Structure and cohomology of 3-Lie-Rinehart superalgebras

Abstract

We introduce a concept of 3-Lie-Rinehart superalgebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we study the relationships between a Lie-Rinehart superalgebra and its induced 3-Lie-Rinehart superalgebra. The deformations of 3-Lie-Rinehart superalgebra are considered via a cohomology theory.

Keywords:

• Cohomology
• deformation
• Lie-Rinehart superalgebra
• 3-Lie-Rinehart superalgebra

2020 Mathematics subject classification:

• 17A40
• 17B05
• 17B60
• 17B66
• 53D17
• 17B61

1. Introduction

The notion of Lie-Rinehart algebras was introduced by J. Herz in [32] and mainly developed in [49, 50]. A Lie-Rinehart algebra can be thought as a Lie $\mathbb{K}$-algebra, which is also an A-module, where A is an associative and commutative $\mathbb{K}$-algebra, in such a way that both structures are related by a compatibility condition. A first approach to this class of algebras can be found in [34, 36].

G. Rinehart developed in [50] a formalism of differential forms for general commutative algebras which relies on the notion of $(\mathbb{K},A)$-Lie algebra where $\mathbb{K}$ is a commutative ring with unit and A is a commutative $\mathbb{K}$-algebra. The notion of Lie-Rinehart algebra includes an abstract algebraic characterization of the algebraic structure which underlies a Lie algebroid [33, 34, 44, 45]. Thus a Lie-Rinehart algebra is an algebraic generalization of the notion of a Lie algebroid: the space of sections of a vector bundle is replaced by a module over a ring, a vector field by a derivation of the ring. For further details and a history of the notion of Lie-Rinehart algebra, we refer to [37]. Lie-Rinehart algebras have been investigated further in many papers [19, 22, 26, 34–36, 38–40, 44].

Some generalizations of Lie-Rinehart algebras, such as Lie-Rinehart superalgebras [25] or restricted Lie-Rinehart algebras [27], have been recently studied. The cohomology $H_{\mathit{Rin}}^{*}(L;M)$ of a Lie-Rinehart algebra L with coefficients in a Lie-Rinehart module M was first defined by Rinehart [50] and further developed in [34].

In [22–24], the authors introduced a notion of cross modules of Lie-Rinehart algebras. Hom-Lie-Rinehart algebras and their extensions in the small dimension cohomology space was introduced and studied in [46]. Hom-Rinehart algebras have close relations with Hom-Gerstenhaber Algebras and Hom-Lie Algebroids [47, 48].

The study of 3-Lie algebras [29] gets a lot of attention since it has close relationships with Lie algebras, Hom-Lie algebras, commutative associative algebras, and cubic matrices [14–16, 20]. For example, it is applied to the study of Nambu mechanics and the study of supersymmetry and gauge symmetry transformations of the world-volume theory of multiple coincident M2-branes [10, 46, 51, 52]. A notion of n-Lie Rinehart algebras was introduced recently in [18] and extensions and crossed modules were developed for such algebras.

In 1996, the concept of n-Lie superalgebras was firstly introduced by Daletskii and Kushnirevich in [28]. Moreover, Cantarini and Kac gave a more general concept of n-Lie superalgebras again in 2010 (see [21]). n-Lie superalgebras are more general structures including n-Lie algebras, n-ary Nambu-Lie superalgebras, and Lie superalgebras. The construction of (n + 1)-Lie algebras induced by n-Lie algebras using a combination of bracket multiplication with a trace, motivated by the works on the quantization of the Nambu brackets [11], was generalized using the brackets of general Hom-Lie algebra or n-Hom-Lie algebra and trace-like linear forms satisfying some conditions depending on the linear maps defining the Hom-Lie or n-Hom-Lie algebras in [8, 9]. The structure of 3-Lie algebras induced by Lie algebras, classification of 3-Lie algebras and application to constructions of B.R.S. algebras have been considered in [1, 3, 4]. Interesting constructions of ternary Lie superalgebras in connection to superspace extension of Nambu-Hamilton equation is considered in [5]. In [17], a method was demonstrated of how to construct n-ary multiplications from the binary multiplication of a Hom-Lie algebra and a $(n-2)$-linear function satisfying certain compatibility conditions. Solvability and Nilpotency for n-Hom-Lie algebras and (n + 1)-Hom-Lie algebras induced by n-Hom-Lie algebras have been considered in [43]. In [1, 2, 5], 3-Lie superalgebras were constructed starting with a Lie superalgebras and various properties of such algebras considered. Related constructions for n-ary hom-Lie algebras and for n-ary hom-Lie superalgebras can be found in [6–9, 17, 30, 42]. The ternary case of (Hom-)Lie Rinehart algebras was developed in [12, 13, 31].

This paper is organized as follows. In Section 1, we recall the notion of Lie-Rinehart superalgebras and define their cohomology groups. We generalize the Lie-Rinehart superalgebra to the ternary case. Section 2 is devoted to some construction results. We begin by constructing 3-Lie-Rinehart superalgebras starting with a Lie-Rinehart superalgebras. We construct some new 3-Lie-Rinehart superalgebras from a given 3-Lie-Rinehart superalgebra. The notion of a module for a 3-Lie-Rinehart superalgebras appears in Section 3, and subsequently we introduce a cochain complex and cohomology of a 3-Lie-Rinehart superalgebras with coefficients in a module. Then we study relations between 1-cocycles and 2-cocycles of a Lie-Rinehart superalgebra and the induced 3-Lie-Rinehart superalgebra. At the end of this section, we study the deformation of 3-Lie-Rinehart superalgebras.

2. Definitions and notations

In this section, we review basic definitions of Lie superalgebras, 3-Lie superalgebras, Lie-Rinehart superalgebras and generalize the notion of 3-Lie-Rinhehart algebras to the super case.

Let $V=V_{\overline{0}}\oplus V_{\overline{1}}$ be a ${\mathbb{Z}}_2$-graded vector space. If $v\in V$ is a homogenous element, then its degree will be denoted by $\overline{v},$ where $\overline{v}\in {\mathbb{Z}}_2$ and ${\mathbb{Z}}_2=\{\overline{0},\overline{1}\}.$ Let End(V) be the ${\mathbb{Z}}_2$-graded vector space of endomorphisms of a ${\mathbb{Z}}_2$-graded vector space $V=V_{\overline{0}}\oplus V_{\overline{1}}.$ Denoted by $\mathcal{H}(V)$ the set of homogenous elements of V. For the composition of endomorphisms $a°b,$ the graded commutator, defined by $[a,b]=a°b-{(-1)}^{\overline{a}\overline{b}}b°a$ for homogeneous $a,b\in \mathit{End}(V),$ induces the structure of Lie superalgebra on the ${\mathbb{Z}}_2$-graded vector space End(V).

2.1. Lie-Rinehart superalgebras

Definition 1.1

([41]). A Lie superalgebra is a pair $(\mathfrak{g},[·,·])$ consisting of a ${\mathbb{Z}}_2$-graded vector space $\mathfrak{g}={\mathfrak{g}}_{\overline{0}}\oplus {\mathfrak{g}}_{\overline{1}},$ an even bilinear map $[·,·]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$ satisfying the following identities: (1.1) $$[x,y]=-{(-1)}^{\overline{x}\overline{y}}[y,x],(\mathit{\text{super}}\mathrm{ }\mathit{\text{skew}}-\mathit{\text{symetry}})$$(1.1) (1.2) $${\circlearrowleft }_{x,y,z}{(-1)}^{\overline{x}\overline{z}}[x,[y,z\left]\right]=0,(\mathit{\text{super}}-\mathit{\text{Jacobi}}\mathrm{ }\mathit{\text{identity}})$$(1.2) where x, y and z are homogeneous elements in $\mathfrak{g}.$

Definition 1.2.

