Cohomology and deformations of 3-dimensional Heisenberg Hom-Lie superalgebras

TL;DR

This paper investigates 3-dimensional Heisenberg Hom-Lie superalgebras by describing their structures, computing cohomology spaces, and analyzing their infinitesimal deformations to understand their algebraic properties.

Contribution

It provides a detailed classification of Hom-Lie super structures and cohomology for 3D Heisenberg superalgebras, including deformation analysis, which is a novel contribution.

Findings

Classification of Hom-Lie super structures

Computation of cohomology spaces

Characterization of infinitesimal deformations

Abstract

In this paper, we study Hom-Lie superalgebras of Heisenberg type. For 3-dimensional Heisenberg Hom-Lie superalgebras, we describe their Hom-Lie super structures, compute the cohomology spaces and characterize their infinitesimal deformations.

Tables

Table 2
(a) $\left({\mu }_0-1\right){\mu }_{11}=0.$
${[\cdot ,\cdot ]}_t$	base change	Hom-Lie superalgebra
$\begin{array}{ccc}{[u,v]}_t=h& & \\ {[u,h]}_t=a_{26}v+a_{36}h& & \end{array}$
$a_{26}{a}_{36}\ne 0$	$\begin{array}{ccc}\tau ={\left(k_1-k_2\right)}^{-1}& & \\ k_1\ne k_2,k_1{k}_2=-a_{26}^{}{}_{}{}^{-1}& & \\ k_1+k_2=-a_{36}a_{26}^{}{}_{}{}^{-1}& & \end{array}$	$\begin{array}{ccc}\lambda =k_{1}k_2^{}{}_{}{}^{-1},\lambda \ne 0,-1& & \\ \left({\mu }_0-1\right){\mu }_{11}=0& & \end{array}$
$\begin{array}{ccc}{[u,v]}_t=h& & \\ {[u,v]}_t=a_{36}h& & \end{array}$
$a_{36}\ne 0$	$a_{36}\ne 0$	$\left({\mu }_0-1\right){\mu }_{11}=0$
$\begin{array}{ccc}{[u,v]}_t=h& & \\ {[u,v]}_t=a_{26}v& & \end{array}$
$a_{26}\ne 0$	$\begin{array}{c}\kappa ={\left(1+a_{26}\right)}^{-1}\\ a_{26}\ne 0\end{array}$	$\left({\mu }_0-1\right){\mu }_{11}=0$
(b) ${\mu }_0=-1.$
${[\cdot ,\cdot ]}_t$	base change	Hom-Lie superalgebra
$\begin{array}{ccc}{[u,v]}_t=h& & \\ {[u,v]}_t=a_{36}h& & \end{array}$
$a_{36}\ne 0$	$a_{36}\ne 0$
(c) ${\mu }_0=0.$
${[\cdot ,\cdot ]}_t$	base change	Hom-Lie superalgebra
$\begin{array}{ccc}{[u,v]}_t=h& & \\ {[u,v]}_t=a_{26}v& & \end{array}$
$a_{26}\ne 0$	$\begin{array}{c}\rho ={\left(1+a_{26}\right)}^{-1}\\ a_{26}\ne 0\end{array}$	$\xi =-a_{26}^{}{}_{}{}^{-\frac{1}{2}}$

Table 3
(a) ${\mu }_0={\mu }_{11}=0.$
${[\cdot ,\cdot ]}_t$	base change	Hom-Lie superalgebra
$\begin{array}{ccc}{[u,v]}_t=h& & \\ {[u,v]}_t=a_{36}h& & \end{array}$
$a_{36}\ne 0$	$a_{36}\ne 0$	${\mu }_{22}=a_{36}$
(b) ${\mu }_0=1.$
${[\cdot ,\cdot ]}_t$	base change	Hom-Lie superalgebra
$\begin{array}{ccc}{[u,v]}_t=a_{36}v+h& & \\ {[u,v]}_t=a_{36}h& & \end{array}$
$a_{36}\ne 0$	$a_{36}\ne 0$
(c) ${\mu }_0\ne 0,1,{\mu }_{11}=0.$
${[\cdot ,\cdot ]}_t$	base change	Hom-Lie superalgebra
$\begin{array}{ccc}{[u,v]}_t={\mu }_0{a}_{36}v+h& & \\ {[u,v]}_t=a_{36}h& & \end{array}$
$a_{36}\ne 0$	$a_{36}\ne 0$	$\lambda \ne 0,1$

Equations

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

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

$$Z(\mathfrak{g})=\{x\in V:[x,y]=0,\forall y\in V\}.$$

$$\varphi ([x,y{]}_1)=[\varphi (x),\varphi (y){]}_2,\forall x,y\in V$$

$$\varphi \circ \alpha =\beta \circ \varphi .$$

$$(\mathfrak{h}_{2},\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&0&\mu_{0}\mu_{11}\\ \end{array}\right)).$$

$$\phi=\left(\begin{array}[]{ccc}b_{0}&0&0\\ 0&b_{11}&0\\ 0&b_{21}&b_{0}b_{11}\\ \end{array}\right),\ \phi^{-1}=\left(\begin{array}[]{ccc}{b_{0}}^{-1}&0&0\\ 0&{b_{11}}^{-1}&0\\ 0&-{b_{0}}^{-1}{b_{11}}^{-2}{b_{21}}&{b_{0}}^{-1}{b_{11}}^{-1}\\ \end{array}\right).$$

$$\begin{split}\phi\alpha\phi^{-1}&=\left(\begin{array}[]{ccc}b_{0}&0&0\\ 0&b_{11}&0\\ 0&b_{21}&b_{0}b_{11}\\ \end{array}\right)\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&\mu_{21}&\mu_{0}\mu_{11}\\ \end{array}\right)\left(\begin{array}[]{ccc}{b_{0}}^{-1}&0&0\\ 0&{b_{11}}^{-1}&0\\ 0&-{b_{0}}^{-1}{b_{11}}^{-2}{b_{21}}&{b_{0}}^{-1}{b_{11}}^{-1}\\ \end{array}\right)\\ &=\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&(1-\mu_{0})\mu_{11}{b_{11}}^{-1}b_{21}+\mu_{21}b_{0}&\mu_{0}a_{11}\\ \end{array}\right).\end{split}$$

$$\phi\alpha\phi^{-1}=\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&0&\mu_{0}\mu_{11}\\ \end{array}\right),$$

$$(\mathfrak{h}_{2},\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&1&\mu_{11}\\ \end{array}\right)),\ (\mu_{0}-1)\mu_{11}=0.$$

$$(\mathfrak{h}_{1},\left(\begin{array}[]{ccc}\mu_{11}\mu_{22}&0&0\\ 0&\mu_{11}&0\\ 0&0&\mu_{22}\\ \end{array}\right))(\mu_{11}\mu_{22}\neq 0),\ (\mathfrak{h}_{1},\left(\begin{array}[]{ccc}\mu_{12}\mu_{21}&0&0\\ 0&0&\mu_{12}\\ 0&\mu_{21}&0\\ \end{array}\right))(\mu_{12}\mu_{21}\neq 0)$$

