)\right(|x_1|+|x_2\left|\right)}D({\alpha }^2\left(x_1\right),{\alpha }^2\left(x_2\right))f(y,x_3,x_4,x_5)\hfill \\ \hfill & +{(-1)}^{\left(\right|f|+|y|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)}D({\alpha }^2\left(x_3\right),{\alpha }^2\left(x_4\right))f(y,x_1,x_2,x_5)\hfill \\ \hfill & +f(\alpha \left(y\right),[x_1,x_2,x_3],\alpha \left(x_4\right),\alpha \left(x_5\right))\hfill \\ \hfill & +{(-1)}^{|x_3\left|\right(|x_1|+|x_2\left|\right)}f(\alpha \left(y\right),\alpha \left(x_3\right),[x_1,x_2,x_4],\alpha \left(x_5\right))\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)}f(\alpha \left(y\right),\alpha \left(x_3\right),\alpha \left(x_4\right),[x_1,x_2,x_5])\hfill \\ \hfill & -f(\alpha \left(y\right),\alpha \left(x_1\right),\alpha \left(x_2\right),[x_3,x_4,x_5]).\hfill \end{array}$$

Theorem 1.

The coboundary operator $d_{hom}^n$ defined above satisfies $d_{hom}^{n+2}{d}_{hom}^n=0,$ $n=1,2.$

Proof. 

From the definition of the coboundary operator it follows immediately that $d_{hom}^3{d}_{hom}^1=0$ implies $d_{hom}^4{d}_{hom}^2=0$. Then we only need to prove $d_{hom}^3{d}_{hom}^1=0.$ In fact, by (7)–(10), we get