A representation of a Lie superalgebra $(\mathfrak{g},[·,·])$ on a ${\mathbb{Z}}_2$-graded vector space $V=V_{\overline{0}}\oplus V_{\overline{1}}$ is an even linear map $\mu :\mathfrak{g}\to \mathfrak{g}\mathfrak{l}(V),$ such that the following identity is satisfied for all homogeneous $x,y\in \mathfrak{g}:$ (1.3) $$\mu ([x,y])=\mu (x)\mu (y)-{(-1)}^{\overline{x}\overline{y}}\mu (y)\mu (x).$$(1.3)

Now, we recall the definition of Lie-Rinehart superalgebra [25].

Definition 1.3.

A Lie-Rinehart superalgebra L over (an associative supercommutative superalgebra) A is a Lie superalgebra over $\mathbb{K}$ with an A-module structure and an even linear map $\mu :L\to \mathit{Der}(A),$ such that that following conditions hold:

1. μ is a representation of $(L,[·,·])$ on A.

2. $\mu (ax)=a\mu (x)$ for all $a\in A,x\in L.$

3. The compatibility condition: (1.4) $$\forall a\in \mathcal{H}(A),x,y\in \mathcal{H}(L):\hspace{1em}[x,ay]=\mu (x)ay+{(-1)}^{\overline{a}\overline{x}}a[x,y].$$(1.4)

Example 1.4.

Let us observe that Lie-Rinehart superalgebras over A with trivial map $\mu :L\to \mathit{Der}(A)$ are exactly Lie superalgebras. If $A=\mathbb{K},$ then Der(A) = 0 and there is no difference between Lie and Lie-Rinehart superalgebras. Therefore, the concept of Lie-Rinehart superalgebras generalizes the concept of Lie superalgebras.

Example 1.5.

Let A an associative supercommutative superalgebra. Then, the A-module $\mathit{Der}(A)\oplus A$ is a Lie-Rinehart superalgebra over A with the bracket $$[(D,a),(D\prime ,a\prime )]=([D,D\prime ],D(a\prime )-{(-1)}^{\overline{D}\prime \overline{a}}D\prime (a)),$$ and an even map $p_1:\mathit{Der}(A)\oplus A\to \mathit{Der}(A),$ the projection onto the first factor.

Now, we define the cohomology of a Lie-Rinehart superalgebra. First we introduce the notion of left module over a Lie-Rinehart superalgebra.

Definition 1.6.

Let M be an A-module. Then M is a left module over a Lie-Rinehart superalgebra L if there exits an even map $\theta :L\otimes M\to M$ such that:

1. θ is a representation of the Lie superalgebra $(L,[·,·])$ on M.

2. $\theta (ax,m)=a\theta (x,m)$ for all $a\in A,x\in L,m\in M.$

3. $\theta (x,am)={(-1)}^{\overline{a}\overline{x}}a\theta (x,m)+\mu (x)(a)m$ for all $x\in \mathcal{H}(L),a\in \mathcal{H}(A),m\in M.$

Let $(L,A,[·,·],\mu )$ be a Lie-Rinehart superalgebra, and let M be a left module over L. For Lie-Rinehart superalgebra L with coefficients in M, consider the ${\mathbb{Z}}_{+}$-graded space of $\mathbb{K}$-modules $$C^{*}(L;M):={\oplus }_{n\ge 0}{C}^n(L;M)$$ where $C^n(L;M)\subset \mathit{Hom}({\wedge }^{n}L,M)$ consisting of elements satisfying $$f(x_1,\dots ,a{x}_i,\dots ,x_n)={(-1)}^{\overline{a}({\overline{x}}_1+{\overline{x}}_2+\dots +{\overline{x}}_{i-1}+\overline{f})}af(x_1,\dots ,x_i,\dots ,x_n),$$ for all $x_i\in \mathcal{H}(L),i=1,\dots ,n$ and $a\in \mathcal{H}(A).$

Define the even linear map ${\delta }_{LR}:C^n(L;M)\to C^{n+1}(L;M)$ given by $$\begin{array}{c}{\delta }_{LR}f(x_1,\dots ,x_{n+1}):=\sum _{i=1}^{n+1}{(-1)}^{i+1+{\overline{x}}_i(\overline{f}+{\overline{x}}_1+\dots +{\overline{x}}_{i-1})}\theta (x_i,f(x_1,\dots ,{\widehat{x}}_i,\dots ,x_{n+1}))\\ +\sum _{1\le i<j\le n+1}{(-1)}^{({\overline{x}}_i+{\overline{x}}_j)({\overline{x}}_1+\dots +{\overline{x}}_{i-1}+{\overline{x}}_{i+1}+\dots +{\overline{x}}_{j-1})}f([x_i,x_j],x_1,\dots ,{\widehat{x}}_i,\dots ,{\widehat{x}}_j,\dots ,x_{n+1})\end{array}$$ for all $f\in C^n(L;M),x_i\in \mathcal{H}(L),$ where $1\le i\le n+1.$

With the above notation, the map δ_LR gives rise to a coboundary map.

Proposition 1.7.

If $f\in C^n(L;M)$, then ${\delta }_{LR}^{2}f\in C^{n+1}(L;M)$ and ${\delta }_{LR}^2=0.$

By the above proposition, $(C^{*}(L,M),{\delta }_{LR})$ is a cochain complex. The resulting cohomology of the cochain complex we define to be the cohomology space of Lie-Rinehart superalgebra $(L,A,[·,·],\mu )$ with coefficients in M, and we denote this cohomology as $H^{*}(L,M).$

2.2. 3-Lie-Rinehart superalgebras

Definition 1.8

([21]). A ${\mathbb{Z}}_2$-graded vector space $\mathfrak{g}={\mathfrak{g}}_{\overline{0}}\oplus {\mathfrak{g}}_{\overline{1}}$ is said to be a 3-Lie superalgebra, if it is endowed with an even trilinear map (bracket) $[·,·,·]:\mathfrak{g}\times \mathfrak{g}\times \mathfrak{g}\to \mathfrak{g},$ satisfying the following conditions: (1.5) $$[x,y,z]=-{(-1)}^{\overline{x}\overline{y}}[y,x,z],\hspace{1em}[x,y,z]=-{(-1)}^{\overline{y}\overline{z}}[x,z,y],$$(1.5) (1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) where $x,y,z,u,v\in \mathfrak{g}$ are homogeneous elements.

Proposition 1.9.

Let $\mathfrak{g}$ be a ${\mathbb{Z}}_2$-graded vector space together with a super skew-symmetric even linear map $[·,·,·]:{\otimes }^3\mathfrak{g}\to \mathfrak{g}$. Then $(\mathfrak{g},[·,·,·])$ is a 3-Lie superalgebra if and only if the following identities hold: (1.7) $$\begin{array}{c}\left[\right[x_1,x_2,x_3],x_4,x_5]-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}\left[\right[x_1,x_2,x_4],x_3,x_5]+{(-1)}^{{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_4)}\left[\right[x_1,x_3,x_4],x_2,x_5]\hfill \\ -{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3+{\overline{x}}_4)}\left[\right[x_2,x_3,x_4],x_1,x_5]=0,\hfill \end{array}$$(1.7) (1.8) $$\begin{array}{c}{(-1)}^{({\overline{x}}_4+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_2+{\overline{x}}_3)}\left[\right[x_4,x_5,x_1],x_2,x_3]+{(-1)}^{({\overline{x}}_4+{\overline{x}}_5)({\overline{x}}_2+{\overline{x}}_3)}[x_1,[x_4,x_5,x_2],x_3]\hfill \\ +{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}\left[\right[x_3,x_5,x_1],x_2,x_4]-{(-1)}^{({\overline{x}}_3+{\overline{x}}_5){\overline{x}}_2+{\overline{x}}_4{\overline{x}}_5}[x_1,[x_5,x_3,x_2],x_4]\hfill \\ +{(-1)}^{{\overline{x}}_3({\overline{x}}_4+{\overline{x}}_5)}[x_1,x_2,[x_4,x_5,x_3\left]\right]=0,\hfill \end{array}$$(1.8) for all $x_i\in \mathcal{H}(\mathfrak{g}),1\le i\le 5.$

Proof.