$$\phi (x_1,\cdots ,x_{i+1},x_i\cdots ,x_k)=-(-1{)}^{|x_i||x_{i+1}|}\phi (x_1,\cdots ,x_i,x_{i+1}\cdots ,x_k),$$

$$\alpha (\phi (x_1,\cdots ,x_k))=\phi (\alpha (x_1),\cdots ,\alpha (x_k))$$

$$\begin{array}{rl}{\delta }^k(\phi )(x_0,\cdots ,x_k)=& \sum _{0⩽s<t⩽k}(-1{)}^{t+|x_t|(|x_{s+1}|+\cdots +|x_{t-1}|)}\\ & \times \phi (\alpha (x_0),\cdots ,\alpha (x_{s-1}),[x_s,x_t],\alpha (x_{s+1}),\cdots ,\widehat{x_t},\cdots ,\alpha (x_k))\\ & +\sum _{s=1}^k(-1{)}^{s+|x_s|(|\phi |+|x_0|+\cdots +|x_{s-1}|)}[{\alpha }^{k-1}(x_s),\phi (x_0,\cdots ,\widehat{x_s},\cdots ,x_k)],\end{array}$$

$$B^{2}(\mathfrak{g},\mathfrak{g})_{\overline{0}}=\{\delta^{1}\varphi_{0}|\varphi_{0}\in C^{1}_{\alpha}(\mathfrak{g},\mathfrak{g})_{\overline{0}}\}=\left\langle\left(\begin{array}[]{cccccc}0&2a_{32}&a_{22}+a_{33}-a_{11}&2a_{23}&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ \end{array}\right)\right\rangle.$$

$$[x,y{]}_t=-[y,x{]}_t,$$

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

$$\phi \circ \psi (x,y,z)={\circlearrowleft }_{x,y,z}(-1{)}^{|x||z|}\phi (\alpha (x),\psi (y,z)),$$

$$.\begin{array}{rl}0& =[\cdot ,\cdot {]}_t\circ [\cdot ,\cdot {]}_t\\ & ={\circlearrowleft }_{x,y,z}(-1{)}^{|x||z|}([\alpha (x),[y,z{]}_t{]}_0+t\phi (\alpha (x),[y,z{]}_t))\\ & ={\circlearrowleft }_{x,y,z}(-1{)}^{|x||z|}\left[t([\alpha (x),\phi (y,z){]}_0+\phi (\alpha (x),[y,z{]}_0))+t^2\phi (\alpha (x),\phi (y,z))\right].\end{array}$$

$${\circlearrowleft }_{x,y,z}(-1{)}^{|x||z|}([\alpha (x),\phi (y,z){]}_0+\phi (\alpha (x),[y,z{]}_0))=(-1{)}^{|x||z|}{\delta }^2\phi .$$

$${\Phi }_t([x,y{]}_t)=[{\Phi }_t(x),{\Phi }_t(y){]}_t^{{}^{\prime }},\forall x,y\in V$$

$$\begin{array}[]{l}L_{1}:[e_{1},V_{\overline{1}}]=0,[e_{2},e_{2}]=e_{1},[e_{3},e_{3}]=[e_{2},e_{3}]=0;\\ L_{2}:[e_{1},V_{\overline{1}}]=0,[e_{2},e_{2}]=[e_{3},e_{3}]=e_{1},[e_{2},e_{3}]=0;\\ L_{3}^{\lambda}:[e_{1},e_{2}]=e_{2},[e_{1},e_{3}]=\lambda e_{3},[V_{\overline{1}},V_{\overline{1}}]=0;\\ L_{4}:[e_{1},e_{2}]=e_{2},[e_{1},e_{3}]=e_{2}+e_{3},[V_{\overline{1}},V_{\overline{1}}]=0;\\ L_{5}:[e_{1},e_{2}]=0,[e_{1},e_{3}]=e_{2},[V_{\overline{1}},V_{\overline{1}}]=0.\end{array}$$

$$\varphi (h)=e_1,\varphi (v_1)=e_2+i{e}_3,\varphi (v_2)=\frac{1}{2}{e}_2-\frac{1}{2}i{e}_3$$

$$({L_{2}}^{{}^{\prime}},\left(\begin{array}[]{ccc}0&0&0\\ 0&\mu_{11}&\mu_{12}\\ 0&0&0\\ \end{array}\right)).$$

$$\alpha=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&\mu_{12}\\ 0&0&0\\ \end{array}\right)(\mu_{12}\neq 0),\ \varphi=\left(\begin{array}[]{cccccc}0&0&0&0&0&0\\ 0&0&0&0&0&a_{26}\\ 0&0&0&0&0&0\\ \end{array}\right)(a_{26}\neq 0).$$

$$\alpha=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&0\\ 0&1&0\\ \end{array}\right),\ \varphi=\left(\begin{array}[]{cccccc}0&a_{12}&a_{13}&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ \end{array}\right).$$

Full text

Key words: Hom-Lie superalgebras, Lie superalgebras, Heisenberg, cohomology, deformations

000Corresponding author(L. Chen): chenly640@nenu.edu.cn.000Supported by NNSF of China (No. 117701069), NSF of Jilin province(No. 20170101048JC) and the project of jilin province department of education (No. JJKH20180005K).

1 Introduction

In recent years, Hom-Lie algebras and other Hom-algebras are widely studied, motivated initially by instances appeared in Physics literature when looking for quantum deformations of some algebras of vector fields. Hom-Lie superalgebras, as a generalization of Hom-Lie algebras, are introduced in [3], [4]. Furthermore, the cohomology and deformation theories of Hom-algebras are studied in [1] [2], [6], [9] and so on, while the two theories of Hom-Lie superalgebras can be seen in [3], [5].

We will follow [7], [8] to define Heisenberg Hom-Lie superalgebras, which are a special case of 2-step nilpotent Hom-Lie superalgebras. The main idea of this paper is to characterize the infinitesimal deformations of Heisenberg Hom-Lie superalgebras using cohomology.

The paper proceeds as follows. In Section 2, we recall the definitions of Hom-Lie superalgebras. Section $3$ is dedicated to introduce Heisenberg Hom-Lie superalgebras and classify three-dimensional Heisenberg Hom-Lie superalgebras. In Section $4$, we review the cohomology theory and give the 2-nd cohomology spaces of Heisenberg Hom-Lie superalgebras of dimension three. In the last section, we characterize all the infinitesimal deformations of three-dimensional Heisenberg Hom-Lie superalgebras using cohomology.

2 Preliminaries

Let $V$ be a vector superspace over a field $\mathbb{F}$; that is, a $\mathbb{Z}_{2}$-graded vector space with a direct sum decomposition $V=V_{\overline{0}}\oplus V_{\overline{1}}$. The elements of $V_{\overline{j}}$, $j=0,1$, are called homogeneous of parity $j$. The parity of homogenous element $x$ is denoted by $|x|$. Moreover, the superspace $End(V)$ have a natural direct sum decomposition $End(V)=End(V)_{\overline{0}}\oplus End(V)_{\overline{1}}$, where $End(V)_{\overline{j}}=\{f|f(V_{\overline{i}})\subseteq V_{\overline{i+j}}\}$, $j=0,1$. Elements of $End(V)_{\overline{j}}$ are homogeneous of parity $j$.

We review the definition of Hom-Lie superalgebra in [3].