$$\begin{array}{cc}\hfill & d_{hom}^3\left(d_{hom}^{1}f\right)(x_1,x_2,x_3,x_4,x_5)\hfill \\ \hfill =& {(-1)}^{\left(\right|f|+|x_1|+|x_2|+|x_3\left|\right)\left(\right|x_4|+|x_5\left|\right)}\theta (\alpha \left(x_4\right),\alpha \left(x_5\right))\left(d_{hom}^{1}f\right)(x_1,x_2,x_3)\hfill \\ \hfill & -\delta {(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_5\left|\right)+|x_4\left|\right|x_5|}\theta (\alpha \left(x_3\right),\alpha \left(x_5\right))\left(d_{hom}^{1}f\right)(x_1,x_2,x_4)\hfill \\ \hfill & -\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_2\left|\right)}D(\alpha \left(x_1\right),\alpha \left(x_2\right))\left(d_{hom}^{1}f\right)(x_3,x_4,x_5)\hfill \\ \hfill & +{(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)}D(\alpha \left(x_3\right),\alpha \left(x_4\right))\left(d_{hom}^{1}f\right)(x_1,x_2,x_5)\hfill \\ \hfill & +\left(d_{hom}^{1}f\right)([x_1,x_2,x_3],\alpha \left(x_4\right),\alpha \left(x_5\right))+{(-1)}^{|x_3\left|\right(|x_1|+|x_2\left|\right)}\left(d_{hom}^{1}f\right)(\alpha \left(x_3\right),[x_1,x_2,x_4],\alpha \left(x_5\right))\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)}\left(d_{hom}^{1}f\right)(\alpha \left(x_3\right),\alpha \left(x_4\right),[x_1,x_2,x_5])\hfill \\ \hfill & -\left(d_{hom}^{1}f\right)(\alpha \left(x_1\right),\alpha \left(x_2\right),[x_3,x_4,x_5])\hfill \\ \hfill =& {(-1)}^{\left(\right|f|+|x_1|+|x_2|+|x_3\left|\right)\left(\right|x_4|+|x_5\left|\right)}\theta (\alpha \left(x_4\right),\alpha \left(x_5\right))({(-1)}^{\left(\right|f|+|x_1\left|\right)\left(\right|x_2|+|x_3\left|\right)}\theta (x_2,x_3)f\left(x_1\right)\hfill \\ \hfill & -\delta {(-1)}^{|x_2\left|\right|x_3|+|f\left|\right(|x_1|+|x_3\left|\right)}\theta (x_1,x_3)f\left(x_2\right)+\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_2\left|\right)}D(x_1,x_2)f\left(x_3\right)-f\left([x_1,x_2,x_3]\right))\hfill \\ \hfill & -\delta {(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_5\left|\right)+|x_4\left|\right|x_5|}\theta (\alpha \left(x_3\right),\alpha \left(x_5\right))({(-1)}^{\left(\right|f|+|x_1\left|\right)\left(\right|x_2|+|x_4\left|\right)}\theta (x_2,x_4)f\left(x_1\right)\hfill \\ \hfill & -\delta {(-1)}^{|x_2\left|\right|x_4|+|f\left|\right(|x_1|+|x_4\left|\right)}\theta (x_1,x_4)f\left(x_2\right)+\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_2\left|\right)}D(x_1,x_2)f\left(x_4\right)-f\left([x_1,x_2,x_4]\right))\hfill \\ \hfill & -\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_2\left|\right)}D(\alpha \left(x_1\right),\alpha \left(x_2\right))({(-1)}^{\left(\right|f|+|x_3\left|\right)\left(\right|x_4|+|x_5\left|\right)}\theta (x_4,x_5)f\left(x_3\right)\hfill \\ \hfill & -\delta {(-1)}^{|x_4\left|\right|x_5|+|f\left|\right(|x_3|+|x_5\left|\right)}\theta (x_3,x_5)f\left(x_4\right)+\delta {(-1)}^{\left|f\right|\left(\right|x_3|+|x_4\left|\right)}D(x_3,x_4)f\left(x_5\right)-f\left([x_3,x_4,x_5]\right))\hfill \\ \hfill & +{(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)}D(\alpha \left(x_3\right),\alpha \left(x_4\right))({(-1)}^{\left(\right|f|+|x_1\left|\right)\left(\right|x_2|+|x_5\left|\right)}\theta (x_2,x_5)f\left(x_1\right)\hfill \\ \hfill & -\delta {(-1)}^{|x_2\left|\right|x_5|+|f\left|\right(|x_1|+|x_5\left|\right)}\theta (x_1,x_5)f\left(x_2\right)+\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_2\left|\right)}D(x_1,x_2)f\left(x_5\right)-f\left([x_1,x_2,x_5]\right))\hfill \\ \hfill & +{(-1)}^{\left(\right|f|+|x_1|+|x_2|+|x_3\left|\right)\left(\right|x_4|+|x_5\left|\right)}(\theta (\alpha \left(x_4\right),\alpha \left(x_5\right))f\left([x_1,x_2,x_3]\right)\hfill \\ \hfill & -\delta {(-1)}^{|x_4\left|\right|x_5|+|f\left|\right(|x_1|+|x_2|+|x_3|+|x_5\left|\right)}\theta ([x_1,x_2,x_3],\alpha \left(x_5\right))f\left(\alpha \left(x_4\right)\right)\hfill \\ \hfill & +\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_2|+|x_3|+|x_4\left|\right)}D([x_1,x_2,x_3],\alpha \left(x_4\right))f\left(\alpha \left(x_5\right)\right)-f\left([[x_1,x_2,x_3],\alpha \left(x_4\right),\alpha \left(x_5\right)]\right))\hfill \\ \hfill & +{(-1)}^{|x_3\left|\right(|x_1|+|x_2\left|\right)+\left(\right|f|+|x_3\left|\right)\left(\right|x_1|+|x_2|+|x_4|+|x_5\left|\right)}(\theta ([x_1,x_2,x_4],\alpha \left(x_5\right))f\left(\alpha \left(x_3\right)\right)\hfill \\ \hfill & -\delta {(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_5\left|\right)+|x_4\left|\right|x_5|}\theta (\alpha \left(x_3\right),\alpha \left(x_5\right))f\left([x_1,x_2,x_4]\right)\hfill \\ \hfill & +\delta {(-1)}^{|x_3\left|\right(|x_1|+|x_2\left|\right)+\left|f\right|\left(\right|x_1|+|x_2|+|x_3|+|x_4\left|\right)}D(\alpha \left(x_3\right),[x_1,x_2,x_4])f\left(\alpha \left(x_5\right)\right)\hfill \\ \hfill & -{(-1)}^{|x_3\left|\right(|x_1|+|x_2\left|\right)}f\left([\alpha \left(x_3\right),[x_1,x_2,x_4],\alpha \left(x_5\right)]\right))\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)}{((-1)}^{\left(\right|f|+|x_3\left|\right)\left(\right|x_1|+|x_2|+|x_4|+|x_5\left|\right)}\theta (\alpha \left(x_4\right),[x_1,x_2,x_5])f\left(\alpha \left(x_3\right)\right)\hfill \\ \hfill & -\delta {(-1)}^{|x_4\left|\right(|x_1|+|x_2|+|x_5\left|\right)+\left|f\right|\left(\right|x_1|+|x_2|+|x_3|+|x_5\left|\right)}\theta (\alpha \left(x_3\right),[x_1,x_2,x_5])f\left(\alpha \left(x_4\right)\right)\hfill \\ \hfill & +\delta {(-1)}^{\left|f\right|\left(\right|x_3|+|x_4\left|\right)}D(\alpha \left(x_3\right),\alpha \left(x_4\right))f\left([x_1,x_2,x_5]\right)-f\left([\alpha \left(x_3\right),\alpha \left(x_4\right),[x_1,x_2,x_5]]\right))\hfill \\ \hfill & -({(-1)}^{\left(\right|f|+|x_1\left|\right)\left(\right|x_2|+|x_3|+|x_4|+|x_5\left|\right)}\theta (\alpha \left(x_2\right),[x_3,x_4,x_5])f\left(\alpha \left(x_1\right)\right)\hfill \\ \hfill & -\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_3|+|x_4|+|x_5\left|\right)+|x_2\left|\right(|x_3|+|x_4|+|x_5\left|\right)}\theta (\alpha \left(x_1\right),[x_3,x_4,x_5])f\left(\alpha \left(x_2\right)\right)\hfill \\ \hfill & +\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_2\left|\right)}D(\alpha \left(x_1\right),\alpha \left(x_2\right))f\left([x_3,x_4,x_5]\right)-f\left([\alpha \left(x_1\right),\alpha \left(x_2\right),[x_3,x_4,x_5]]\right))\hfill \\ \hfill =& -f\left([[x_1,x_2,x_3],\alpha \left(x_4\right),\alpha \left(x_5\right)]\right)-{(-1)}^{|x_3\left|\right(|x_1|+|x_2\left|\right)}f\left([\alpha \left(x_3\right),[x_1,x_2,x_4],\alpha \left(x_5\right)]\right)\hfill \\ \hfill & -\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)}f\left([\alpha \left(x_3\right),\alpha \left(x_4\right),[x_1,x_2,x_5]]\right)+f\left([\alpha \left(x_1\right),\alpha \left(x_2\right),[x_3,x_4,x_5]]\right)\hfill \\ \hfill & +{(-1)}^{\left(\right|f|+|x_1|+|x_2|+|x_3\left|\right)\left(\right|x_4|+|x_5\left|\right)+\left(\right|f|+|x_1\left|\right)\left(\right|x_2|+|x_3\left|\right)}\theta (\alpha \left(x_4\right),\alpha \left(x_5\right))\theta (x_2,x_3)f\left(x_1\right)\hfill \\ \hfill & -\delta {(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_5\left|\right)+|x_4\left|\right|x_5|+(\left|f\right|+|x_1\left|\right)\left(\right|x_2|+|x_4\left|\right)}\theta (\alpha \left(x_3\right),\alpha \left(x_5\right))\theta (x_2,x_4)f\left(x_1\right)\hfill \\ \hfill & +{(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)+\left(\right|f|+|x_1\left|\right)\left(\right|x_2|+|x_5\left|\right)}D(\alpha \left(x_3\right),\alpha \left(x_4\right))\theta (x_2,x_5)f\left(x_1\right)\hfill \\ \hfill & -{(-1)}^{\left(\right|f|+|x_1\left|\right)\left(\right|x_2|+|x_3|+|x_4|+|x_5\left|\right)}\theta (\alpha \left(x_2\right),[x_3,x_4,x_5])f\left(\alpha \left(x_1\right)\right)\hfill \\ \hfill & -\delta {(-1)}^{\left(\right|f|+|x_1|+|x_2|+|x_3\left|\right)\left(\right|x_4|+|x_5\left|\right)+|x_2\left|\right|x_3|+|f\left|\right(|x_1|+|x_3\left|\right)}\theta (\alpha \left(x_4\right),\alpha \left(x_5\right))\theta (x_1,x_3)f\left(x_2\right)\hfill \\ \hfill & +{(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_5\left|\right)+|x_4\left|\right(|x_2|+|x_5\left|\right)+\left|f\right|\left(\right|x_1|+|x_4\left|\right)}\theta (\alpha \left(x_3\right),\alpha \left(x_5\right))\theta (x_1,x_4)f\left(x_2\right)\hfill \\ \hfill & -\delta {(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)+|x_2\left|\right|x_5|+|f\left|\right(|x_1|+|x_5\left|\right)}D(\alpha \left(x_3\right),\alpha \left(x_4\right))\theta (x_1,x_5)f\left(x_2\right)\hfill \\ \hfill & +\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_3|+|x_4|+|x_5\left|\right)+|x_2\left|\right(|x_3|+|x_4|+|x_5\left|\right)}\theta (\alpha \left(x_1\right),[x_3,x_4,x_5])f\left(\alpha \left(x_2\right)\right)\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|f|+|x_1|+|x_2|+|x_3\left|\right)\left(\right|x_4|+|x_5\left|\right)+\left|f\right|\left(\right|x_1|+|x_2\left|\right)}\theta (\alpha \left(x_4\right),\alpha \left(x_5\right))D(x_1,x_2)f\left(x_3\right)\hfill \\ \hfill & -\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_2\left|\right)+\left(\right|f|+|x_3\left|\right)\left(\right|x_4|+|x_5\left|\right)}D(\alpha \left(x_1\right),\alpha \left(x_2\right))\theta (x_4,x_5)f\left(x_3\right)\hfill \\ \hfill & +{(-1)}^{|x_3\left|\right(|x_1|+|x_2\left|\right)+\left(\right|f|+|x_3\left|\right)\left(\right|x_1|+|x_2|+|x_4|+|x_5\left|\right)}\theta ([x_1,x_2,x_4],\alpha \left(x_5\right))f\left(\alpha \left(x_3\right)\right)\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)+\left(\right|f|+|x_3\left|\right)\left(\right|x_1|+|x_2|+|x_4|+|x_5\left|\right)}\theta (\alpha \left(x_4\right),[x_1,x_2,x_5])f\left(\alpha \left(x_3\right)\right)\hfill \\ \hfill & -{(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_5\left|\right)+|x_4\left|\right|x_5|+|f\left|\right(|x_1|+|x_2\left|\right)}\theta (\alpha \left(x_3\right),\alpha \left(x_5\right))D(x_1,x_2)f\left(x_4\right)\hfill \\ \hfill & +{(-1)}^{\left|f\right|\left(\right|x_1|+|x_2\left|\right)+|x_4\left|\right|x_5|+|f\left|\right(|x_3|+|x_5\left|\right)}D(\alpha \left(x_1\right),\alpha \left(x_2\right))\theta (x_3,x_5)f\left(x_4\right)\hfill \\ \hfill & -\delta {(-1)}^{|x_4\left|\right|x_5|+|f\left|\right(|x_1|+|x_2|+|x_3|+|x_5\left|\right)}\theta ([x_1,x_2,x_3],\alpha \left(x_5\right))f\left(\alpha \left(x_4\right)\right)\hfill \\ \hfill & -{(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)+|x_4\left|\right(|x_1|+|x_2|+|x_5\left|\right)+\left|f\right|\left(\right|x_1|+|x_2|+|x_3|+|x_5\left|\right)}\theta (\alpha \left(x_3\right),[x_1,x_2,x_5])f\left(\alpha \left(x_4\right)\right)\hfill \\ \hfill & -{(-1)}^{\left|f\right|\left(\right|x_1|+|x_2|+|x_3|+|x_4\left|\right)}D(\alpha \left(x_1\right),\alpha \left(x_2\right))D(x_3,x_4)f\left(x_5\right)\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)+\left|f\right|\left(\right|x_1|+|x_2\left|\right)}D(\alpha \left(x_3\right),\alpha \left(x_4\right))D(x_1,x_2)f\left(x_5\right)\hfill \\ \hfill & +\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_2|+|x_3|+|x_4\left|\right)}D([x_1,x_2,x_3],\alpha \left(x_4\right))f\left(\alpha \left(x_5\right)\right)\hfill \\ \hfill & +\delta {(-1)}^{|x_3\left|\right(|x_1|+|x_2\left|\right)+\left|f\right|\left(\right|x_1|+|x_2|+|x_3|+|x_4\left|\right)}D(\alpha \left(x_3\right),[x_1,x_2,x_4])f\left(\alpha \left(x_5\right)\right)\hfill \\ \hfill & -{(-1)}^{\left(\right|f|+|x_1|+|x_2|+|x_3\left|\right)\left(\right|x_4|+|x_5\left|\right)}\theta (\alpha \left(x_4\right),\alpha \left(x_5\right))f\left([x_1,x_2,x_3]\right)\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_5\left|\right)+|x_4\left|\right|x_5|}\theta (\alpha \left(x_3\right),\alpha \left(x_5\right))f\left([x_1,x_2,x_4]\right)\hfill \\ \hfill & +\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_2\left|\right)}D(\alpha \left(x_1\right),\alpha \left(x_2\right))f\left([x_3,x_4,x_5]\right)\hfill \\ \hfill & -{(-1)}^{\left|f\right|\left(\right|x_3|+|x_4\left|\right)+\left(\right|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)}D(\alpha \left(x_3\right),\alpha \left(x_4\right))f\left([x_1,x_2,x_5]\right)\hfill \\ \hfill & +{(-1)}^{\left(\right|f|+|x_1|+|x_2|+|x_3\left|\right)\left(\right|x_4|+|x_5\left|\right)}\theta (\alpha \left(x_4\right),\alpha \left(x_5\right))f\left([x_1,x_2,x_3]\right)\hfill \\ \hfill & -\delta {(-1)}^{\left(\right|f|+|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_5\left|\right)+|x_4\left|\right|x_5|}\theta (\alpha \left(x_3\right),\alpha \left(x_5\right))f\left([x_1,x_2,x_4]\right)\hfill \\ \hfill & +{(-1)}^{\left|f\right|\left(\right|x_3|+|x_4\left|\right)+\left(\right|x_1|+|x_2\left|\right)\left(\right|x_3|+|x_4\left|\right)}D(\alpha \left(x_3\right),\alpha \left(x_4\right))f\left([x_1,x_2,x_5]\right)\hfill \\ \hfill & -\delta {(-1)}^{\left|f\right|\left(\right|x_1|+|x_2\left|\right)}D(\alpha \left(x_1\right),\alpha \left(x_2\right))f\left([x_3,x_4,x_5]\right)\hfill \\ \hfill =& 0.\hfill \end{array}$$