If $(\mathfrak{g},[·,·,·])$ is a 3-Lie superalgebra, then applying (1.6)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) to the last term in (1.6)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) for $x_i\in \mathcal{H}(\mathfrak{g}),1\le i\le 5$ and using (1.5)(1.5) $$[x,y,z]=-{(-1)}^{\overline{x}\overline{y}}[y,x,z],\hspace{1em}[x,y,z]=-{(-1)}^{\overline{y}\overline{z}}[x,z,y],$$(1.5) , $$\begin{array}{c}[x_1,x_2,[x_3,x_4,x_5\left]\right]=\left[\right[x_1,x_2,x_3],x_4,x_5]+{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2)}[x_3,[x_1,x_2,x_4],x_5]\\ +{(-1)}^{({\overline{x}}_3+{\overline{x}}_4)({\overline{x}}_1+{\overline{x}}_2)}[x_3,x_4,[x_1,x_2,x_5\left]\right]\\ =\left[\right[x_1,x_2,x_3],x_4,x_5]+{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2)}[x_3,[x_1,x_2,x_4],x_5]\\ +{(-1)}^{({\overline{x}}_3+{\overline{x}}_4)({\overline{x}}_1+{\overline{x}}_2)}\left[\right[x_3,x_4,x_1],x_2,x_5]\\ +{(-1)}^{{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_4)}[x_1,[x_3,x_4,x_2],x_5]\\ +[x_1,x_2,[x_3,x_4,x_5\left]\right],\end{array}$$ yields (1.7)(1.7) $$\begin{array}{c}\left[\right[x_1,x_2,x_3],x_4,x_5]-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}\left[\right[x_1,x_2,x_4],x_3,x_5]+{(-1)}^{{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_4)}\left[\right[x_1,x_3,x_4],x_2,x_5]\hfill \\ -{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3+{\overline{x}}_4)}\left[\right[x_2,x_3,x_4],x_1,x_5]=0,\hfill \end{array}$$(1.7) .

Similarly, applying (1.6)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) to the first and second terms on the right hand side of (1.6)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) for $x_i\in \mathcal{H}(\mathfrak{g}),1\le i\le 5$ and using (1.5)(1.5) $$[x,y,z]=-{(-1)}^{\overline{x}\overline{y}}[y,x,z],\hspace{1em}[x,y,z]=-{(-1)}^{\overline{y}\overline{z}}[x,z,y],$$(1.5) , $$\begin{array}{c}[x_1,x_2,[x_3,x_4,x_5\left]\right]={(-1)}^{({\overline{x}}_4+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_2+{\overline{x}}_3)}[x_4,x_5,[x_1,x_2,x_3]\\ -{(-1)}^{({\overline{x}}_3+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_2)+{\overline{x}}_4{\overline{x}}_5}[x_3,x_5,[x_1,x_2,x_4\left]\right]\\ +{(-1)}^{({\overline{x}}_3+{\overline{x}}_4)({\overline{x}}_1+{\overline{x}}_2)}[x_3,x_4,[x_1,x_2,x_5\left]\right]\\ ={(-1)}^{({\overline{x}}_4+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_2+{\overline{x}}_3)}\left[\right[x_4,x_5,x_1],x_2,x_3]\\ +{(-1)}^{({\overline{x}}_4+{\overline{x}}_5)({\overline{x}}_2+{\overline{x}}_3)}[x_1,[x_4,x_5,x_2],x_3]\\ +{(-1)}^{{\overline{x}}_3({\overline{x}}_4+{\overline{x}}_5)}[x_1,x_2,[x_4,x_5,x_3\left]\right]\\ -{(-1)}^{({\overline{x}}_3+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_2)+{\overline{x}}_4{\overline{x}}_5}\left[\right[x_3,x_5,x_1],x_2,x_4]\\ -{(-1)}^{({\overline{x}}_3+{\overline{x}}_5){\overline{x}}_2+{\overline{x}}_4{\overline{x}}_5}[x_1,[x_3,x_5,x_2],x_4]\\ -{(-1)}^{{\overline{x}}_4{\overline{x}}_5}[x_1,x_2,[x_3,x_5,x_4\left]\right]\\ +{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}[x_3,x_4,[x_1,x_2,x_5\left]\right],\end{array}$$ yields (1.8)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) .

Conversely, suppose that (1.7)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) and (1.8)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) hold. First (1.7)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) gives $$\begin{array}{c}\left[\right[x_1,x_2,x_3],x_4,x_5]={(-1)}^{{\overline{x}}_3{\overline{x}}_4}\left[\right[x_1,x_2,x_4],x_3,x_5]\\ \hspace{1em}-{(-1)}^{{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_4)}\left[\right[x_1,x_3,x_4],x_2,x_5]+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3+{\overline{x}}_4)}\left[\right[x_2,x_3,x_4],x_1,x_5]\\ ={(-1)}^{({\overline{x}}_3+{\overline{x}}_4+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_2)}\left[\right[x_3,x_4,x_5],x_1,x_2]\\ \hspace{1em}+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_4+{\overline{x}}_5)+{\overline{x}}_3({\overline{x}}_4+{\overline{x}}_5)}\left[\right[x_2,x_4,x_5],x_1,x_3]+{(-1)}^{({\overline{x}}_2+{\overline{x}}_3)({\overline{x}}_4+{\overline{x}}_4)}\left[\right[x_1,x_4,x_5],x_2,x_3]\end{array}$$

by (1.8)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) . Thus, $\mathfrak{g}$ is a 3-Lie superalgebra. □

In the following we recall that given a Lie superalgebra analogue of supertrace one can construct a 3-Lie superalgebra [1]. Let $(\mathfrak{g},[·,·])$ be a Lie superalgebra and $\tau :\mathfrak{g}\to \mathbb{K}$ an even linear form. We say that τ is a supertrace of $\mathfrak{g}$ if $\tau ([·,·])=0.$ For any $x_1,x_2,x_3\in \mathcal{H}(\mathfrak{g}),$ we define the 3-ary bracket by (1.9) $${[x_1,x_2,x_3]}_{\tau }=\tau (x_1)[x_2,x_3]-{(-1)}^{{\overline{x}}_1{\overline{x}}_2}\tau (x_2)[x_1,x_3]+{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2)}\tau (x_3)[x_1,x_2].$$(1.9)

Theorem 1.10.

For any Lie superalgebra $(\mathfrak{g},[·,·])$ and supertrace τ, the pair $(\mathfrak{g},{[·,·,·]}_{\tau })$ is a 3-Lie superalgebra.

Definition 1.11.

Let $\mathfrak{g}$ be a 3-Lie superalgebra, V be a graded vector space and $\rho :\mathfrak{g}\wedge \mathfrak{g}\to \mathfrak{g}\mathfrak{l}(V)$ be an even linear mapping. If ρ satisfies (1.10) $$\begin{array}{c}\rho (x_1,x_2)\rho (x_3,x_4)-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}\rho (x_3,x_4)\rho (x_1,x_2)\hfill \\ =\rho ([x_1,x_2,x_3],x_4)-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}\rho ([x_1,x_2,x_4],x_3),\hfill \end{array}$$(1.10) (1.11) $$\begin{array}{c}\rho ([x_1,x_2,x_3],x_4)=\rho (x_1,x_2)\rho (x_3,x_4)+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)}\rho (x_2,x_3)\rho (x_1,x_4)\hfill \\ +{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2)}\rho (x_3,x_1)\rho (x_2,x_4),\hfill \end{array}$$(1.11) for all $x_i\in \mathcal{H}(\mathfrak{g}),1\le i\le 4,$ then $(V,\rho )$ is called a representation of $\mathfrak{g},$ or $(V,\rho )$ is an $\mathfrak{g}$-module.

Define (1.12) $$\text{ad}:\mathfrak{g}\wedge \mathfrak{g}\to \mathfrak{g}\mathfrak{l}(\mathfrak{g}), {\text{ad}}_{x,y}(z)=[x,y,z].$$(1.12)

Using (1.6)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) , we can see that $(\mathfrak{g},\text{ad})$ is a representation of the 3-Lie superalgebra $\mathfrak{g}$ and it is called the adjoint representation of $\mathfrak{g}.$

Proposition 1.12.