Definition 2.1**.**

A Hom-Lie superalgebra $\mathfrak{g}=(V,[\cdot,\cdot],\alpha)$ is a triple consisting of a superspace $V$ over a field $\mathbb{F}$, an even bilinear map $[\cdot,\cdot]:V\times V\longrightarrow V$ and an even superspace homomorphism $\alpha:V\longrightarrow V$ satisfying

[TABLE]

[TABLE]

for all homogenous elements $x,y,z\in V$, where $\circlearrowleft_{x,y,z}$ denotes the cyclic summation over $x,y,z$.

We denote $\mathfrak{g}_{\overline{0}}=\mathfrak{g}|_{V_{\overline{0}}}$, $\mathfrak{g}_{\overline{1}}=\mathfrak{g}|_{V_{\overline{1}}}$ and then $\mathfrak{g}=\mathfrak{g}_{\overline{0}}\oplus\mathfrak{g}_{\overline{1}}$. It follows that $\mathfrak{g}$ is a Hom-Lie algebra when $\mathfrak{g}_{\overline{1}}=0$. The classical Lie superalgebra can be obtained when $\alpha=\rm{id}$.

A hom-Lie superalgebra is called multiplicative, if $\alpha([x,y])=[\alpha(x),\alpha(y)],\forall x,y$. It is obvious that the classical Lie superalgebras are a special case of multiplicative hom-Lie superalgebras.

The center of Hom-Lie superalgebra $\mathfrak{g}=(V,[\cdot,\cdot],\alpha)$ is defined by

[TABLE]

Two hom-Lie superalgebras $(V,[\cdot,\cdot]_{1},\alpha)$ and $(V,[\cdot,\cdot]_{2},\beta)$ are said to be isomorphic if there exists an even bijective homomorphism $\phi:(V,[\cdot,\cdot]_{1})\rightarrow(V,[\cdot,\cdot]_{2})$ satisfying

[TABLE]

[TABLE]

In particular, $(V,[\cdot,\cdot],\alpha)$ and $(V,[\cdot,\cdot],\beta)$ are isomorphic if and only if there exists an even automorphism $\phi$ such that $\beta=\phi\alpha\phi^{-1}$.

Let $V$ be a vector superspace as before. A bilinear form $\mathcal{B}$ on $V$ is called homogeneous of parity $j$ if it satisfies $\mathcal{B}(x,y)=0,\forall x,y\in V,|x|\neq|y|+j$; skew-supersymmetric if $\mathcal{B}(x,y)=-(-1)^{|x||y|}\mathcal{B}(y,x)$ for all homogenous elements $x,y\in V$; non-degenerate if from $\mathcal{B}(x,y)=0$ for all $x\in V$, it follows that $y=0$.

In this paper, we only discuss multiplicative Hom-Lie superalgebras over the complex field $\mathbb{C}$ and the elements mentioned are homogenous.

3 Heisenberg Hom-Lie superalgebras

Let $\mathfrak{g}$ be a finite-dimensional Hom-Lie superalgebra with a 1-dimensional homogenous derived ideal such that $[\mathfrak{g},\mathfrak{g}]\subset Z(\mathfrak{g})$. Let $h\in Z(\mathfrak{g})$ be the homogenous generator of $[\mathfrak{g},\mathfrak{g}]$. Then a homogenous skew-supersymmetric bilinear form $\bar{{\mathcal{B}}}$ can be defined on $\mathfrak{g}$ via $[x,y]=\bar{{\mathcal{B}}}(x,y)h$, $\forall x,y\in\mathfrak{g}$. This induces a homogenous skew-supersymmetric bilinear form $\mathcal{B}$ on $\mathfrak{g}/Z(\mathfrak{g})$ via ${\mathcal{B}}(x+Z(\mathfrak{g}),y+Z(\mathfrak{g}))=\bar{{\mathcal{B}}}(x,y)$.

Definition 3.1**.**

A Hom-Lie superalgebra $\mathfrak{g}$ is called a Heisenberg Hom-Lie superalgebra if the derived ideal $[\mathfrak{g},\mathfrak{g}]$ is generated by a homogenous element $h\in Z(\mathfrak{g})$ and $\mathcal{B}$ is non-degenerate.

From now on, we will also denote a Hom-Lie superalgebra by $(\mathfrak{h},\alpha)$, where $\mathfrak{h}=(V,[\cdot,\cdot]_{\mathfrak{h}})$ is a superalgebra and $\alpha$ is an even linear map. All brackets unmentioned in the following are zero.

Let $\mathfrak{g}=(V,[\cdot,\cdot],\alpha)$ be a 3-dimensional Heisenberg Hom-Lie superalgebra with a direct sum decomposition $\mathfrak{g}=\mathfrak{g}_{\overline{0}}\oplus\mathfrak{g}_{\overline{1}}$. Let $h\in Z(\mathfrak{g})$ be the homogenous generator of the derived ideal $[\mathfrak{g},\mathfrak{g}]$, We analyze the cases $h\in\mathfrak{g}_{\overline{0}}$ and $h\in\mathfrak{g}_{\overline{1}}$ separately.

Case $1$. If $h\in\mathfrak{g}_{\overline{0}}$, we have two subcases:

(1.1) There are $u_{1},u_{2}\in\mathfrak{g}_{\overline{0}}$ such that $\{u_{1},u_{2},h\}$ is a basis of $\mathfrak{g}$ and $[u_{1},u_{2}]=h$, which implies that $\mathfrak{g}$ is a Hom-Lie algebra.

(1.2) There are $v_{1},v_{2}\in\mathfrak{g}_{\overline{1}}$ such that $\{h|v_{1},v_{2}\}$ is a basis of $\mathfrak{g}$ and $[v_{1},v_{2}]=h$. Then the Hom-Lie superalgebra will be denoted by $(\mathfrak{h}_{1},\alpha)$.

Case $2$. If $h\in\mathfrak{g}_{\overline{1}}$, there exist $u\in\mathfrak{g}_{\overline{0}}$, $v\in\mathfrak{g}_{\overline{1}}$ such that $\{u|v,h\}$ is a basis of $\mathfrak{g}$ and $[u,v]=h$. In this case, we denote the Hom-Lie superalgebra by $(\mathfrak{h}_{2},\alpha)$.

Theorem 3.2**.**

Let $\mathfrak{g}$ be a multiplicative Heisenberg Hom-Lie(non-Lie) superalgebra of dimension three. Then $\mathfrak{g}$ must be isomorphic to one of the following:

$(1)$* (\mathfrak{h}_{1},\left(\begin{array}[]{ccc}\mu_{11}\mu_{22}&0&0\\ 0&\mu_{11}&0\\ 0&0&\mu_{22}\\ \end{array}\right)), $\mu_{11}\mu_{22}\neq 0$;*

$(2)$* (\mathfrak{h}_{1},\left(\begin{array}[]{ccc}\mu_{12}\mu_{21}&0&0\\ 0&0&\mu_{12}\\ 0&\mu_{21}&0\\ \end{array}\right)), $\mu_{12}\mu_{21}\neq 0$;*

$(3)$* (\mathfrak{h}_{1},\left(\begin{array}[]{ccc}0&0&0\\ 0&\mu_{11}&\mu_{12}\\ 0&0&0\\ \end{array}\right));*

$(4)$* (\mathfrak{h}_{2},\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&0&\mu_{0}\mu_{11}\\ \end{array}\right));*

$(5)$* (\mathfrak{h}_{2},\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&1&\mu_{11}\\ \end{array}\right)),(\mu_{0}-1)\mu_{11}=0,*

where $\mu_{0},\mu_{ij}\in\mathbb{C}$, $i,j=1,2$.