Therefore, the proof is complete. □

For $n=1,2,3,4$, the map $f\in C_{\alpha ,A}^n(T,V)$ is called an n-Hom-cocycle if $d_{hom}^{n}f=0$. We denote by $Z_{\alpha ,A}^n(T,V)$ the subspace spanned by n-Hom-cocycles and $B_{\alpha ,A}^n(T,V)=d_{hom}^{n-2}{C}_{\alpha ,A}^{n-2}(T,V)$.

Since $d_{hom}^{n+2}{d}_{hom}^n=0$, $B_{\alpha ,A}^n(T,V)$ is a subspace of $Z_{\alpha ,A}^n(T,V)$. Hence, we can define a cohomology space $H_{\alpha ,A}^n(T,V)$ of $(T,[·,·,·],\alpha )$ as the factor space $Z_{\alpha ,A}^n(T,V)/B_{\alpha ,A}^n(T,V).$

3. Nijenhuis Operators of Hom-$\delta$-Jordan Lie Supertriple Systems

In this section, we study infinitesimal deformations of Hom-$\delta$-Jordan Lie supertriple systems. We introduce the notion of Nijenhuis operators for Hom-$\delta$-Jordan Lie supertriple systems, and obtain trivial deformations using this kind of Nijenhuis operators.

Let $(T,[·,·,·],\alpha )$ be a Hom-$\delta$-Jordan Lie supertriple system and $\psi :T\times T\times T\to T$ be an even trilinear map. Consider a $\lambda$-parametrized family of linear operations:

$${[x_1,x_2,x_3]}_{\lambda }=[x_1,x_2,x_3]+\lambda \psi (x_1,x_2,x_3),$$

where $\lambda$ is a formal variable, and $\lambda \ne 0.$

If ${[·,·,·]}_{\lambda }$ endow T with Hom-$\delta$-Jordan Lie supertriple system structure, which is denoted by $T_{\lambda }$, then we call that $\psi$ generates a $\lambda$-parameter infinitesimal deformation of Hom-$\delta$-Jordan Lie supertriple system.

Theorem 2.

ψ generates a λ-parameter infinitesimal deformation of Hom-δ-Jordan Lie supertriple system T is equivalent to (i) ψ itself defines a Hom-δ-Jordan Lie supertriple system structure on T and (ii) ψ is a 3-Hom-cocycle of T.

Proof. 

$${[x_1,x_2,x_3]}_{\lambda }=[x_1,x_2,x_3]+\lambda \psi (x_1,x_2,x_3),$$

and

$$-\delta {(-1)}^{|x_1\left|\right|x_2|}{[x_2,x_1,x_3]}_{\lambda }=-\delta {(-1)}^{|x_1\left|\right|x_2|}[x_2,x_1,x_3]-\delta \lambda {(-1)}^{|x_1\left|\right|x_2|}\psi (x_2,x_1,x_3),$$

we have

$$\begin{array}{c}\hfill \psi (x_1,x_2,x_3)=-\delta {(-1)}^{|x_1\left|\right|x_2|}\psi (x_2,x_1,x_3).\end{array}$$

(11)

From the equality

$$\begin{array}{cc}\hfill 0=& {(-1)}^{|x_1\left|\right|x_3|}{[x_1,x_2,x_3]}_{\lambda }+{(-1)}^{|x_2\left|\right|x_1|}{[x_2,x_3,x_1]}_{\lambda }+{(-1)}^{|x_3\left|\right|x_2|}{[x_3,x_1,x_2]}_{\lambda }\hfill \\ \hfill =& {(-1)}^{|x_1\left|\right|x_3|}[x_1,x_2,x_3]+{(-1)}^{|x_2\left|\right|x_1|}[x_2,x_3,x_1]+{(-1)}^{|x_3\left|\right|x_2|}[x_3,x_1,x_2]\hfill \\ \hfill & +\lambda ({(-1)}^{|x_1\left|\right|x_3|}\psi (x_1,x_2,x_3)+{(-1)}^{|x_2\left|\right|x_1|}\psi (x_2,x_3,x_1)+{(-1)}^{|x_3\left|\right|x_2|}\psi (x_3,x_1,x_2)),\hfill \end{array}$$

it follows that

$$\begin{array}{c}\hfill {(-1)}^{|x_1\left|\right|x_3|}\psi (x_1,x_2,x_3)+{(-1)}^{|x_2\left|\right|x_1|}\psi (x_2,x_3,x_1)+{(-1)}^{|x_3\left|\right|x_2|}\psi (x_3,x_1,x_2)=0.\end{array}$$

(12)

For the equality

$$\begin{array}{cc}\hfill & {[\alpha \left(x_1\right),\alpha \left(x_2\right),{[y_1,y_2,y_3]}_{\lambda }]}_{\lambda }\hfill \\ \hfill & ={[{[x_1,x_2,y_1]}_{\lambda },\alpha \left(y_2\right),\alpha \left(y_3\right)]}_{\lambda }+{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}{[\alpha \left(y_1\right),{[x_1,x_2,y_2]}_{\lambda },\alpha \left(y_3\right)]}_{\lambda }\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}{[\alpha \left(y_1\right),\alpha \left(y_2\right),{[x_1,x_2,y_3]}_{\lambda }]}_{\lambda },\hfill \end{array}$$

the left hand side is equal to

$$\begin{array}{cc}\hfill & {[\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,y_2,y_3]+\lambda \psi (y_1,y_2,y_3)]}_{\lambda }\hfill \\ \hfill =& [\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,y_2,y_3]]+\lambda \psi (\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,y_2,y_3])\hfill \\ \hfill & +[\alpha \left(x_1\right),\alpha \left(x_2\right),\lambda \psi (y_1,y_2,y_3)]+\lambda \psi (\alpha \left(x_1\right),\alpha \left(x_2\right),\lambda \psi (y_1,y_2,y_3))\hfill \\ \hfill =& [\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,y_2,y_3]]+\lambda (\psi (\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,y_2,y_3])+[\alpha \left(x_1\right),\alpha \left(x_2\right),\psi (y_1,y_2,y_3)])\hfill \\ \hfill & +{\lambda }^2\psi (\alpha \left(x_1\right),\alpha \left(x_2\right),\psi (y_1,y_2,y_3)),\hfill \end{array}$$