Let $(V,\mu )$ be a representation of a Lie superalgebra $(\mathfrak{g},[·,·])$ and τ be a supertrace of $\mathfrak{g}$. Then $(V,{\rho }_{\tau })$ is a representation of the 3-Lie superalgebra $(\mathfrak{g},{[·,·,·]}_{\tau })$ where ${\rho }_{\tau }:\mathfrak{g}\wedge \mathfrak{g}\to \mathfrak{g}\mathfrak{l}(V)$ is defined by (1.13) $${\rho }_{\tau }(x,y)=\tau (x)\mu (y)-{(-1)}^{\overline{x}\overline{y}}\tau (y)\mu (x),\hspace{1em}\forall x,y\in \mathcal{H}(\mathfrak{g}).$$(1.13)

Proof.

For all $x_1,x_2,x_3,x_4\in \mathcal{H}(\mathfrak{g}),$ the left hand of (1.10)(1.10) $$\begin{array}{c}\rho (x_1,x_2)\rho (x_3,x_4)-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}\rho (x_3,x_4)\rho (x_1,x_2)\hfill \\ =\rho ([x_1,x_2,x_3],x_4)-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}\rho ([x_1,x_2,x_4],x_3),\hfill \end{array}$$(1.10) becomes $$\begin{array}{cc}\hfill & {\rho }_{\tau }(x_1,x_2){\rho }_{\tau }(x_3,x_4)-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}{\rho }_{\tau }(x_3,x_4){\rho }_{\tau }(x_1,x_2)\hfill \\ \hfill & =(\tau (x_1)\mu (x_2)-{(-1)}^{{\overline{x}}_1{\overline{x}}_2}\tau (x_2)\mu (x_1))(\tau (x_3)\mu (x_4)-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}\tau (x_4)\mu (x_3))\hfill \\ \hfill & \hspace{1em}-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}(\tau (x_3)\mu (x_4)-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}\tau (x_4)\mu (x_3))(\tau (x_1)\mu (x_2)-{(-1)}^{{\overline{x}}_1{\overline{x}}_2}\tau (x_2)\mu (x_1))\hfill \\ \hfill & =\tau (x_1)\mu (x_2)\tau (x_3)\mu (x_4)-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}\tau (x_1)\mu (x_2)\tau (x_4)\mu (x_3)\hfill \\ \hfill & \hspace{1em}-{(-1)}^{{\overline{x}}_1{\overline{x}}_2}\tau (x_2)\mu (x_1)\tau (x_3)\mu (x_4)-{(-1)}^{{\overline{x}}_1{\overline{x}}_2+{\overline{x}}_3{\overline{x}}_4}\tau (x_2)\mu (x_1)\tau (x_4)\mu (x_3)\hfill \\ \hfill & \hspace{1em}-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}\tau (x_3)\mu (x_4)\tau (x_1)\mu (x_2)+{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)+{\overline{x}}_1{\overline{x}}_2}\tau (x_3)\mu (x_4)\tau (x_2)\mu (x_1)\hfill \\ \hfill & \hspace{1em}+{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)+{\overline{x}}_3{\overline{x}}_4}\tau (x_4)\mu (x_3)\tau (x_1)\mu (x_2)\hfill \\ \hfill & \hspace{1em}-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)+{\overline{x}}_1{\overline{x}}_2+x_3{\overline{x}}_4}\tau (x_4)\mu (x_3)\tau (x_2)\mu (x_1).\hfill \end{array}$$

Using (1.9)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) and (1.13)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) , the right hand of (1.10)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) can be written as follows: $$\begin{array}{c}{\rho }_{\tau }([x_1,x_2,x_3],x_4)-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}{\rho }_{\tau }([x_1,x_2,x_4],x_3)\\ =-{(-1)}^{{\overline{x}}_4({\overline{x}}_2+{\overline{x}}_3)}\tau (x_1)\tau (x_4)\mu ([x_2,x_3])+{(-1)}^{{\overline{x}}_4({\overline{x}}_2+{\overline{x}}_3)+{\overline{x}}_1{\overline{x}}_2}\tau (x_2)\tau (x_4)\mu ([x_1,x_3])\\ -{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}\tau (x_3)\tau (x_4)\mu ([x_1,x_2])+{(-1)}^{{\overline{x}}_2{\overline{x}}_3}\tau (x_1)\tau (x_3)\mu ([x_2,x_4])\\ +{(-1)}^{{\overline{x}}_1{\overline{x}}_2+{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_4)}\tau (x_2)\tau (x_3)\mu ([x_1,x_4])-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}\tau (x_4)\tau (x_3)\mu ([x_1,x_2]).\end{array}$$

By direct identification, we proof that $$\begin{array}{c}{\rho }_{\tau }(x_1,x_2){\rho }_{\tau }(x_3,x_4)-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}{\rho }_{\tau }(x_3,x_4){\rho }_{\tau }(x_1,x_2)\\ ={\rho }_{\tau }([x_1,x_2,x_3],x_4)-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}{\rho }_{\tau }([x_1,x_2,x_4],x_3).\end{array}$$

Similarly, we can proof (1.11)(1.11) $$\begin{array}{c}\rho ([x_1,x_2,x_3],x_4)=\rho (x_1,x_2)\rho (x_3,x_4)+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)}\rho (x_2,x_3)\rho (x_1,x_4)\hfill \\ +{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2)}\rho (x_3,x_1)\rho (x_2,x_4),\hfill \end{array}$$(1.11) . □

We generalize the Lie-Rinehart superalgebra to the ternary case.

Definition 1.13.

A 3-Lie-Rinehart superalgebra over A is a tuple $(L,A,[·,·,·],\rho ),$ where A is an associative supercommutative superalgebra, L is an $A-$module, $[·,·,·]:L\times L\times L\to L$ is an even super skew-symmetric trilinear map, and the $\mathbb{K}$-map $\rho :L\times L\to \mathit{Der}(A)$ such that the following conditions hold:

1. $(L,[·,·,·])$ is a 3-Lie superalgebra.

2. ρ is a representation of $(L,[·,·,·])$ on A.

3. For all $x,y\in \mathcal{H}(L),a\in \mathcal{H}(A),$ (1.14) $$\rho (ax,y)={(-1)}^{\overline{a}\overline{x}}\rho (x,ay)=a\rho (x,y).$$(1.14)

4. The compatibility condition: (1.15) $$[x,y,az]={(-1)}^{\overline{a}(\overline{x}+\overline{y})}a[x,y,z]+\rho (x,y)az, \forall x,y,z\in \mathcal{H}(L),a\in \mathcal{H}(A).$$(1.15)

Remark 1.14.

If ρ = 0, then $(L,A,[·,·,·])$ is called a 3-Lie A-superalgebra.

Remark 1.15.

If the condition (1.14)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) is not satisfied, then we call $(L,A,[·,·,·],\rho )$ a weak 3-Lie-Rinehart superalgebra over A.

Definition 1.16.

Let $(L,A,{[·,·,·]}_L,\rho )$ and $(L\prime ,A\prime ,{[·,·,·]}_{L\prime },\rho \prime )$ be two 3-Lie-Rinehart superalgebras, then a 3-Lie-Rinehart superalgebra homomorphism is defined as a pair of maps (g, f), where $g:A\to A\prime$ and $f:L\to L\prime$ are two $\mathbb{K}$-algebra homomorphisms such that

1. $f(ax)=g(a)f(x)$ for all $x\in L,a\in A,$

2. $g(\rho (x,y)(a))=\rho \prime (f(x),f(y))(g(a))$ for all $x\in L,a\in A.$

3. Some constructions of 3-Lie-Rinehart superalgebras

In this section, we give some construction results. We begin by constructing 3-Lie-Rinehart superalgebras starting with a Lie-Rinehart superalgebras. We also construct Lie-Rinehart superalgebras starting from 3-Lie-Rinehart superalgebras. At the end of this section, we construct some new 3-Lie-Rinehart superalgebras from a given 3-Lie-Rinehart superalgebra.

The following Theorem generalizes results in [2] to Lie-Rinehart case.

Theorem 2.1.