Proof. .

We analyze the cases $h\in\mathfrak{g}_{\overline{1}}$ and $h\in\mathfrak{g}_{\overline{0}}$ separately.

Case $1$. If $h\in\mathfrak{g}_{\overline{1}}$, there exists a basis $\{u|v,h\}$ of $\mathfrak{g}$ such that $[u,v]=h$. Suppose that \alpha=\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&\mu_{12}\\ 0&\mu_{21}&\mu_{22}\\ \end{array}\right), $\mu_{0},\mu_{ij}\in\mathbb{C}$, $i,j=1,2$.

We have that $\mathfrak{g}$ is multiplicative if and only if $\alpha([e_{i},e_{j}])=[\alpha(e_{i}),\alpha(e_{j})]$ for $i,j=1,2,3$, which implies $\mu_{12}=0$ and $\mu_{22}=\mu_{0}\mu_{11}$. Then we obtain that \alpha=\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&\mu_{21}&\mu_{0}\mu_{11}\\ \end{array}\right).

(a)If $\mu_{21}=0$, we obtain a Heisenberg Hom-Lie superalgebra

[TABLE]

(b)If $\mu_{21}\neq 0$, let

[TABLE]

Then

[TABLE]

If $\mu_{0}\neq 1$ and $\mu_{11}\neq 0$, then $b_{21}=-(1-\mu_{0})^{-1}{\mu_{11}}^{-1}\mu_{21}b_{0}{b_{11}}$ yields

[TABLE]

which induces a Heisenberg Hom-Lie superalgebra like the one in (a).

Otherwise, i.e.$\mu_{0}=1$ or $\mu_{11}=0$, then $b_{0}={\mu_{21}}^{-1}$ yields \phi\alpha\phi^{-1}=\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&1&\mu_{11}\\ \end{array}\right). We can obtain a new Heisenberg Hom-Lie superalgebra

[TABLE]

Case $2$. If $h\in\mathfrak{g}_{\overline{0}}$, there exist $v_{1},v_{2}\in\mathfrak{g}_{\overline{1}}$ such that $\{h|v_{1},v_{2}\}$ is a basis of $\mathfrak{g}$ and $[v_{1},v_{2}]=h$. In this case, we can get three Heisenberg Hom-Lie superalgebras:

[TABLE]

and (\mathfrak{h}_{1},\left(\begin{array}[]{ccc}0&0&0\\ 0&\mu_{11}&\mu_{12}\\ 0&0&0\\ \end{array}\right)).∎

4 The adjoint cohomology of Heisenberg Hom-Lie superalgebras

Let $\mathfrak{g}=(V,[\cdot,\cdot],\alpha)$ be a Hom-Lie superalgebra. Let $x_{1},\cdots,x_{k}$ be $k$ homogeneous elements of $V$ and $(x_{1},\cdots,x_{k})\in\wedge^{k}V$. Then we denote by $|(x_{1},\cdots,x_{k})|=|x_{1}|+\cdots+|x_{k}|$ the parity of $(x_{1},\cdots,x_{k})$. The set $C^{k}_{\alpha}(\mathfrak{g},\mathfrak{g})$ of $k$-hom-cochains of $\mathfrak{g}=(V,[\cdot,\cdot],\alpha)$ is the set of $k$-linear maps $\varphi:\wedge^{k}V\rightarrow V$ satisfying

[TABLE]

[TABLE]

for $x_{1},\cdots,x_{k}\in V$, $1\leqslant i\leqslant k-1$. In particular, $C^{0}_{\alpha}(\mathfrak{g},\mathfrak{g})=\{x\in\mathfrak{g}|\alpha(x)=x\}$. Denote by $|\varphi|$ the parity of $\varphi$ and $|\varphi(x_{1},\cdots,x_{k})|=|(x_{1},\cdots,x_{k})|+|\varphi|$. We immediately get a direct sum decomposition $C^{k}_{\alpha}(\mathfrak{g},\mathfrak{g})=C^{k}_{\alpha}(\mathfrak{g},\mathfrak{g})_{\overline{0}}\oplus C^{k}_{\alpha}(\mathfrak{g},\mathfrak{g})_{\overline{1}}$.

A $k$-coboundary operator $\delta^{k}(\varphi):C^{k}_{\alpha}(\mathfrak{g},\mathfrak{g})\rightarrow C^{k+1}_{\alpha}(\mathfrak{g},\mathfrak{g})$ is defined by

[TABLE]

where $\hat{x_{i}}$ means that $x_{i}$ is omitted.

The $k$-cocycles space, $k$-coboundaries space and $k$-th cohomology space are defined as:

(1) $Z^{k}(\mathfrak{g},\mathfrak{g})=ker\delta^{k}$, $Z^{k}(\mathfrak{g},\mathfrak{g})_{\overline{j}}=Z^{k}(\mathfrak{g},\mathfrak{g})\cap C^{k}_{\alpha}(\mathfrak{g},\mathfrak{g})_{\overline{j}},\ j=0,1$;

(2) $B^{k}(\mathfrak{g},\mathfrak{g})=Im\delta^{k-1}$, $B^{k}(\mathfrak{g},\mathfrak{g})_{\overline{j}}=B^{k}(\mathfrak{g},\mathfrak{g})\cap C^{k}_{\alpha}(\mathfrak{g},\mathfrak{g})_{\overline{j}},\ j=0,1$;

(3)$H^{k}(\mathfrak{g},\mathfrak{g})=\frac{Z^{k}(\mathfrak{g},\mathfrak{g})}{B^{k}(\mathfrak{g},\mathfrak{g})}=H^{k}(\mathfrak{g},\mathfrak{g})_{\overline{0}}\oplus H^{k}(\mathfrak{g},\mathfrak{g})_{\overline{1}}$, where $H^{k}(\mathfrak{g},\mathfrak{g})_{\overline{j}}=\frac{Z^{k}(\mathfrak{g},\mathfrak{g})_{\overline{j}}}{B^{k}(\mathfrak{g},\mathfrak{g})_{\overline{j}}},\ j=0,1$.