and the right hand side is equal to

$$\begin{array}{cc}\hfill & {[[x_1,x_2,y_1]+\lambda \psi (x_1,x_2,y_1),\alpha \left(y_2\right),\alpha \left(y_3\right)]}_{\lambda }+{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}[\alpha \left(y_1\right),[x_1,x_2,y_2]\hfill \\ \hfill & +\lambda \psi (x_1,x_2,y_2),\alpha \left(y_3\right){]}_{\lambda }+\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}{[\alpha \left(y_1\right),\alpha \left(y_2\right),[x_1,x_2,y_3]+\lambda \psi (x_1,x_2,y_3)]}_{\lambda }\hfill \\ \hfill =& [[x_1,x_2,y_1],\alpha \left(y_2\right),\alpha \left(y_3\right)]+{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}[\alpha \left(y_1\right),[x_1,x_2,y_2],\alpha \left(y_3\right)]\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}[\alpha \left(y_1\right),\alpha \left(y_2\right),[x_1,x_2,y_3]]+\lambda (\psi ([x_1,x_2,y_1],\alpha \left(y_2\right),\alpha \left(y_3\right))\hfill \\ \hfill & +[\psi (x_1,x_2,y_1),\alpha \left(y_2\right),\alpha \left(y_3\right)]+{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}\psi (\alpha \left(y_1\right),[x_1,x_2,y_2],\alpha \left(y_3\right))\hfill \\ \hfill & +{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}[\alpha \left(y_1\right),\psi (x_1,x_2,y_2),\alpha \left(y_3\right)]\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}\psi (\alpha \left(y_1\right),\alpha \left(y_2\right),[x_1,x_2,y_3])\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}[\alpha \left(y_1\right),\alpha \left(y_2\right),\psi (x_1,x_2,y_3)])+{\lambda }^2(\psi (\psi (x_1,x_2,y_1),\alpha \left(y_2\right),\alpha \left(y_3\right))\hfill \\ \hfill & +{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}\psi (\alpha \left(y_1\right),\psi (x_1,x_2,y_2),\alpha \left(y_3\right))\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}\psi (\alpha \left(y_1\right),\alpha \left(y_2\right),\psi (x_1,x_2,y_3))).\hfill \end{array}$$

Thus, we have

$$\begin{array}{cc}\hfill & \psi (\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,y_2,y_3])+\delta D(\alpha \left(x_1\right),\alpha \left(x_2\right))\psi (y_1,y_2,y_3)\hfill \\ \hfill =& \psi ([x_1,x_2,y_1],\alpha \left(y_2\right),\alpha \left(y_3\right))+{(-1)}^{\left(\right|x_1|+|x_2|+|y_1\left|\right)\left(\right|y_2|+|y_3\left|\right)}\theta (\alpha \left(y_2\right),\alpha \left(y_3\right))\psi (x_1,x_2,y_1)\hfill \\ \hfill & +{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}\psi (\alpha \left(y_1\right),[x_1,x_2,y_2],\alpha \left(y_3\right))\hfill \\ \hfill & -\delta {(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}\theta (\alpha \left(y_1\right),\alpha \left(y_3\right))\psi (x_1,x_2,y_2)\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}\psi (\alpha \left(y_1\right),\alpha \left(y_2\right),[x_1,x_2,y_3])\hfill \\ \hfill & +{(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}D(\alpha \left(y_1\right),\alpha \left(y_2\right))\psi (x_1,x_2,y_3).\hfill \end{array}$$

(13)

and

$$\begin{array}{cc}\hfill & \psi (\alpha \left(x_1\right),\alpha \left(x_2\right),\psi (y_1,y_2,y_3))\hfill \\ \hfill =& \psi (\psi (x_1,x_2,y_1),\alpha \left(y_2\right),\alpha \left(y_3\right))+{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}\psi (\alpha \left(y_1\right),\psi (x_1,x_2,y_2),\alpha \left(y_3\right))\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}\psi (\alpha \left(y_1\right),\alpha \left(y_2\right),\psi (x_1,x_2,y_3)).\hfill \end{array}$$

(14)

Therefore, $\psi$ defines a Hom-$\delta$-Jordan Lie supertriple system structure on T by (11)–(14). Furthermore, by (13) $\psi$ is a 3-Hom-cocycle. □

A deformation is said to be trivial if there exists an even linear map $N:T\to T$ such that for ${\phi }_{\lambda }=\mathrm{id}+\lambda N:T_{\lambda }\to T$ we have

$$\begin{array}{c}\hfill {\phi }_{\lambda }{[x_1,x_2,x_3]}_{\lambda }=[{\phi }_{\lambda }{x}_1,{\phi }_{\lambda }{x}_2,{\phi }_{\lambda }{x}_3].\end{array}$$

(15)

It is clear that

$$\begin{array}{cc}\hfill {\phi }_{\lambda }{[x_1,x_2,x_3]}_{\lambda }=& [x_1,x_2,x_3]+\lambda \psi (x_1,x_2,x_3)+\lambda N([x_1,x_2,x_3]+\lambda \psi (x_1,x_2,x_3))\hfill \\ \hfill =& [x_1,x_2,x_3]+\lambda (\psi (x_1,x_2,x_3)+N[x_1,x_2,x_3])+{\lambda }^{2}N\psi (x_1,x_2,x_3),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill [{\phi }_{\lambda }{x}_1,{\phi }_{\lambda }{x}_2,{\phi }_{\lambda }{x}_3]=& [x_1+\lambda N{x}_1,x_2+\lambda N{x}_2,x_3+\lambda N{x}_3]\hfill \\ \hfill =& [x_1,x_2,x_3]+\lambda ([N{x}_1,x_2,x_3]+[x_1,N{x}_2,x_3]+[x_1,x_2,N{x}_3])\hfill \\ \hfill & +{\lambda }^2([N{x}_1,N{x}_2,x_3]+[N{x}_1,x_2,N{x}_3]+[x_1,N{x}_2,N{x}_3])\hfill \\ \hfill & +{\lambda }^3[N{x}_1,N{x}_2,N{x}_3].\hfill \end{array}$$

Thus, we have

$$\begin{array}{cc}\hfill \psi (x_1,x_2,x_3)=& [N{x}_1,x_2,x_3]+[x_1,N{x}_2,x_3]+[x_1,x_2,N{x}_3]-N[x_1,x_2,x_3]\hfill \\ \hfill =& {(-1)}^{|x_1\left|\right(|x_2|+|x_3\left|\right)}\theta (x_2,x_3)N\left(x_1\right)-\delta {(-1)}^{|x_2\left|\right|x_3|}\theta (x_1,x_3)N\left(x_2\right)\hfill \\ \hfill & +\delta D(x_1,x_2)N\left(x_3\right)-N[x_1,x_2,x_3]\hfill \end{array}$$

(16)

$$\begin{array}{c}\hfill N\psi (x_1,x_2,x_3)=[N{x}_1,N{x}_2,x_3]+[N{x}_1,x_2,N{x}_3]+[x_1,N{x}_2,N{x}_3]\end{array}$$

(17)

$$\begin{array}{c}\hfill 0=[N{x}_1,N{x}_2,N{x}_3]\end{array}$$

(18)

From the cohomology theory discussed in Section 2, (16) can be represented in terms of 1-coboundary as $\psi =d_{hom}^{1}N$. Moreover, it follows from (16) and (17) that N must satisfy the following condition

$$\begin{array}{cc}\hfill N^2[x_1,x_2,x_3]=& N[N{x}_1,x_2,x_3]+N[x_1,N{x}_2,x_3]+N[x_1,x_2,N{x}_3]\hfill \\ \hfill & -([N{x}_1,N{x}_2,x_3]+[N{x}_1,x_2,N{x}_3]+[x_1,N{x}_2,N{x}_3]).\hfill \end{array}$$

(19)

In the following, we denote by

$$\begin{array}{c}\hfill \psi (x_1,x_2,x_3)={[x_1,x_2,x_3]}_N,\end{array}$$

(20)

then (17) is equivalent to

$$\begin{array}{c}\hfill N{[x_1,x_2,x_3]}_N=[N{x}_1,N{x}_2,x_3]+[N{x}_1,x_2,N{x}_3]+[x_1,N{x}_2,N{x}_3].\end{array}$$

(21)

Definition 6.