Let $(L,A,[·,·],\mu )$ be a Lie-Rinehart superalgebra and τ is a supertrace. If the condition (2.1) $$\tau (ax)y=\tau (x)ay$$(2.1) is satisfied for any $x,y\in \mathcal{H}(L),a\in \mathcal{H}(A)$, then $(L,A,{[·,·,·]}_{\tau },{\rho }_{\tau })$ is a 3-Lie-Rinehart superalgebra, where ${[·,·,·]}_{\tau }$ and ${\rho }_{\tau }$ are defined in (1.9)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) and (1.12)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) respectively. We say that $(L,A,{[·,·,·]}_{\tau },{\rho }_{\tau })$ is induced by $(L,A,[·,·],\mu )$ and denote it by $L_{\tau }.$

Proof.

Proposition 1.12 gives that $(A,{\rho }_{\tau })$ is a representation of the 3-Lie superalgebra $(L,{[·,·,·]}_{\tau })$ and ${\rho }_{\tau }(x,y)\in \mathit{Der}(A).$ For all $x,y,z\in \mathcal{H}(L)$ and $a\in \mathcal{H}(A),$ $$\begin{array}{c}{[x,y,az]}_{\tau }=\tau (x)[y,az]-{(-1)}^{\overline{x}\overline{y}}\tau (y)[x,az]+{(-1)}^{(\overline{a}+\overline{z})(\overline{x}+\overline{y})}\tau (az)[x,y]\\ ={(-1)}^{\overline{a}(\overline{x}+\overline{y})}a\tau (x)[y,z]+\tau (x)\mu (y)az-{(-1)}^{\overline{a}(\overline{x}+\overline{y})+\overline{x}\overline{y}}a\tau (y)[x,z]\\ -{(-1)}^{\overline{x}\overline{y}}\tau (y)\mu (x)az+{(-1)}^{(\overline{a}+\overline{z})(\overline{x}+\overline{y})}a\tau (z)[x,y]\\ ={(-1)}^{\overline{a}(\overline{x}+\overline{y})}a{[x,y,z]}_{\tau }+{\rho }_{\tau }(x,y)az.\end{array}$$

So, we obtain (1.15)(1.5) $$[x,y,z]=-{(-1)}^{\overline{x}\overline{y}}[y,x,z],\hspace{1em}[x,y,z]=-{(-1)}^{\overline{y}\overline{z}}[x,z,y],$$(1.5) . Since (2.1)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) is satisfied and $$\begin{array}{c}{\rho }_{\tau }(ax,y)=\tau (ax)\mu (y)-{(-1)}^{(\overline{a}+\overline{x})\overline{y}}\tau (y)\mu (ax),\\ {(-1)}^{\overline{a}\overline{x}}{\rho }_{\tau }(x,ay)={(-1)}^{\overline{a}\overline{x}}\tau (x)\mu (ay)-{(-1)}^{(\overline{a}+\overline{y})\overline{x}+\overline{a}\overline{x}}\tau (ay)\mu (x),\\ a{\rho }_{\tau }(x,y)=a\tau (x)\mu (y)-{(-1)}^{\overline{x}\overline{y}}a\tau (y)\mu (x),\end{array}$$ the condition (1.14)(1.4) $$\forall a\in \mathcal{H}(A),x,y\in \mathcal{H}(L):\hspace{1em}[x,ay]=\mu (x)ay+{(-1)}^{\overline{a}\overline{x}}a[x,y].$$(1.4) holds. □

Conversely, we can construct a Lie-Rinehart superalgebra structure from a given 3-Lie-Rinehart superalgebra.

Proposition 2.2.

Let $(L,A,[·,·,·],\rho )$ be a 3-Lie-Rinehart superalgebra. Let $x_0\in L_{\overline{0}}$. Define the bracket ${[·,·]}_{x_0}=[x_0,·,·]$ and ${\rho }_{x_0}(x)(v)=\rho (x_0,x)v$. Then $(L,A,{[·,·]}_{x_0},{\rho }_{x_0})$ is a Lie-Rinehart superalgebra.

Proof.

It is easy to check that the bracket is super skew-symmetric. For any $x,y,z\in \mathcal{H}(L),$ we have $$\begin{array}{c}{[x,{[y,z]}_{x_0}]}_{x_0}=[x_0,x,[x_0,y,z\left]\right]\\ =\left[\right[x_0,x,x_0],y,z]+[x_0,[x_0,x,y],z]+{(-1)}^{\overline{x}\overline{y}}[x_0,y,[x_0,x,z\left]\right]\\ =[x_0,[x_0,x,y],z]+{(-1)}^{\overline{x}\overline{y}}[x_0,y,[x_0,x,z\left]\right]\\ ={[{[x,y]}_{x_0},z]}_{x_0}+{(-1)}^{\overline{x}\overline{y}}{[y,{[x,z]}_{x_0}]}_{x_0}.\end{array}$$

It is obvious to see that ${\rho }_{x_0}$ is a representation of L on A and ${\rho }_{x_0}(x)\in \mathit{Der}(A).$ It remains to prove (1.4)(1.4) $$\forall a\in \mathcal{H}(A),x,y\in \mathcal{H}(L):\hspace{1em}[x,ay]=\mu (x)ay+{(-1)}^{\overline{a}\overline{x}}a[x,y].$$(1.4) . Using (1.15)(1.5) $$[x,y,z]=-{(-1)}^{\overline{x}\overline{y}}[y,x,z],\hspace{1em}[x,y,z]=-{(-1)}^{\overline{y}\overline{z}}[x,z,y],$$(1.5) , we obtain $$\begin{array}{c}{[x,ay]}_{x_0}=[x_0,x,ay]={(-1)}^{\overline{a}\overline{x}}a[x_0,x,y]+\rho (x_0,x)ay\\ ={(-1)}^{\overline{a}\overline{x}}a{[x,y]}_{x_0}+{\rho }_{x_0}(x)ay.\end{array}$$ which completes the proof. □

Definition 2.3.

Let $(L,A,[·,·,·],\rho )$ be a 3-Lie-Rinehart superalgebra.

1. If S is a subalgebra of the 3-Lie superalgebra $(L,[·,·,·])$ satisfying $AS\subset S,$ then $(S,A,[·,·,·],\rho {|}_{S\wedge S})$ is a 3-Lie-Rinehart superalgebra, which is called a subalgebra of the 3-Lie-Rinehart superalgebra $(L,A,[·,·,·],\rho ).$

2. If I is an ideal of the 3-Lie superalgebra $(L,[·,·,·])$ and satisfies $AI\subset I$ and $\rho (I,L)(A)L\subset I,$ then $(I,A,[·,·,·],\rho {|}_{I\wedge I})$ is a 3-Lie-Rinehart superalgebra, which is called an ideal of the 3-Lie-Rinehart superalgebra $(L,A,[·,·,·],\rho ).$

3. If a 3-Lie-Rinehart superalgebra $(L,A,[·,·,·],\rho )$ cannot be decomposed into the direct sum of two nonzero ideals, then L is called an indecomposable 3-Lie-Rinehart superalgebra.

Proposition 2.4.

If $(L,A,[·,·,·],\rho )$ is a 3-Lie-Rinehart superalgebra. Then $$\mathit{Ker}\rho =\{x\in \mathcal{H}(L)|\rho (x,L)=0\}$$ is an ideal, which is called the kernel of the representation ρ.

Proof.