Theorem 4.1**.**

The cohomology spaces of Heisenberg Hom-Lie superalgebras are:

$(1)$* For \mathfrak{g}=(\mathfrak{h}_{1},\left(\begin{array}[]{ccc}\mu_{11}\mu_{22}&0&0\\ 0&\mu_{11}&0\\ 0&0&\mu_{22}\\ \end{array}\right))(\mu_{11}\mu_{22}\neq 0),*

H^{1}(\mathfrak{g},\mathfrak{g})=\left\langle\left(\begin{array}[]{ccc}a_{22}+a_{33}&a_{12}\delta_{\mu_{22},1}&a_{13}\delta_{\mu_{11},1}\\ 0&a_{22}&0\\ 0&0&a_{33}\\ \end{array}\right)\right\rangle,**

H^{2}(\mathfrak{g},\mathfrak{g})=\left\langle\left(\begin{array}[]{cccccc}0&0&0&0&-\frac{1}{2}\mu_{22}a_{22}\delta_{\mu_{11},1}&-\frac{1}{2}\mu_{11}a_{34}\delta_{\mu_{22},1}\\ 0&\delta_{\mu_{11},1}a_{22}&0&0&0&0\\ 0&0&0&\delta_{\mu_{22},1}a_{34}&0&0\\ \end{array}\right)\right\rangle;**

$(2)$* For \mathfrak{g}=(\mathfrak{h}_{1},\left(\begin{array}[]{ccc}\mu_{12}\mu_{21}&0&0\\ 0&0&\mu_{12}\\ 0&\mu_{21}&0\\ \end{array}\right))(\mu_{12}\mu_{21}\neq 0),*

H^{1}(\mathfrak{g},\mathfrak{g})=\left\langle\left(\begin{array}[]{ccc}2a_{22}&a_{12}\delta_{\mu_{12}\mu_{21},1}&a_{12}\delta_{\mu_{12}\mu_{21},1}\\ 0&a_{22}&0\\ 0&0&a_{22}\\ \end{array}\right)\right\rangle,**

H^{2}(\mathfrak{g},\mathfrak{g})=\left\langle\left(\begin{array}[]{cccccc}0&0&0&0&0&0\\ 0&0&\mu_{12}a_{23}\delta_{\mu_{12}\mu_{21},1}&-2{\mu_{12}}^{2}a_{23}\delta_{\mu_{12}\mu_{21},1}&0&0\\ 0&-2\mu_{21}a_{23}\delta_{\mu_{12}\mu_{21},1}&a_{23}\delta_{\mu_{12}\mu_{21},1}&0&0&0\\ \end{array}\right)\right\rangle;**

$(3)$* For \mathfrak{g}=(\mathfrak{h}_{1},\left(\begin{array}[]{ccc}0&0&0\\ 0&\mu_{11}&\mu_{12}\\ 0&0&0\\ \end{array}\right)),*

H^{1}(\mathfrak{g},\mathfrak{g})=\left\langle\left(\begin{array}[]{ccc}(a_{22}+a_{33})\delta_{\mu_{12},0}&0&a_{13}\\ 0&a_{22}\delta_{\mu_{12},0}&0\\ 0&0&a_{33}\delta_{\mu_{12},0}\\ \end{array}\right)\right\rangle,**

$H^{2}(\mathfrak{g},\mathfrak{g})=$**

\left\langle\left(\begin{array}[]{cccccc}0&0&0&\delta_{\mu_{12},0}a_{14}&0&a_{16}\\ 0&a_{22}\delta_{\mu_{11}(\mu_{11}-1),0}&a_{23}\delta_{\mu_{11},0}+\mu_{12}a_{22}\delta_{\mu_{11},1}&a_{24}\delta_{\mu_{11},0}+\epsilon a_{22}\delta_{\mu_{11},1}&0&\delta_{\mu_{11},0}a_{26}\\ 0&0&0&0&0&0\\ \end{array}\right)\right\rangle,

where $\epsilon={\mu_{11}}^{-1}{\mu_{12}}^{2}$;

$(4)$* For \mathfrak{g}=(\mathfrak{h}_{2},\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&0&\mu_{0}\mu_{11}\\ \end{array}\right)),*

H^{1}(\mathfrak{g},\mathfrak{g})=\left\langle\left(\begin{array}[]{ccc}a_{11}&0&0\\ a_{21}\delta_{\mu_{0},\mu_{11}}&a_{22}&0\\ a_{31}\delta_{\mu_{0}(\mu_{11}-1),0}&a_{32}\delta_{(\mu_{0}-1)\mu_{11},0}&a_{11}+a_{22}\\ \end{array}\right)\right\rangle,**

$H^{2}(\mathfrak{g},\mathfrak{g})=$**

\left\langle\left(\begin{array}[]{cccccc}0&0&0&0&\frac{1}{2}\mu_{0}a_{22}\delta_{\mu_{11},1}&-\mu_{0}a_{34}\delta_{\mu_{0}\mu_{11},1}\\ 0&a_{22}\delta_{\mu_{11},1}&0&0&0&a_{26}\delta_{({\mu_{0}}^{2}-1)\mu_{11},0}\\ 0&a_{32}\delta_{\mu_{11},0}&a_{33}\delta_{\mu_{11},0}&a_{34}\delta_{\mu_{0}\mu_{11}(\mu_{0}\mu_{11}-1),0}&0&a_{36}\delta_{\mu_{0}(\mu_{0}-1)\mu_{11},0}\\ \end{array}\right)\right\rangle;**

$(5)$* For \mathfrak{g}=(\mathfrak{h}_{2},\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&1&\mu_{11}\\ \end{array}\right)),(\mu_{0}-1)\mu_{11}=0,*

H^{1}(\mathfrak{g},\mathfrak{g})=\left\langle\left(\begin{array}[]{ccc}0&0&0\\ 0&a_{33}&0\\ a_{31}\delta_{\mu_{0},\mu_{11}}&a_{32}\delta_{(\mu_{0}-1)\mu_{11},0}&a_{33}\\ \end{array}\right)\right\rangle,**

$H^{2}(\mathfrak{g},\mathfrak{g})=$**

\left\langle\left(\begin{array}[]{cccccc}0&a_{12}\delta_{\mu_{0},0}\delta_{\mu_{11},0}&a_{13}\delta_{\mu_{0},0}\delta_{\mu_{11},0}&0&0&a_{16}\delta_{\mu_{0},0}\\ 0&2a_{33}\delta_{\mu_{11},1}+a_{34}\delta_{\mu_{11},0}&0&0&\mu_{0}a_{36}&0\\ 0&a_{32}\delta_{\mu_{11}(\mu_{11}-1),0}&a_{33}\delta_{\mu_{11}(\mu_{11}-1),0}&a_{34}\delta_{\mu_{0},2}\delta_{\mu_{11},0}&0&a_{36}\\ \end{array}\right)\right\rangle.**

Proof. .

It is easy to obtain $C^{k}_{\alpha}(\mathfrak{g},\mathfrak{g})$ for $k=1,2$ by Eq.(4.1) and Eq.(4.2).

Taking \mathfrak{g}=(\mathfrak{h}_{1},\left(\begin{array}[]{ccc}\mu_{11}\mu_{22}&0&0\\ 0&\mu_{11}&0\\ 0&0&\mu_{22}\\ \end{array}\right),\ \mu_{11}\mu_{22}\neq 0 for example,

C^{1}_{\alpha}(\mathfrak{g},\mathfrak{g})=\left(\begin{array}[]{ccc}a_{11}&0&0\\ 0&a_{22}&a_{23}\delta_{\mu_{11},\mu_{22}}\\ 0&a_{32}\delta_{\mu_{11},\mu_{22}}&a_{33}\\ \end{array}\right),

C^{2}_{\alpha}(\mathfrak{g},\mathfrak{g})=\left(\begin{array}[]{cccccc}0&a_{12}\delta_{\mu_{11},\mu_{22}}&a_{13}&a_{14}\delta_{\mu_{11},\mu_{22}}&0&0\\ 0&0&0&0&a_{25}\delta_{\mu_{11}\mu_{22},1}&a_{26}\delta_{{\mu_{22}}^{2},1}\\ 0&0&0&0&a_{35}\delta_{{\mu_{11}}^{2},1}&a_{36}\delta_{\mu_{11}\mu_{22},1}\\ \end{array}\right).