An even linear operator $N:T\to T$ is called a Nijenhuis operator if and only if $N\circ \alpha =\alpha \circ N$ and (18) and (19) hold.

Theorem 3.

Let N be a Nijenhuis operator for T. Then, a deformation of T can be obtained by putting

$$\begin{array}{cc}\hfill \psi (x_1,x_2,x_3)=& {(-1)}^{|x_1\left|\right(|x_2|+|x_3\left|\right)}\theta (x_2,x_3)N\left(x_1\right)-\delta {(-1)}^{|x_2\left|\right|x_3|}\theta (x_1,x_3)N\left(x_2\right)\hfill \\ \hfill & +\delta D(x_1,x_2)N\left(x_3\right)-N[x_1,x_2,x_3].\hfill \end{array}$$

Furthermore, this deformation is a trivial one.

Proof. 

It is obvious that $\psi =dN$ and $d\psi =d^{2}N=0$. Thus, $\psi$ is a 3-Hom-cocycle of T. Now, we check that (3) holds for $\psi$. Considering (16), (20) and (21), it follows that

$$\begin{array}{cc}\hfill & \psi (\alpha \left(x_1\right),\alpha \left(x_2\right),\psi (y_1,y_2,y_3))\hfill \\ \hfill =& {[\alpha \left(x_1\right),\alpha \left(x_2\right),[N{y}_1,y_2,y_3]+[y_1,N{y}_2,y_3]+[y_1,y_2,N{y}_3]-N[y_1,y_2,y_3]]}_N\hfill \\ \hfill =& {[\alpha \left(x_1\right),\alpha \left(x_2\right),[N{y}_1,y_2,y_3]]}_N+{[\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,N{y}_2,y_3]]}_N\hfill \\ \hfill & +{[\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,y_2,N{y}_3]]}_N-{[\alpha \left(x_1\right),\alpha \left(x_2\right),N[y_1,y_2,y_3]]}_N\hfill \\ \hfill =& [N\alpha \left(x_1\right),\alpha \left(x_2\right),[N{y}_1,y_2,y_3]]+[\alpha \left(x_1\right),N\alpha \left(x_2\right),[N{y}_1,y_2,y_3]]\hfill \\ \hfill & +[\alpha \left(x_1\right),\alpha \left(x_2\right),N[N{y}_1,y_2,y_3]]-N[\alpha \left(x_1\right),\alpha \left(x_2\right),[N{y}_1,y_2,y_3]]\hfill \\ \hfill & +[N\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,N{y}_2,y_3]]+[\alpha \left(x_1\right),N\alpha \left(x_2\right),[y_1,N{y}_2,y_3]]\hfill \\ \hfill & +[\alpha \left(x_1\right),\alpha \left(x_2\right),N[y_1,N{y}_2,y_3]]-N[\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,N{y}_2,y_3]]\hfill \\ \hfill & +[N\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,y_2,N{y}_3]]+[\alpha \left(x_1\right),N\alpha \left(x_2\right),[y_1,y_2,N{y}_3]]\hfill \\ \hfill & +[\alpha \left(x_1\right),\alpha \left(x_2\right),N[y_1,y_2,N{y}_3]]-N[\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,y_2,N{y}_3]]\hfill \\ \hfill & -[N\alpha \left(x_1\right),\alpha \left(x_2\right),N[y_1,y_2,y_3]]-[\alpha \left(x_1\right),N\alpha \left(x_2\right),N[y_1,y_2,y_3]]\hfill \\ \hfill & -[\alpha \left(x_1\right),\alpha \left(x_2\right),N^2[y_1,y_2,y_3]]+N[\alpha \left(x_1\right),\alpha \left(x_2\right),N[y_1,y_2,y_3]]\hfill \\ \hfill =& [N\alpha \left(x_1\right),\alpha \left(x_2\right),[N{y}_1,y_2,y_3]]+[\alpha \left(x_1\right),N\alpha \left(x_2\right),[N{y}_1,y_2,y_3]]\hfill \\ \hfill & -N[\alpha \left(x_1\right),\alpha \left(x_2\right),[N{y}_1,y_2,y_3]]+[N\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,N{y}_2,y_3]]\hfill \\ \hfill & +[\alpha \left(x_1\right),N\alpha \left(x_2\right),[y_1,N{y}_2,y_3]]-N[\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,N{y}_2,y_3]]\hfill \\ \hfill & +[N\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,y_2,N{y}_3]]+[\alpha \left(x_1\right),N\alpha \left(x_2\right),[y_1,y_2,N{y}_3]]\hfill \\ \hfill & -N[\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,y_2,N{y}_3]]-[N\alpha \left(x_1\right),\alpha \left(x_2\right),N[y_1,y_2,y_3]]\hfill \\ \hfill & -[\alpha \left(x_1\right),N\alpha \left(x_2\right),N[y_1,y_2,y_3]]+N[\alpha \left(x_1\right),\alpha \left(x_2\right),N[y_1,y_2,y_3]]\hfill \\ \hfill & +[\alpha \left(x_1\right),\alpha \left(x_2\right),[N{y}_1,N{y}_2,y_3]+[\alpha \left(x_1\right),\alpha \left(x_2\right),[N{y}_1,y_2,N{y}_3]\hfill \\ \hfill & +[\alpha \left(x_1\right),\alpha \left(x_2\right),[y_1,N{y}_2,N{y}_3]].\hfill \end{array}$$