By (1.10)(1.10) $$\begin{array}{c}\rho (x_1,x_2)\rho (x_3,x_4)-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}\rho (x_3,x_4)\rho (x_1,x_2)\hfill \\ =\rho ([x_1,x_2,x_3],x_4)-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}\rho ([x_1,x_2,x_4],x_3),\hfill \end{array}$$(1.10) and (1.11)(1.11) $$\begin{array}{c}\rho ([x_1,x_2,x_3],x_4)=\rho (x_1,x_2)\rho (x_3,x_4)+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)}\rho (x_2,x_3)\rho (x_1,x_4)\hfill \\ +{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2)}\rho (x_3,x_1)\rho (x_2,x_4),\hfill \end{array}$$(1.11) , for all homogeneous elements $x\in \mathit{Ker}\rho ,y,z,w\in \mathcal{H}(L),$ $$\rho ([x,y,z],w)=\rho (x,y)\rho (z,w)+{(-1)}^{\overline{x}(\overline{y}+\overline{z})}\rho (y,z)\rho (x,w)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\rho (z,x)\rho (y,w)=0.$$

Therefore, $[x,L,L]\subseteq \mathit{Ker}\rho .$ By (1.14)(1.14) $$\rho (ax,y)={(-1)}^{\overline{a}\overline{x}}\rho (x,ay)=a\rho (x,y).$$(1.14) , for all $a\in \mathcal{H}(A),$ $$\rho (ax,y)={(-1)}^{\overline{x}\overline{a}}\rho (x,ay)=a\rho (x,y)=0,\hspace{1em}\rho (\rho (x,y)az,w)=\rho (x,y)(a)\rho (z,w)=0.$$

We get $ax\in \mathit{Ker}\rho ,\rho (x,y)az\in \mathit{Ker}\rho ,$ that is, $\mathit{AKer}\rho \subset \mathit{Ker}\rho ,\rho (\mathit{Ker}\rho ,L)(A)L\subset \mathit{Ker}\rho .$

Therefore, $\mathit{Ker}\rho$ is an ideal of the 3-Lie-Rinehart superalgebra $(L,A,[·,·,·],\rho ).$ □

Theorem 2.5.

Let $(L,A,[·,·,·],\rho )$ be a 3-Lie-Rinehart superalgebra. Then the following identities hold, for all $a,b,c\in \mathcal{H}(A),x_i\in \mathcal{H}(L),1\le i\le 5,$ (2.2) $$\begin{array}{cc}\hfill & \rho (x_2,x_3)a[x_1,x_4,x_5]+{(-1)}^{({\overline{x}}_1+{\overline{x}}_4)({\overline{x}}_2+{\overline{x}}_3)+\overline{a}({\overline{x}}_1+{\overline{x}}_4+{\overline{x}}_2+{\overline{x}}_3)}\rho (x_1,x_4)a[x_2,x_3,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_1)+\overline{a}({\overline{x}}_1+{\overline{x}}_2)}\rho (x_3,x_1)a[x_2,x_4,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_4({\overline{x}}_3+{\overline{x}}_1)+\overline{a}({\overline{x}}_3+{\overline{x}}_4)}\rho (x_2,x_4)a[x_3,x_1,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)+\overline{a}({\overline{x}}_1+{\overline{x}}_3)}\rho (x_1,x_2)a[x_3,x_4,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_2({\overline{x}}_1+{\overline{x}}_3+{\overline{x}}_4)+{\overline{x}}_4{\overline{x}}_1+\overline{a}({\overline{x}}_2+{\overline{x}}_4))}\rho (x_3,x_4)a[x_1,x_2,x_5]=0,\hfill \end{array}$$(2.2) (2.3) $$\begin{array}{cc}\hfill & \rho (x_2,x_3)(a_4{a}_5)[x_1,x_4,x_5]+{(-1)}^{{\overline{x}}_2({\overline{a}}_1+{\overline{a}}_4+{\overline{a}}_5+{\overline{x}}_3)}\rho (x_3,x_1)(a_4{a}_5)[x_2,x_4,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_1({\overline{a}}_4+{\overline{a}}_5+{\overline{x}}_2+{\overline{x}}_3)}\rho (x_1,x_2)(a_4{a}_5)[x_3,x_4,x_5]\hfill \\ \hfill & +{(-1)}^{({\overline{a}}_4+{\overline{x}}_5)({\overline{x}}_2+{\overline{x}}_3)+({\overline{x}}_1+{\overline{x}}_4)({\overline{x}}_2+{\overline{x}}_3+{\overline{a}}_5)}{a}_4\rho (x_1,x_4)a_5[x_5,x_2,x_3]\hfill \\ \hfill & +{(-1)}^{{\overline{a}}_4({\overline{x}}_2+{\overline{x}}_3)+({\overline{x}}_4+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_3)+{\overline{x}}_4{\overline{a}}_5}{a}_4\rho (x_2,x_4)a_5[x_5,x_3,x_1]\hfill \\ \hfill & +{(-1)}^{{\overline{a}}_4({\overline{x}}_2+{\overline{x}}_3)+({\overline{x}}_4+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_2)+{\overline{x}}_4{\overline{a}}_5}{a}_4\rho (x_3,x_4)a_5[x_5,x_1,x_2]\hfill \\ \hfill & -{(-1)}^{({\overline{x}}_1+{\overline{a}}_5+{\overline{x}}_5)({\overline{a}}_4+{\overline{x}}_2+{\overline{x}}_3)+{\overline{x}}_4({\overline{x}}_2+{\overline{x}}_3)}{a}_5\rho (x_1,x_5)a_4[x_4,x_2,x_3]\hfill \\ \hfill & -{(-1)}^{({\overline{a}}_5+{\overline{x}}_5)({\overline{x}}_2+{\overline{x}}_3)+({\overline{a}}_4+{\overline{x}}_5)({\overline{x}}_3+{\overline{x}}_5)}{a}_5\rho (x_2,x_5)a_4[x_4,x_3,x_1]\hfill \\ \hfill & -{(-1)}^{{\overline{a}}_5({\overline{x}}_2+{\overline{x}}_3+{\overline{a}}_4)+({\overline{x}}_2+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_4+{\overline{a}}_4)+{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_5)+{\overline{x}}_1{\overline{x}}_4}{a}_5\rho (x_3,x_5)a_4[x_4,x_1,x_2])=0,\hfill \end{array}$$(2.3) (2.4) $$\begin{array}{cc}\hfill & \rho (x_1,x_2)\rho (x_3,x_4)+{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}\rho (x_3,x_4)\rho (x_1,x_2)\hfill \\ \hfill & +{(-1)}^{x_1({\overline{x}}_2+{\overline{x}}_3)}\rho (x_2,x_3)\rho (x_1,x_4)+{(-1)}^{{\overline{x}}_4({\overline{x}}_2+{\overline{x}}_3)}\rho (x_1,x_4)\rho (x_2,x_3)\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2)}\rho (x_3,x_1)\rho (x_2,x_4)+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3+{\overline{x}}_4)+{\overline{x}}_3{\overline{x}}_4}\rho (x_2,x_4)\rho (x_3,x_1)=0.\hfill \end{array}$$(2.4) (2.5) $$\begin{array}{cc}\hfill & (\rho (x_1,x_2)b)\rho (x_3,x_4)+{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2+\overline{b})+\overline{b}{\overline{x}}_2}(\rho (x_3,x_1)b)\rho (x_2,x_4)\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3+\overline{b})+{\overline{x}}_1\overline{b}}(\rho (x_2,x_3)b)\rho (x_1,x_4)=0.\hfill \end{array}$$(2.5)

Proof.