Let \varphi_{0}=\left(\begin{array}[]{ccc}a_{11}&0&0\\ 0&a_{22}&a_{23}\\ 0&a_{32}&a_{33}\\ \end{array}\right)\in C^{1}_{\alpha}(\mathfrak{g},\mathfrak{g})_{\overline{0}}. Then

[TABLE]

Moreover, we have $\varphi_{0}\in Z^{1}(\mathfrak{g},\mathfrak{g})_{\overline{0}}$ if and only if $\delta^{1}(\varphi_{0})=0$.

In the same way, we suppose \varphi_{1}=\left(\begin{array}[]{ccc}0&a_{12}&a_{13}\\ a_{21}&0&0\\ a_{31}&0&0\\ \end{array}\right)\in C^{1}(\mathfrak{g},\mathfrak{g})_{\overline{1}} and immediately get $Z^{1}(\mathfrak{g},\mathfrak{g})_{\overline{1}}$ and $B^{2}(\mathfrak{g},\mathfrak{g})_{\overline{1}}$.

Now suppose \alpha=\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&\mu_{12}\\ 0&\mu_{21}&\mu_{22}\\ \end{array}\right) and \psi_{0}=\left(\begin{array}[]{cccccc}0&a_{12}&a_{13}&a_{14}&0&0\\ 0&0&0&0&a_{25}&a_{26}\\ 0&0&0&0&a_{35}&b_{36}\\ \end{array}\right)\in C^{2}_{\alpha}(\mathfrak{g},\mathfrak{g})_{\overline{0}}(\psi_{1}=\left(\begin{array}[]{cccccc}0&0&0&0&a_{15}&a_{16}\\ 0&a_{22}&a_{23}&a_{24}&0&0\\ 0&a_{32}&a_{33}&a_{34}&0&0\\ \end{array}\right)\in C^{2}_{\alpha}(\mathfrak{g},\mathfrak{g})_{\overline{1}}). We know that $\psi_{0}\in Z^{2}_{\alpha}(g,g)_{\overline{0}}$

($\psi_{1}\in Z^{2}_{\alpha}(g,g)_{\overline{1}}$) if and only if $\delta^{1}(\psi_{0})=0$($\delta^{1}(\psi_{1})=0$).∎

5 Infinitesimal deformations of Heisenberg Hom-Lie superalgebras

Let $\mathfrak{g}=(V,[\cdot,\cdot]_{0},\alpha)$ be a Hom-Lie superalgebra and $\varphi:V\times V\longrightarrow V$ be an even bilinear map commuting with $\alpha$. A bilinear map $[\cdot,\cdot]_{t}=[\cdot,\cdot]_{0}+t\varphi(\cdot,\cdot)$ is called an infinitesimal deformation of $\mathfrak{g}$ if $\varphi$ satisfies

[TABLE]

[TABLE]

for $x,y,z\in V$. The previous E.q.(5.1) imply $\varphi$ is skew-supersymmetric. We denote

[TABLE]

and then Eq.(5.2) can be denoted by $[\cdot,\cdot]_{t}\circ[\cdot,\cdot]_{t}=0$.

Lemma 5.1**.**

Let $\mathfrak{g}=(V,[\cdot,\cdot]_{0},\alpha)$ be a Hom-Lie superalgebra and $[\cdot,\cdot]_{t}=[\cdot,\cdot]_{0}+t\varphi(\cdot,\cdot)$ be an infinitesimal deformation of $g=(V,[\cdot,\cdot]_{0},\alpha)$. Then $\varphi\in Z^{2}(\mathfrak{g},\mathfrak{g})_{\overline{0}}$.

Proof. .

By Eq.(5.2), we have

[TABLE]

Note that

[TABLE]

Hence $\varphi\in Z^{2}(\mathfrak{g},\mathfrak{g})_{\overline{0}}$.∎

By Eq.(5.3), we can see that $[\cdot,\cdot]_{t}$ is an infinitesimal deformation if and only if $\varphi\circ\varphi=0$.

Let $\mathfrak{g}_{t}=(V,[\cdot,\cdot]_{t},\alpha)$ and $\mathfrak{g}_{t}^{\prime}=(V,[\cdot,\cdot]_{t}^{\prime},\alpha^{\prime})$ be two deformations of $\mathfrak{g}$, where $[\cdot,\cdot]_{t}=[\cdot,\cdot]_{0}+t\varphi(\cdot,\cdot)$ and $[\cdot,\cdot]_{t}^{\prime}=[\cdot,\cdot]_{0}+t\psi(\cdot,\cdot)$. If there exists a linear automorphism $\Phi_{t}=\rm{id}+t\phi(\phi\in C^{1}_{\alpha}(g,g)_{\overline{0}})$, satisfying

[TABLE]

we call the deformations $\mathfrak{g}_{t}$ and $\mathfrak{g}_{t}^{{}^{\prime}}$ are equivalent. It is obvious that $\mathfrak{g}_{t}$ and $\mathfrak{g}_{t}^{{}^{\prime}}$ are equivalent if and only if $\varphi_{1}-\psi_{1}\in B^{2}(\mathfrak{g},\mathfrak{g})_{\overline{0}}$. Therefore, the set of infinitesimal deformations of $\mathfrak{g}$ can be parameterized by $H^{2}(\mathfrak{g},\mathfrak{g})_{\overline{0}}$.

A deformation $\mathfrak{g}_{t}$ of Hom-Lie superalgebras $\mathfrak{g}$ is called trivial if it is equivalent to $\mathfrak{g}$.

Corollary 5.2**.**

All the infinitesimal deformations of the following Heisenberg Hom-Lie superalgebras are trivial:

$(1)$* (\mathfrak{h}_{1},\left(\begin{array}[]{ccc}\mu_{11}\mu_{22}&0&0\\ 0&\mu_{11}&0\\ 0&&\mu_{22}\\ \end{array}\right)),\ \mu_{11}\mu_{22}\neq 0;*

$(2)$* (\mathfrak{h}_{1},\left(\begin{array}[]{ccc}\mu_{12}\mu_{21}&0&0\\ 0&0&\mu_{12}\\ 0&\mu_{21}&0\\ \end{array}\right)), $\mu_{12}\mu_{21}\neq 0$;*

$(3)$* (\mathfrak{h}_{2},\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&0&\mu_{0}\mu_{11}\\ \end{array}\right)),\ \mu_{0}({\mu_{0}}^{2}-1)\mu_{11}\neq 0.*

Proof. .

All the infinitesimal deformations of a Heisenberg Hom-Lie superalgebras are trivial if and only if $H^{2}(\mathfrak{g},\mathfrak{g})_{\overline{0}}=0$. ∎

In the following, we discuss the non-trivial infinitesimal deformations of Heisenberg Hom-Lie superalgebras. We will distinguish two separate cases: the ones that are also Lie superalgebras and those are not.