Similarly, a direct computation shows that

$$\begin{array}{cc}\hfill & \psi (\psi (x_1,x_2,y_1),\alpha \left(y_2\right),\alpha \left(y_3\right))\hfill \\ \hfill =& [[N{x}_1,N{x}_2,y_1],\alpha \left(y_2\right),\alpha \left(y_3\right)]+[[N{x}_1,x_2,N{y}_1],\alpha \left(y_2\right),\alpha \left(y_3\right)]\hfill \\ \hfill & +[[x_1,N{x}_2,N{y}_1],\alpha \left(y_2\right),\alpha \left(y_3\right)]\hfill \\ \hfill & +[[N{x}_1,x_2,y_1],N\alpha \left(y_2\right),\alpha \left(y_3\right)]+[[N{x}_1,x_2,y_1],\alpha \left(y_2\right),N\alpha \left(y_3\right)]\hfill \\ \hfill & -N[[N{x}_1,x_2,y_1],\alpha \left(y_2\right),\alpha \left(y_3\right)]+[[x_1,N{x}_2,y_1],N\alpha \left(y_2\right),\alpha \left(y_3\right)]\hfill \\ \hfill & +[[x_1,N{x}_2,y_1],\alpha \left(y_2\right),N\alpha \left(y_3\right)]-N[[x_1,N{x}_2,y_1],\alpha \left(y_2\right),\alpha \left(y_3\right)]\hfill \\ \hfill & +[[x_1,x_2,N{y}_1],N\alpha \left(y_2\right),\alpha \left(y_3\right)]+[[x_1,x_2,N{y}_1],\alpha \left(y_2\right),N\alpha \left(y_3\right)]\hfill \\ \hfill & -N[[x_1,x_2,N{y}_1],\alpha \left(y_2\right),\alpha \left(y_3\right)]-[N[x_1,x_2,y_1],N\alpha \left(y_2\right),\alpha \left(y_3\right)]\hfill \\ \hfill & -[N[x_1,x_2,y_1],\alpha \left(y_2\right),N\alpha \left(y_3\right)]+N[N[x_1,x_2,y_1],\alpha \left(y_2\right),\alpha \left(y_3\right)],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}\psi (\alpha \left(y_1\right),\psi (x_1,x_2,y_2),\alpha \left(y_3\right))\hfill \\ \hfill =& {(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}([N\alpha \left(y_1\right),[N{x}_1,x_2,y_2],\alpha \left(y_3\right)]+[\alpha \left(y_1\right),[N{x}_1,x_2,y_2],N\alpha \left(y_3\right)]\hfill \\ \hfill & -N[\alpha \left(y_1\right),[N{x}_1,x_2,y_2],\alpha \left(y_3\right)]+[N\alpha \left(y_1\right),[x_1,N{x}_2,y_2],\alpha \left(y_3\right)]\hfill \\ \hfill & +[\alpha \left(y_1\right),[x_1,N{x}_2,y_2],N\alpha \left(y_3\right)]-N[\alpha \left(y_1\right),[x_1,N{x}_2,y_2],\alpha \left(y_3\right)]\hfill \\ \hfill & +[N\alpha \left(y_1\right),[x_1,x_2,N{y}_2],\alpha \left(y_3\right)]+[\alpha \left(y_1\right),[x_1,x_2,N{y}_2],N\alpha \left(y_3\right)]\hfill \\ \hfill & -N[\alpha \left(y_1\right),[x_1,x_2,N{y}_2],\alpha \left(y_3\right)]-[N\alpha \left(y_1\right),N[x_1,x_2,y_2],\alpha \left(y_3\right)]\hfill \\ \hfill & -[\alpha \left(y_1\right),N[x_1,x_2,y_2],N\alpha \left(y_3\right)]+N[\alpha \left(y_1\right),N[x_1,x_2,y_2],\alpha \left(y_3\right)]\hfill \\ \hfill & +[\alpha \left(y_1\right),[N{x}_1,N{x}_2,y_2],\alpha \left(y_3\right)]+[\alpha \left(y_1\right),[N{x}_1,x_2,N{y}_2],\alpha \left(y_3\right)]\hfill \\ \hfill & +[\alpha \left(y_1\right),[x_1,N{x}_2,N{y}_2],\alpha \left(y_3\right)]),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}\psi (\alpha \left(y_1\right),\alpha \left(y_2\right),\psi (x_1,x_2,y_3))\hfill \\ \hfill =& \delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}([N\alpha \left(y_1\right),\alpha \left(y_2\right),[N{x}_1,x_2,y_3]]+[\alpha \left(y_1\right),N\alpha \left(y_2\right),[N{x}_1,x_2,y_3]]\hfill \\ \hfill & -N[\alpha \left(y_1\right),\alpha \left(y_2\right),[N{x}_1,x_2,y_3]]+[N\alpha \left(y_1\right),\alpha \left(y_2\right),[x_1,N{x}_2,y_3]]\hfill \\ \hfill & +[\alpha \left(y_1\right),N\alpha \left(y_2\right),[x_1,N{x}_2,y_3]]-N[\alpha \left(y_1\right),\alpha \left(y_2\right),[x_1,N{x}_2,y_3]]\hfill \\ \hfill & +[N\alpha \left(y_1\right),\alpha \left(y_2\right),[x_1,x_2,N{y}_3]]+[\alpha \left(y_1\right),N\alpha \left(y_2\right),[x_1,x_2,N{y}_3]]\hfill \\ \hfill & -N[\alpha \left(y_1\right),\alpha \left(y_2\right),[x_1,x_2,N{y}_3]]-[N\alpha \left(y_1\right),\alpha \left(y_2\right),N[x_1,x_2,y_3]]\hfill \\ \hfill & -[\alpha \left(y_1\right),N\alpha \left(y_2\right),N[x_1,x_2,y_3]]+N[\alpha \left(y_1\right),\alpha \left(y_2\right),N[x_1,x_2,y_3]]\hfill \\ \hfill & +[\alpha \left(y_1\right),\alpha \left(y_2\right),[N{x}_1,N{x}_2,y_3]]+[\alpha \left(y_1\right),\alpha \left(y_2\right),[N{x}_1,x_2,N{y}_3]]\hfill \\ \hfill & +[\alpha \left(y_1\right),\alpha \left(y_2\right),[x_1,N{x}_2,N{y}_3]]).\hfill \end{array}$$

By the definition of N, we know that $N\circ \alpha =\alpha \circ N$. Using (3) and (21), and Theorem 2, it follows that

$$\begin{array}{cc}\hfill & \psi (\alpha \left(x_1\right),\alpha \left(x_2\right),\psi (y_1,y_2,y_3))-\psi (\psi (x_1,x_2,y_1),\alpha \left(y_2\right),\alpha \left(y_3\right))\hfill \\ \hfill & -{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}\psi (\alpha \left(y_1\right),\psi (x_1,x_2,y_2),\alpha \left(y_3\right))\hfill \\ \hfill & -\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}\psi (\alpha \left(y_1\right),\alpha \left(y_2\right),\psi (x_1,x_2,y_3))\hfill \\ \hfill & =0.\hfill \end{array}$$

The proof of the theorem is complete. □

4. Abelian Extensions of Hom-$\delta$-Jordan Lie Supertriple Systems

In this section, we show that associated with any abelian extension, there is a representation and a 3-Hom-cocycle.

An ideal of a Hom-$\delta$-Jordan Lie supertriple system T is a subspace I such that $[I,T,T]\subseteq I.$ An ideal I of a Hom-$\delta$-Jordan Lie supertriple system is called an abelian ideal if, moreover, $[T,I,I]=0.$ Notice that $[T,I,I]=0$ implies that $[I,T,I]=0$ and $[I,I,T]=0.$

Definition 7.

Let $(T,{[·,·,·]}_T,\delta )$, $(V,{[·,·,·]}_V,\delta )$, and $(\widehat{T},{[·,·,·]}_{\widehat{T}},\delta )$ be Hom-δ-Jordan Lie supertriple systems and $i:V\to \widehat{T}$, $p:\widehat{T}\to T$ be homomorphisms. The following sequence of Hom-δ-Jordan Lie supertriple systems is a short exact sequence if $\mathrm{Im}\left(i\right)=\mathrm{Ker}\left(p\right),$ $\mathrm{Ker}\left(i\right)=0$ and $\mathrm{Im}\left(p\right)=T,$

$$\begin{array}{c}\hfill 0⟶V\stackrel{i}{⟶}\widehat{T}\stackrel{p}{⟶}T⟶0.\end{array}$$

(22)

In this case, we call $\widehat{T}$ an extension of T by V, and denote it by $E_{\widehat{T}}$. It is called an abelian extension if V is an abelian ideal of $\widehat{T}$, i.e., ${[u,v,·]}_{\widehat{T}}={[u,·,v]}_{\widehat{T}}={[·,u,v]}_{\widehat{T}}=0,$ for all $u,v\in V.$

A section $\sigma :T\to \widehat{T}$ of $p:\widehat{T}\to T$ consists of linear maps $\sigma :T\to \widehat{T}$ such that $p\circ \sigma ={\mathrm{id}}_T,$ $\widehat{\alpha }\circ \sigma =\sigma \circ \alpha .$

Definition 8.

Two extensions of Hom-δ-Jordan Lie supertriple systems $E_{\widehat{T}}:0⟶V\stackrel{i}{⟶}\widehat{T}\stackrel{p}{⟶}T⟶0$ and $E_{\tilde{T}}:0⟶V\stackrel{j}{⟶}\tilde{T}\stackrel{q}{⟶}T⟶0$ are equivalent. If there exists a Hom-δ-Jordan Lie supertriple system homomorphism $F:\widehat{T}\to \tilde{T}$ such that the following diagram commutes

Let $\widehat{T}$ be an abelian extension of T by V, and a linear mapping $\sigma :T\to \widehat{T}$ be a section. Define maps $T\otimes T\to End\left(V\right)$ by

$$\begin{array}{c}\hfill D(x_1,x_2)\left(u\right)=\delta {[\sigma \left(x_1\right),\sigma \left(x_2\right),u]}_{\widehat{T}},\end{array}$$

(23)

$$\begin{array}{c}\hfill \theta (x_1,x_2)\left(u\right)={(-1)}^{\left|u\right|\left(\right|x_1|+|x_2\left|\right)}{[u,\sigma \left(x_1\right),\sigma \left(x_2\right)]}_{\widehat{T}}\end{array}$$

(24)

Clearly, the following fact holds, i.e.,

$$D(x_1,x_2)\left(u\right)={(-1)}^{|x_1\left|\right|x_2|}\theta (x_2,x_1)\left(u\right)-\delta \theta (x_1,x_2)\left(u\right),$$

for all $(x_1,x_2)\in T\otimes T,u\in V.$

Let $\sigma :T\to \widehat{T}$ be a section of the abelian extension. Define the following map $\omega :T\times T\times T\to \widehat{T}$:

$$\begin{array}{c}\hfill \omega (x_1,x_2,x_3)={[\sigma \left(x_1\right),\sigma \left(x_2\right),\sigma \left(x_3\right)]}_{\widehat{T}}-\sigma \left({[x_1,x_2,x_3]}_T\right),\end{array}$$

(25)

for all $x_1,x_2,x_3\in T.$

Theorem 4.