Using (1.15)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) we have $$\begin{array}{c}\rho (x_2,x_3)a[x_1,x_4,x_5]+{(-1)}^{({\overline{x}}_1+{\overline{x}}_4)({\overline{x}}_2+{\overline{x}}_3)+\overline{a}({\overline{x}}_1+{\overline{x}}_4+{\overline{x}}_2+{\overline{x}}_3)}\rho (x_1,x_4)a[x_2,x_3,x_5]\\ +{(-1)}^{{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_1)+\overline{a}({\overline{x}}_1+{\overline{x}}_2)}\rho (x_3,x_1)a[x_2,x_4,x_5]\\ +{(-1)}^{{\overline{x}}_4({\overline{x}}_3+{\overline{x}}_1)+\overline{a}({\overline{x}}_2+{\overline{x}}_4)}\rho (x_2,x_4)a[x_3,x_1,x_5]\\ +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)+\overline{a}({\overline{x}}_1+{\overline{x}}_3)}\rho (x_1,x_2)a[x_3,x_4,x_5]\\ +{(-1)}^{{\overline{x}}_2({\overline{x}}_1+{\overline{x}}_3+{\overline{x}}_4)+{\overline{x}}_4{\overline{x}}_1+\overline{a}({\overline{x}}_2+{\overline{x}}_4))}\rho (x_3,x_4)a[x_1,x_2,x_5]\\ =[x_2,x_3,a[x_1,x_4,x_5\left]\right]-\underset{I_1}{\underbrace{{(-1)}^{\overline{a}({\overline{x}}_2+{\overline{x}}_3)}a[x_2,x_3,[x_1,x_4,x_5\left]\right]}}\\ +{(-1)}^{({\overline{x}}_1+{\overline{x}}_4)({\overline{x}}_2+{\overline{x}}_3)+\overline{a}({\overline{x}}_1+{\overline{x}}_4+{\overline{x}}_2+{\overline{x}}_3)}([x_1,x_4,a[x_2,x_3,x_5]]\underset{I_2}{\underbrace{-{(-1)}^{\overline{a}({\overline{x}}_1+{\overline{x}}_4)}a[x_1,x_4,[x_2,x_3,x_5\left]\right]}})\\ +{(-1)}^{{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_1)+\overline{a}({\overline{x}}_1+{\overline{x}}_2)}([x_3,x_1,a[x_2,x_4,x_5]]\underset{I_3}{\underbrace{-{(-1)}^{\overline{a}({\overline{x}}_3+{\overline{x}}_1)}a[x_4,x_1,[x_2,x_4,x_5\left]\right]}})\\ +{(-1)}^{{\overline{x}}_4({\overline{x}}_3+{\overline{x}}_1)+\overline{a}({\overline{x}}_3+{\overline{x}}_4)}([x_2,x_4,a[x_3,x_1,x_5]]\underset{I_4}{\underbrace{-{(-1)}^{\overline{a}({\overline{x}}_2+{\overline{x}}_4)}a[x_2,x_4,[x_3,x_1,x_5\left]\right]}})\\ +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)+\overline{a}({\overline{x}}_1+{\overline{x}}_3)}([x_1,x_2,a[x_3,x_4,x_5]]\underset{I_5}{\underbrace{-{(-1)}^{\overline{a}({\overline{x}}_1+{\overline{x}}_2)}a[x_1,x_2,[x_3,x_4,x_5\left]\right]}})\\ +{(-1)}^{{\overline{x}}_2({\overline{x}}_1+{\overline{x}}_3+{\overline{x}}_4)+{\overline{x}}_4{\overline{x}}_1+\overline{a}({\overline{x}}_2+{\overline{x}}_4))}([x_3,x_4,a[x_1,x_2,x_5]]-\underset{I_6}{\underbrace{{(-1)}^{\overline{a}({\overline{x}}_3+{\overline{x}}_4)}a[x_3,x_4,[x_1,x_2,x_5\left]\right]}}).\end{array}$$

Thanks to Proposition 1.9, we have ${\sum }_{i=1}^6{I}_i=0.$ Using (1.15)(1.15) $$[x,y,az]={(-1)}^{\overline{a}(\overline{x}+\overline{y})}a[x,y,z]+\rho (x,y)az, \forall x,y,z\in \mathcal{H}(L),a\in \mathcal{H}(A).$$(1.15) , (1.10)(1.10) $$\begin{array}{c}\rho (x_1,x_2)\rho (x_3,x_4)-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}\rho (x_3,x_4)\rho (x_1,x_2)\hfill \\ =\rho ([x_1,x_2,x_3],x_4)-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}\rho ([x_1,x_2,x_4],x_3),\hfill \end{array}$$(1.10) and (1.11)(1.11) $$\begin{array}{c}\rho ([x_1,x_2,x_3],x_4)=\rho (x_1,x_2)\rho (x_3,x_4)+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)}\rho (x_2,x_3)\rho (x_1,x_4)\hfill \\ +{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2)}\rho (x_3,x_1)\rho (x_2,x_4),\hfill \end{array}$$(1.11) , we obtain (2.2)(2.2) $$\begin{array}{cc}\hfill & \rho (x_2,x_3)a[x_1,x_4,x_5]+{(-1)}^{({\overline{x}}_1+{\overline{x}}_4)({\overline{x}}_2+{\overline{x}}_3)+\overline{a}({\overline{x}}_1+{\overline{x}}_4+{\overline{x}}_2+{\overline{x}}_3)}\rho (x_1,x_4)a[x_2,x_3,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_1)+\overline{a}({\overline{x}}_1+{\overline{x}}_2)}\rho (x_3,x_1)a[x_2,x_4,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_4({\overline{x}}_3+{\overline{x}}_1)+\overline{a}({\overline{x}}_3+{\overline{x}}_4)}\rho (x_2,x_4)a[x_3,x_1,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)+\overline{a}({\overline{x}}_1+{\overline{x}}_3)}\rho (x_1,x_2)a[x_3,x_4,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_2({\overline{x}}_1+{\overline{x}}_3+{\overline{x}}_4)+{\overline{x}}_4{\overline{x}}_1+\overline{a}({\overline{x}}_2+{\overline{x}}_4))}\rho (x_3,x_4)a[x_1,x_2,x_5]=0,\hfill \end{array}$$(2.2) . Similarly, (2.3)(2.3) $$\begin{array}{cc}\hfill & \rho (x_2,x_3)(a_4{a}_5)[x_1,x_4,x_5]+{(-1)}^{{\overline{x}}_2({\overline{a}}_1+{\overline{a}}_4+{\overline{a}}_5+{\overline{x}}_3)}\rho (x_3,x_1)(a_4{a}_5)[x_2,x_4,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_1({\overline{a}}_4+{\overline{a}}_5+{\overline{x}}_2+{\overline{x}}_3)}\rho (x_1,x_2)(a_4{a}_5)[x_3,x_4,x_5]\hfill \\ \hfill & +{(-1)}^{({\overline{a}}_4+{\overline{x}}_5)({\overline{x}}_2+{\overline{x}}_3)+({\overline{x}}_1+{\overline{x}}_4)({\overline{x}}_2+{\overline{x}}_3+{\overline{a}}_5)}{a}_4\rho (x_1,x_4)a_5[x_5,x_2,x_3]\hfill \\ \hfill & +{(-1)}^{{\overline{a}}_4({\overline{x}}_2+{\overline{x}}_3)+({\overline{x}}_4+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_3)+{\overline{x}}_4{\overline{a}}_5}{a}_4\rho (x_2,x_4)a_5[x_5,x_3,x_1]\hfill \\ \hfill & +{(-1)}^{{\overline{a}}_4({\overline{x}}_2+{\overline{x}}_3)+({\overline{x}}_4+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_2)+{\overline{x}}_4{\overline{a}}_5}{a}_4\rho (x_3,x_4)a_5[x_5,x_1,x_2]\hfill \\ \hfill & -{(-1)}^{({\overline{x}}_1+{\overline{a}}_5+{\overline{x}}_5)({\overline{a}}_4+{\overline{x}}_2+{\overline{x}}_3)+{\overline{x}}_4({\overline{x}}_2+{\overline{x}}_3)}{a}_5\rho (x_1,x_5)a_4[x_4,x_2,x_3]\hfill \\ \hfill & -{(-1)}^{({\overline{a}}_5+{\overline{x}}_5)({\overline{x}}_2+{\overline{x}}_3)+({\overline{a}}_4+{\overline{x}}_5)({\overline{x}}_3+{\overline{x}}_5)}{a}_5\rho (x_2,x_5)a_4[x_4,x_3,x_1]\hfill \\ \hfill & -{(-1)}^{{\overline{a}}_5({\overline{x}}_2+{\overline{x}}_3+{\overline{a}}_4)+({\overline{x}}_2+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_4+{\overline{a}}_4)+{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_5)+{\overline{x}}_1{\overline{x}}_4}{a}_5\rho (x_3,x_5)a_4[x_4,x_1,x_2])=0,\hfill \end{array}$$(2.3) -(2.5)(2.5) $$\begin{array}{cc}\hfill & (\rho (x_1,x_2)b)\rho (x_3,x_4)+{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2+\overline{b})+\overline{b}{\overline{x}}_2}(\rho (x_3,x_1)b)\rho (x_2,x_4)\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3+\overline{b})+{\overline{x}}_1\overline{b}}(\rho (x_2,x_3)b)\rho (x_1,x_4)=0.\hfill \end{array}$$(2.5) can be verified by a direct computation according to (2.2)(2.2) $$\begin{array}{cc}\hfill & \rho (x_2,x_3)a[x_1,x_4,x_5]+{(-1)}^{({\overline{x}}_1+{\overline{x}}_4)({\overline{x}}_2+{\overline{x}}_3)+\overline{a}({\overline{x}}_1+{\overline{x}}_4+{\overline{x}}_2+{\overline{x}}_3)}\rho (x_1,x_4)a[x_2,x_3,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_1)+\overline{a}({\overline{x}}_1+{\overline{x}}_2)}\rho (x_3,x_1)a[x_2,x_4,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_4({\overline{x}}_3+{\overline{x}}_1)+\overline{a}({\overline{x}}_3+{\overline{x}}_4)}\rho (x_2,x_4)a[x_3,x_1,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)+\overline{a}({\overline{x}}_1+{\overline{x}}_3)}\rho (x_1,x_2)a[x_3,x_4,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_2({\overline{x}}_1+{\overline{x}}_3+{\overline{x}}_4)+{\overline{x}}_4{\overline{x}}_1+\overline{a}({\overline{x}}_2+{\overline{x}}_4))}\rho (x_3,x_4)a[x_1,x_2,x_5]=0,\hfill \end{array}$$(2.2) and Definition 1.13. □