We recall the classification of three-dimensional Lie superalgebras([11]):

Theorem 5.3**.**

Let $L=(V,[\cdot,\cdot])$ be a Lie superalgebras with a direct sum decomposition $V=V_{\overline{0}}\oplus V_{\overline{1}}$, where $dimV_{\overline{0}}=1$ and $dimV_{\overline{1}}=2$. There are $e_{1}\in V_{\overline{0}}$ and $e_{2},e_{3}\in V_{\overline{1}}$ such that $\{e_{1}|e_{2},e_{3}\}$ is a basis of $V$ . Then $L$ must be isomorphic to one of the following:

[TABLE]

We construct a new Lie superalgebra ${L_{2}}^{{}^{\prime}}=(V,[\cdot,\cdot]_{0})$. Let $\{h|v_{1},v_{2}\}$ be a basis of $V$ satisfying $[v_{1},v_{2}]_{0}=h$. Ther is an even bijective morphism $\phi:(V,[\cdot,\cdot]_{0})\rightarrow L_{2}$

[TABLE]

such that $\phi([x,y]_{0})=[\phi(x),\phi(y)],\forall x,y\in V.$ Then ${L_{2}}^{{}^{\prime}}$ is isomorphic to ${L_{2}}$ and we shall replace ${L_{2}}$ with it in Theorem 5.3.

Proposition 5.4**.**

A non-trivial infinitesimal deformation of Heisenberg Hom-Lie superalgebra $(\mathfrak{h}_{1},\alpha)$, that is also a Lie superalgebra, is isomorphism to

[TABLE]

Proof. .

Denote by $(\mathfrak{h}_{1},\alpha)=(V,[\cdot,\cdot]_{0},\alpha)$. There is a basis $\{h|v_{1},v_{2}\}$ such that $[v_{1},v_{2}]_{0}=h$ and others are zero.

If \alpha=\left(\begin{array}[]{ccc}\mu_{11}\mu_{22}&0&0\\ 0&\mu_{11}&0\\ 0&0&\mu_{22}\\ \end{array}\right)($\mu_{11}\mu_{22}\neq 0$) or \left(\begin{array}[]{ccc}\mu_{12}\mu_{21}&0&0\\ 0&0&\mu_{12}\\ 0&\mu_{21}&0\\ \end{array}\right)($\mu_{12}\mu_{21}\neq 0$), all the infinitesimal deformations of $(\mathfrak{g}_{1},\alpha)$ are trivial.

Consider \alpha=\left(\begin{array}[]{ccc}0&0&0\\ 0&\mu_{11}&\mu_{12}\\ 0&0&0\\ \end{array}\right). Let \varphi=\left(\begin{array}[]{cccccc}0&0&0&a_{14}\delta_{\mu_{12},0}&0&0\\ 0&0&0&0&0&a_{26}\delta_{\mu_{11},0}\\ 0&0&0&0&0&0\\ \end{array}\right) be an even 2-cocycle and $[\cdot,\cdot]_{t}=[\cdot,\cdot]_{0}+t\varphi(\cdot,\cdot)$. Then $\varphi\circ\varphi=0$ and $\mathfrak{g}_{t}=(V,[\cdot,\cdot]_{t},\alpha^{\prime})$ is an infinitesimal deformation of $(\mathfrak{h}_{1},\alpha)$. Moreover, $\mathfrak{g}_{t}$ is a Lie superalgebra if and only if $[\cdot,\cdot]_{t}\circ[\cdot,\cdot]_{t}=0$, i.e. $a_{26}=0$. All deformations are given in Table $1$. ∎

[TABLE]

Proposition 5.5**.**

The non-trivial infinitesimal deformations of $(\mathfrak{h}_{2},\alpha)$, that are also Lie superalgebras, are isomorphism to:

$(1)$* For \alpha=\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&0&\mu_{0}\mu_{11}\\ \end{array}\right)(\mu_{0}({\mu_{0}}^{2}-1)\mu_{11}=0),*

$(a)$* \left({L_{3}}^{\lambda},\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&0&\mu_{11}\\ \end{array}\right)\right)(\mu_{0}-1)\mu_{11}=0,  $(b)$ \left({L_{3}}^{0},\left(\begin{array}[]{ccc}-1&0&0\\ 0&\mu_{11}&-\mu_{11}\\ 0&0&-2\mu_{11}\\ \end{array}\right)\right),*

$(c)$* \left({L_{3}}^{-1},\left(\begin{array}[]{ccc}0&0&0\\ 0&\frac{1}{2}\mu_{11}&\frac{1}{2}\xi\mu_{11}\\ 0&\frac{1}{2}\xi^{-1}\mu_{11}&\frac{1}{2}\mu_{11}\\ \end{array}\right)\right);*

$(2)$* For \alpha=\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&1&\mu_{11}\\ \end{array}\right),*

$(a)$* \left({L_{3}}^{0},\left(\begin{array}[]{ccc}0&0&0\\ 0&0&0\\ 0&0&\eta\\ \end{array}\right)\right),  $(b)$ \left(L_{4},\left(\begin{array}[]{ccc}1&0&0\\ 0&\mu_{11}&0\\ 0&0&\mu_{11}\\ \end{array}\right)\right),*

$(c)$* \left({L_{3}}^{\mu_{0}},\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&0&{\mu_{0}}^{-1}\\ 0&0&0\\ \end{array}\right)\right),\mu_{0}\neq 0,1.*

Proof. .

Denote by $(\mathfrak{h}_{2},\alpha)=(V,[\cdot,\cdot]_{0},\alpha)$. There is a basis $\{u|v,h\}$ such that $[u,v]_{0}=h$ and others are zero.

$(1)$ Consider \alpha=\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&0&\mu_{0}\mu_{11}\\ \end{array}\right), $\mu_{0}({\mu_{0}}^{2}-1)\mu_{11}=0$.

Let \varphi=\left(\begin{array}[]{ccccc}0&0&0&0&0\\ 0&0&0&0&a_{26}\delta_{({\mu_{0}}^{2}-1)\mu_{11},0}\\ 0&0&0&0&a_{36}\delta_{{\mu_{0}}({\mu_{0}}-1)\mu_{11},0}\\ \end{array}\right) is an even 2-cocycle. Then $\varphi\circ\varphi=0$ and we obtain an infinitesimal deformation $\mathfrak{g}_{t}=((V,[\cdot,\cdot]_{0},\alpha)^{\prime})$ is an infinitesimal deformation, where $[\cdot,\cdot]_{t}=[\cdot,\cdot]_{0}+t\varphi(\cdot,\cdot)$. It is easy to see that is also a Lie algebra. We analyze the cases $(a)(\mu_{0}-1)\mu_{11}=0$, $(b)\mu_{0}=-1$ and $(c)\mu_{0}=0$ separately, which are given in Table 2.

$(2)$ Consider \alpha=\left(\begin{array}[]{ccc}\mu_{0}&0&0\\ 0&\mu_{11}&0\\ 0&1&\mu_{11}\\ \end{array}\right),\ (\mu_{0}-1)\mu_{11}=0.