Let $0⟶V⟶\widehat{T}⟶T⟶0$ be an abelian extension of T by V. Then, ω defined by (25) is a 3-Hom-cocycle of T with coefficients in V, where the representation θ is given by (24).

Proof. 

By the equality

$$\begin{array}{cc}\hfill & {[\widehat{\alpha }\left(\sigma \left(x_1\right)\right),\widehat{\alpha }\left(\sigma \left(x_2\right)\right),{[\sigma \left(y_1\right),\sigma \left(y_2\right),\sigma \left(y_3\right)]}_{\widehat{T}}]}_{\widehat{T}}\hfill \\ \hfill =& {[{[\sigma \left(x_1\right),\sigma \left(x_2\right),\sigma \left(y_1\right)]}_{\widehat{T}},\widehat{\alpha }\left(\sigma \left(y_2\right)\right),\widehat{\alpha }\left(\sigma \left(y_3\right)\right)]}_{\widehat{T}}\hfill \\ \hfill & +{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}{[\widehat{\alpha }\left(\sigma \left(y_1\right)\right),{[\sigma \left(x_1\right),\sigma \left(x_2\right),\sigma \left(y_2\right)]}_{\widehat{T}},\widehat{\alpha }\left(\sigma \left(y_3\right)\right)]}_{\widehat{T}}\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}{[\widehat{\alpha }\left(\sigma \left(y_1\right)\right),\widehat{\alpha }\left(\sigma \left(y_2\right)\right),{[\sigma \left(x_1\right),\sigma \left(x_2\right),\sigma \left(y_3\right)]}_{\widehat{T}}]}_{\widehat{T}}.\hfill \end{array}$$

The left hand side shows that

$$\begin{array}{cc}\hfill & {[\widehat{\alpha }\left(\sigma \left(x_1\right)\right),\widehat{\alpha }\left(\sigma \left(x_2\right)\right),{[\sigma \left(y_1\right),\sigma \left(y_2\right),\sigma \left(y_3\right)]}_{\widehat{T}}]}_{\widehat{T}}\hfill \\ \hfill =& {[\widehat{\alpha }\left(\sigma \left(x_1\right)\right),\widehat{\alpha }\left(\sigma \left(x_2\right)\right),\omega (y_1,y_2,y_3)+\sigma \left({[y_1,y_2,y_3]}_T\right)]}_{\widehat{T}}\hfill \\ \hfill =& \delta D(\widehat{\alpha }\left(x_1\right),\widehat{\alpha }\left(x_2\right))\omega (y_1,y_2,y_3)+{[\widehat{\alpha }\left(\sigma \left(x_1\right)\right),\widehat{\alpha }\left(\sigma \left(x_2\right)\right),\sigma \left({[y_1,y_2,y_3]}_T\right)]}_{\widehat{T}}\hfill \\ \hfill =& \delta D(\widehat{\alpha }\left(x_1\right),\widehat{\alpha }\left(x_2\right))\omega (y_1,y_2,y_3)+[\sigma \left(\alpha \left(x_1\right)\right),\sigma \left(\alpha \left(x_2\right)\right),\sigma \left([y_1,y_2,y_3]\right)]\hfill \\ \hfill =& \delta D(\alpha \left(x_1\right),\alpha \left(x_2\right))\omega (y_1,y_2,y_3)+\omega (\alpha \left(x_1\right),\alpha \left(x_2\right),{[y_1,y_2,y_3]}_T)+\sigma \left({[\alpha \left(x_1\right),\alpha \left(x_2\right),{[y_1,y_2,y_3]}_T]}_T\right).\hfill \end{array}$$

Similarly, the right side is equal to

$$\begin{array}{cc}\hfill & {[{[\sigma \left(x_1\right),\sigma \left(x_2\right),\sigma \left(y_1\right)]}_{\widehat{T}},\widehat{\alpha }\left(\sigma \left(y_2\right)\right),\widehat{\alpha }\left(\sigma \left(y_3\right)\right)]}_{\widehat{T}}\hfill \\ \hfill & +{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}{[\widehat{\alpha }\left(\sigma \left(y_1\right)\right),{[\sigma \left(x_1\right),\sigma \left(x_2\right),\sigma \left(y_2\right)]}_{\widehat{T}},\widehat{\alpha }\left(\sigma \left(y_3\right)\right)]}_{\widehat{T}}\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1+x_2\left|\right)\left(\right|y_1+y_2\left|\right)}{[\widehat{\alpha }\left(\sigma \left(y_1\right)\right),\widehat{\alpha }\left(\sigma \left(y_2\right)\right),{[\sigma \left(x_1\right),\sigma \left(x_2\right),\sigma \left(y_3\right)]}_{\widehat{T}}]}_{\widehat{T}}\hfill \\ \hfill & ={[\omega (x_1,x_2,y_1)+\sigma \left({[x_1,x_2,y_1]}_T\right),\widehat{\alpha }\left(\sigma \left(y_2\right)\right),\widehat{\alpha }\left(\sigma \left(y_3\right)\right)]}_{\widehat{T}}\hfill \\ \hfill & +{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}({[\widehat{\alpha }\left(\sigma \left(y_1\right)\right),\omega (x_1,x_2,y_2)+\sigma \left({[x_1,x_2,y_2]}_T\right),\widehat{\alpha }\left(\sigma \left(y_3\right)\right)]}_{\widehat{T}}\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1+x_2\left|\right)\left(\right|y_1+y_2\left|\right)}{[\widehat{\alpha }\left(\sigma \left(y_1\right)\right),\widehat{\alpha }\left(\sigma \left(y_2\right)\right),\omega (x_1,x_2,y_3)+\sigma \left({[x_1,x_2,y_3]}_T\right)]}_{\widehat{T}}\hfill \\ \hfill & ={[\omega (x_1,x_2,y_1),\widehat{\alpha }\left(\sigma \left(y_2\right)\right),\widehat{\alpha }\left(\sigma \left(y_3\right)\right)]}_{\widehat{T}}+{[\sigma \left({[x_1,x_2,y_1]}_T\right),\widehat{\alpha }\left(\sigma \left(y_2\right)\right),\widehat{\alpha }\left(\sigma \left(y_3\right)\right)]}_{\widehat{T}}\hfill \\ \hfill & +{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}{[\widehat{\alpha }\left(\sigma \left(y_1\right)\right),\omega (x_1,x_2,y_2),\widehat{\alpha }\left(\sigma \left(y_3\right)\right)]}_{\widehat{T}}\hfill \\ \hfill & +{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}{[\widehat{\alpha }\left(\sigma \left(y_1\right)\right),\sigma \left({[x_1,x_2,y_2]}_T\right),\widehat{\alpha }\left(\sigma \left(y_3\right)\right)]}_{\widehat{T}}\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1+x_2\left|\right)\left(\right|y_1+y_2\left|\right)}{[\widehat{\alpha }\left(\sigma \left(y_1\right)\right),\widehat{\alpha }\left(\sigma \left(y_2\right)\right),\omega (x_1,x_2,y_3)]}_{\widehat{T}}\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1+x_2\left|\right)\left(\right|y_1+y_2\left|\right)}{[\widehat{\alpha }\left(\sigma \left(y_1\right)\right),\widehat{\alpha }\left(\sigma \left(y_2\right)\right),\sigma \left({[x_1,x_2,y_3]}_T\right)]}_{\widehat{T}}\hfill \\ \hfill & ={[\omega (x_1,x_2,y_1),\sigma \left(\alpha \left(y_2\right)\right),\sigma \left(\alpha \left(y_3\right)\right)]}_{\widehat{T}}+{[\sigma \left({[x_1,x_2,y_1]}_T\right),\sigma \left(\alpha \left(y_2\right)\right),\sigma \left(\alpha \left(y_3\right)\right)]}_{\widehat{T}}\hfill \\ \hfill & +{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}{[\sigma \left(\alpha \left(y_1\right)\right),\omega (x_1,x_2,y_2),\sigma \left(\alpha \left(y_3\right)\right)]}_{\widehat{T}}\hfill \\ \hfill & +{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}{[\sigma \left(\alpha \left(y_1\right)\right),\sigma \left({[x_1,x_2,y_2]}_T\right),\sigma \left(\alpha \left(y_3\right)\right)]}_{\widehat{T}}\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1+x_2\left|\right)\left(\right|y_1+y_2\left|\right)}{[\sigma \left(\alpha \left(y_1\right)\right),\sigma \left(\alpha \left(y_2\right)\right),\omega (x_1,x_2,y_3)]}_{\widehat{T}}\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1+x_2\left|\right)\left(\right|y_1+y_2\left|\right)}{[\sigma \left(\alpha \left(y_1\right)\right),\sigma \left(\alpha \left(y_2\right)\right),\sigma \left({[x_1,x_2,y_3]}_T\right)]}_{\widehat{T}}\hfill \\ \hfill & ={(-1)}^{\left(\right|x_1|+|x_2|+|y_1\left|\right)\left(\right|y_2|+|y_3\left|\right)}\theta (\alpha \left(y_2\right),\alpha \left(y_3\right))\omega (x_1,x_2,y_1)+\sigma \left({[{[x_1,x_2,y_1]}_T,\alpha \left(y_2\right),\alpha \left(y_3\right)]}_T\right)\hfill \\ \hfill & +\omega \left({[{[x_1,x_2,y_1]}_T,\alpha \left(y_2\right),\alpha \left(y_3\right)]}_T\right)-\delta {(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)+|y_3\left|\right(|x_1|+|x_2|+|y_2\left|\right)}\theta (\alpha \left(y_1\right),\alpha \left(y_3\right))\omega (x_1,x_2,y_2)\hfill \\ \hfill & +{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}\omega (\alpha \left(y_1\right),{[x_1,x_2,y_2]}_T,\alpha \left(y_3\right))+{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}\sigma \left({[\alpha \left(y_1\right),{[x_1,x_2,y_2]}_T,\alpha \left(y_3\right)]}_T\right)\hfill \\ \hfill & +{(-1)}^{\left(\right|x_1+x_2\left|\right)\left(\right|y_1+y_2\left|\right)}D(\alpha \left(y_1\right),\alpha \left(y_2\right))\omega (x_1,x_2,y_3)+\delta {(-1)}^{\left(\right|x_1+x_2\left|\right)\left(\right|y_1+y_2\left|\right)}\omega (\alpha \left(y_1\right),\alpha \left(y_2\right),{[x_1,x_2,y_3]}_T)\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1+x_2\left|\right)\left(\right|y_1+y_2\left|\right)}\sigma \left({[\alpha \left(y_1\right),\alpha \left(y_2\right),{[x_1,x_2,y_3]}_T]}_T\right).\hfill \end{array}$$