Let \varphi=\left(\begin{array}[]{cccccc}0&a_{12}\delta_{\mu_{0},0}\delta_{\mu_{11},0}&a_{13}\delta_{\mu_{0},0}\delta_{\mu_{11},0}&0&0&0\\ 0&0&0&0&\mu_{0}a_{36}&0\\ 0&0&0&0&0&a_{36}\\ \end{array}\right) be an even 2-cocycle. Then $\mathfrak{g}_{t}=((V,[\cdot,\cdot]_{0},\alpha)^{\prime})$ is an infinitesimal deformation, where $[\cdot,\cdot]_{t}=[\cdot,\cdot]_{0}+t\varphi(\cdot,\cdot)$, if and only if $\varphi\circ\varphi=0$. We analyze the cases $(a)\mu_{0}=\mu_{11}=0$, $(b)\mu_{0}=1$ and $(c)\mu_{0}\neq 0,1,\ \mu_{11}=0$ separately.

For case (a), $\varphi\circ\varphi=0$ implies $a_{12}=a_{13}=0$ or $a_{36}=0$. Furthermore, if $a_{12}=a_{13}=0$, the deformation $\mathfrak{g}_{t}$ is also a Lie superalgebra.

For cases (b) and (c), $\mathfrak{g}_{t}$ is an infinitesimal deformation for all $\varphi$. The deformation is also a Lie superalgebra if $a_{12}=a_{13}=0$.

The deformations of $(a),\ (b)$ and $(c)$ are given in Table 3.

∎

Proposition 5.4 and Proposition 5.5 give the infinitesimal deformations of Heisenberg Hom-Lie superalgebras that are also Lie superalgebras. Before discussing the rest deformations, we will recall some multiplicative Hom-Lie superalgebras and those can be find in the classification of multiplicative Hom-Lie superalgebras of [10]. Let $V$ be a superspace with a direct sum decomposition $V=V_{\overline{0}}\oplus V_{\overline{1}}$, $[\cdot,\cdot]$ be an even bilinear map and $\sigma$ be an even linear map on $V$. Let $\{e_{1}|e_{2},e_{3}\}$ be a basis of $V$. The following are three Hom-Lie superalgebras on $V$:

L_{1,2}^{43,a}:[e_{1},e_{2}]=0,[e_{1},e_{3}]=\beta e_{2},[e_{2},e_{2}]=0,[e_{2},e_{3}]=0,[e_{3},e_{3}]=\gamma e_{1},\sigma=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&a\\ 0&0&0\\ \end{array}\right) $(a\neq 0),$

L_{1,2}^{45,a}:[e_{1},e_{2}]=0,[e_{1},e_{3}]=\beta e_{2},[e_{2},e_{2}]=0,[e_{2},e_{3}]=\nu e_{1},[e_{3},e_{3}]=\gamma e_{1},\sigma=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&a\\ 0&0&0\\ \end{array}\right) $(a\neq 0),$

L_{1,2}^{46,a,b}:[e_{1},e_{2}]=0,[e_{1},e_{3}]=\beta e_{2},[e_{2},e_{2}]=0,[e_{2},e_{3}]=\mu e_{1},[e_{3},e_{3}]=0,\sigma=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&a\\ 0&0&b\\ \end{array}\right) $(a^{2}+b^{2}\neq 0)$.

The following Proposition 5.6 and Proposition 5.7 will characterize the infinitesimal deformations of Heisenberg Hom-Lie superalgebras that are not Lie superalgebras.

Proposition 5.6**.**

An infinitesimal deformation of Heisenberg Hom-Lie superalgebra $(\mathfrak{h}_{1},\alpha)$, which is not a Lie superalgebra, is isomorphic to $L_{1,2}^{46,a,0}(a\neq 0)$.

Proof. .

By the proof of Proposition 5.4, we know that $(\mathfrak{h}_{1},\alpha)$ have an infinitesimal deformation(not a Lie superalgebra) if and only if

[TABLE]

There is a basis $\{h|v_{1},v_{2}\}$ of $V$ such that $[v_{1},v_{2}]_{0}=h$. Therefore $\mathfrak{g}_{t}=(V,[\cdot,\cdot]_{t},\alpha^{\prime})$ is an infinitesimal deformation, where $[\cdot,\cdot]_{t}=[\cdot,\cdot]_{0}+t\varphi$ and $[v_{1},v_{2}]_{t}=h$, $[h,v_{2}]_{t}=a_{26}v_{1}(a_{26}\neq 0).$ Then the deformation $\mathfrak{g}_{t}$ is isomorphic to $L_{1,2}^{46,\mu_{12},0}:$

[e_{1},e_{2}]=0,[e_{1},e_{3}]=a_{26}e_{2},[e_{2},e_{2}]=0,[e_{2},e_{3}]=e_{1},[e_{3},e_{3}]=0,\sigma=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&\mu_{12}\\ 0&0&0\\ \end{array}\right). ∎

Proposition 5.7**.**

An infinitesimal deformation of $(\mathfrak{h}_{2},\alpha)$, which is not a Lie superalgebra, is isomorphic to $L_{1,2}^{43,1}$, $L_{1,2}^{45,1}$, or $L_{1,2}^{46,1,0}$.

Proof. .

By the proof of Proposition 5.5, Heisenberg Hom-Lie superalgebra $(\mathfrak{h}_{2},\alpha)$ have an infinitesimal deformations if and only if

[TABLE]

There is a basis $\{u|v,h\}$ of $V$ such that $[u,v]_{0}=h$. Therefore $\mathfrak{g}_{t}=(V,[\cdot,\cdot]_{t},\alpha^{\prime})$ is an infinitesimal deformation, where $[\cdot,\cdot]_{t}=[\cdot,\cdot]_{0}+t\varphi$ and $[u,v]_{t}=h$, $[v,v]_{t}=a_{12}u$, $[v,h]_{t}=a_{13}u$. We analyze into three cases.

$(a)$ If $a_{12}\neq 0$ and $a_{13}\neq 0$, the infinitesimal deformation $\mathfrak{g}_{t}$ is isomorphic to

L_{1,2}^{45,1}:[e_{1},e_{2}]=0,[e_{1},e_{3}]=e_{2},[e_{2},e_{2}]=0,[e_{2},e_{3}]=a_{13}e_{1},[e_{3},e_{3}]=a_{12}e_{3},\sigma=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&1\\ 0&0&0\\ \end{array}\right).

$(b)$ If $a_{12}=0$ and $a_{13}\neq 0$, $\mathfrak{g}_{t}$ is isomorphic to:

L_{1,2}^{46,1,0}:[e_{1},e_{2}]=0,[e_{1},e_{3}]=e_{2},[e_{2},e_{2}]=0,[e_{2},e_{3}]=a_{13}e_{1},[e_{3},e_{3}]=0,\sigma=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&1\\ 0&0&0\\ \end{array}\right).

$(c)$ If $a_{12}\neq 0$ and $a_{13}=0$, $\mathfrak{g}_{t}$ is isomorphic to:

L_{1,2}^{43,1}:[e_{1},e_{2}]=0,[e_{1},e_{3}]=e_{2},[e_{2},e_{2}]=0,[e_{2},e_{3}]=0,[e_{3},e_{3}]=a_{12}e_{1},\sigma=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&1\\ 0&0&0\\ \end{array}\right). ∎