Thus, it follows that

$$\begin{array}{cc}\hfill & \omega ({[x_1,x_2,y_1]}_T,\alpha \left(y_2\right),\alpha \left(y_3\right))+{(-1)}^{|y_1\left|\right(|x_1|+|x_2\left|\right)}\omega (\alpha \left(y_1\right),{[x_1,x_2,y_2]}_T,\alpha \left(y_3\right))\hfill \\ \hfill & +\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}\omega (\alpha \left(y_1\right),\alpha \left(y_2\right),{[x_1,x_2,y_3]}_T)\hfill \\ \hfill & +{(-1)}^{\left(\right|x_1|+|x_2|+|y_1\left|\right)\left(\right|y_2|+|y_3\left|\right)}\theta (\alpha \left(y_2\right),\alpha \left(y_3\right))\omega (x_1,x_2,y_1)\hfill \\ \hfill & -\delta {(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1+y_3\left|\right)+|y_2\left|\right|y_3|}\theta (\alpha \left(y_1\right),\alpha \left(y_3\right))\omega (x_1,x_2,y_2)\hfill \\ \hfill & +{(-1)}^{\left(\right|x_1|+|x_2\left|\right)\left(\right|y_1|+|y_2\left|\right)}D(\alpha \left(y_1\right),\alpha \left(y_2\right))\omega (x_1,x_2,y_3)\hfill \\ \hfill & -\omega (\alpha \left(x_1\right),\alpha \left(x_2\right),{[y_1,y_2,y_3]}_T)-\delta D(\alpha \left(x_1\right),\alpha \left(x_2\right))\omega (y_1,y_2,y_3)\hfill \\ \hfill & =0.\hfill \end{array}$$

Therefore, $\omega$ is a 3-Hom-cocycle. □

Theorem 5.

Let T be a Hom-δ-Jordan Lie surpetriple system, $(V,\theta )$ be a T-module and ω be a 3-Hom-cocycle, then $T\oplus V$ is a Hom-δ-Jordan Lie surpetriple system under the following multiplication:

$$\begin{array}{cc}\hfill & {[x_1+u_1,x_2+u_2,x_3+u_3]}_{\omega }\hfill \\ \hfill & =[x_1,x_2,x_3]+\omega (x_1,x_2,x_3)+\delta D(x_1,x_2)\left(u_3\right)\hfill \\ \hfill & -\delta {(-1)}^{|x_2\left|\right|x_3|}\theta (x_1,x_3)\left(u_2\right)+{(-1)}^{|x_1\left|\right(|x_2|+|x_3\left|\right)}\theta (x_2,x_3)\left(u_1\right).\hfill \end{array}$$

and

$$\begin{array}{c}\hfill (\alpha +A)(x+u)=\alpha \left(x\right)+A\left(u\right).\end{array}$$

Proof. 

Similar to the proof of Proposition 1, we obtain (4)–(6) hold. □

Theorem 6.

Two abelian extensions of $\mathrm{Hom}$-δ-$\mathrm{Jordan}$ Lie supertriple systems $E_{\widehat{T}}:0⟶V\stackrel{i}{⟶}\widehat{T}\stackrel{p}{⟶}T⟶0$ and $E_{\tilde{T}}:0⟶V\stackrel{j}{⟶}\tilde{T}\stackrel{q}{⟶}T⟶0$ are equivalent ⟺ ω and ${\omega }^{\prime }$ are in the same cohomology class.

Proof. 

$\Rightarrow )$ Let $F:T{\oplus }_{\omega }V\to T{\oplus }_{{\omega }^{\prime }}V$ be the corresponding homomorphism. Thus,

$$F{[x_1,x_2,x_3]}_{\omega }={[F\left(x_1\right),F\left(x_2\right),F\left(x_3\right)]}_{{\omega }^{\prime }}.$$

(26)

Note that F is an equivalence of extensions, so there is $\rho :T\to V$ such that

$$F(x_i+u_i)=x_i+\rho \left(x_i\right)+u_i,i=1,2,3.$$

(27)

The left hand side of (26) is equal to

$$\begin{array}{c}\hfill F([x_1,x_2,x_3]+\omega (x_1,x_2,x_3))=[x_1,x_2,x_3]+\omega (x_1,x_2,x_3)+\rho \left([x_1,x_2,x_3]\right),\end{array}$$

and the right hand side of (26) is equal to

$$\begin{array}{cc}\hfill & {[x_1+\rho \left(x_1\right),x_2+\rho \left(x_2\right),x_3+\rho \left(x_3\right)]}_{{\omega }^{\prime }}\hfill \\ \hfill =& [x_1,x_2,x_3]+{\omega }^{\prime }(x_1,x_2,x_3)+\delta D(x_1,x_2)\rho \left(x_3\right)-\delta {(-1)}^{|x_2\left|\right|x_3|}\theta (x_1,x_3)\rho \left(x_2\right)\hfill \\ \hfill & +{(-1)}^{|x_1\left|\right(|x_2|+|x_3\left|\right)}\theta (x_2,x_3)\rho \left(x_1\right).\hfill \end{array}$$

Then we obtain that

$$\begin{array}{cc}\hfill & (\omega -{\omega }^{\prime })(x_1,x_2,x_3)\hfill \\ \hfill =& \delta D(x_1,x_2)\rho \left(x_3\right)-\delta {(-1)}^{|x_2\left|\right|x_3|}\theta (x_1,x_3)\rho \left(x_2\right)\hfill \\ \hfill & +{(-1)}^{|x_1\left|\right(|x_2|+|x_3\left|\right)}\theta (x_2,x_3)\rho \left(x_1\right)-\rho \left([x_1,x_2,x_3]\right).\hfill \end{array}$$

Therefore, $\omega -{\omega }^{\prime }=d^1\rho$, we get $\omega$ and ${\omega }^{\prime }$ are in the same cohomology class.

$\Leftarrow )$ If $\omega$ and ${\omega }^{\prime }$ are in the same cohomology class, we assume that $\omega -{\omega }^{\prime }=d^1\rho$, then F defined by (27) is an equivalence. □