Now we construct some new 3-Lie-Rinehart superalgebra from a given 3-Lie-Rinehart superalgebra $(L,A,[·,·,·],\rho ).$

Theorem 2.6.

Let $(L,A,[·,·,·],\rho )$ be a 3-Lie-Rinehart superalgebra and $B=A\otimes L=\{ax | a\in \mathcal{H}(A),x\in \mathcal{H}(L)\}$. Then $(B,A,[·,·,·]\prime ,\rho \prime )$ is a 3-Lie-Rinehart superalgebra, where the multiplication is defined by, for all $x_1,x_2,x_3\in \mathcal{H}(L),$ $a_1,a_2,a_3\in \mathcal{H}(A),$ (2.6) $$\begin{array}{cc}[a_1{x}_1,a_2{x}_2,a_3{x}_3]\prime =\hfill & {(-1)}^{{\overline{a}}_2{\overline{x}}_1+{\overline{a}}_3({\overline{x}}_1+{\overline{x}}_2)}{a}_1{a}_2{a}_3[x_1,x_2,x_3]+{(-1)}^{{\overline{a}}_2{\overline{x}}_1}{a}_1{a}_2\rho (x_1,x_2)(a_3)x_3\hfill \\ \hfill & +{(-1)}^{{\overline{a}}_3{\overline{x}}_2+({\overline{a}}_1+{\overline{x}}_1)({\overline{a}}_2+{\overline{a}}_3+{\overline{x}}_2+{\overline{x}}_3)}{a}_2{a}_3\rho (x_2,x_3)(a_1)x_1\hfill \\ \hfill & +{(-1)}^{{\overline{a}}_3{\overline{x}}_1+({\overline{a}}_2+x_2)({\overline{a}}_3+{\overline{x}}_3)}{a}_1{a}_3\rho (x_1,x_3)(a_2)x_2,\hfill \end{array}$$(2.6) and $\rho \prime :(A\otimes L)\wedge (A\otimes L)\to \mathit{Der}(A)$ is defined by $\rho \prime (a_1{x}_1,a_2{x}_2)={(-1)}^{{\overline{a}}_2{\overline{x}}_1}{a}_1{a}_2\rho (x_1,x_2).$

Note that $B=B_{\overline{0}}\oplus B_{\overline{1}}$ where $B_{\overline{0}}=A_{\overline{0}}\otimes L_{\overline{0}}\oplus A_{\overline{1}}\otimes L_{\overline{1}}$ and $B_{\overline{1}}=A_{\overline{0}}\otimes L_{\overline{1}}\oplus A_{\overline{1}}\otimes L_{\overline{0}}.$

Proof.

For the super skew-symmetry of the bracket we have $$\begin{array}{c}[a_1{x}_1,a_2{x}_2,a_3{x}_3]\prime ={(-1)}^{{\overline{a}}_2{\overline{x}}_1+{\overline{a}}_3({\overline{x}}_1+{\overline{x}}_2)}{a}_1{a}_2{a}_3[x_1,x_2,x_3]+{(-1)}^{{\overline{a}}_2{\overline{x}}_1}{a}_1{a}_2\rho (x_1,x_2)(a_3)x_3\\ +{(-1)}^{{\overline{a}}_3{\overline{x}}_2+({\overline{a}}_1+{\overline{x}}_1)({\overline{a}}_2+{\overline{a}}_3+{\overline{x}}_2+{\overline{x}}_3)}{a}_2{a}_3\rho (x_2,x_3)(a_1)x_1\\ +{(-1)}^{{\overline{a}}_3{\overline{x}}_1+({\overline{a}}_2+x_2)({\overline{a}}_3+{\overline{x}}_3)}{a}_1{a}_3\rho (x_1,x_3)(a_2)x_2\\ =-{(-1)}^{({\overline{a}}_1+{\overline{x}}_1)({\overline{a}}_2+{\overline{x}}_2)}({(-1)}^{{\overline{a}}_1{\overline{x}}_2+{\overline{a}}_3({\overline{x}}_1+{\overline{x}}_2)}{a}_2{a}_1{a}_3[x_2,x_1,x_3]\\ +{(-1)}^{{\overline{a}}_1{\overline{x}}_2}{a}_2{a}_1\rho (x_2,x_1)(a_3)x_3\\ +{(-1)}^{{\overline{a}}_3{\overline{x}}_1+({\overline{a}}_2+{\overline{x}}_2)({\overline{a}}_1+{\overline{a}}_3+{\overline{x}}_2+{\overline{x}}_3)}{a}_1{a}_3\rho (x_1,x_3)(a_2)x_2\\ +{(-1)}^{{\overline{a}}_3{\overline{x}}_2+({\overline{a}}_1+x_1)({\overline{a}}_3+{\overline{x}}_3)}{a}_2{a}_3\rho (x_2,x_3)(a_1)x_1)\\ =-{(-1)}^{({\overline{a}}_1+{\overline{x}}_1)({\overline{a}}_2+{\overline{x}}_2)}[a_2{x}_2,a_1{x}_1,a_3{x}_3]\prime .\end{array}$$

Similarly, we obtain $[a_1{x}_1,a_2{x}_2,a_3{x}_3]\prime =-{(-1)}^{({\overline{a}}_2+{\overline{x}}_2)({\overline{a}}_3+{\overline{x}}_3)}[a_1{x}_1,a_3{x}_3,a_2{x}_2]\prime .$

Using (2.2)(2.2) $$\begin{array}{cc}\hfill & \rho (x_2,x_3)a[x_1,x_4,x_5]+{(-1)}^{({\overline{x}}_1+{\overline{x}}_4)({\overline{x}}_2+{\overline{x}}_3)+\overline{a}({\overline{x}}_1+{\overline{x}}_4+{\overline{x}}_2+{\overline{x}}_3)}\rho (x_1,x_4)a[x_2,x_3,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_1)+\overline{a}({\overline{x}}_1+{\overline{x}}_2)}\rho (x_3,x_1)a[x_2,x_4,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_4({\overline{x}}_3+{\overline{x}}_1)+\overline{a}({\overline{x}}_3+{\overline{x}}_4)}\rho (x_2,x_4)a[x_3,x_1,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)+\overline{a}({\overline{x}}_1+{\overline{x}}_3)}\rho (x_1,x_2)a[x_3,x_4,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_2({\overline{x}}_1+{\overline{x}}_3+{\overline{x}}_4)+{\overline{x}}_4{\overline{x}}_1+\overline{a}({\overline{x}}_2+{\overline{x}}_4))}\rho (x_3,x_4)a[x_1,x_2,x_5]=0,\hfill \end{array}$$(2.2) and (2.3)(2.3) $$\begin{array}{cc}\hfill & \rho (x_2,x_3)(a_4{a}_5)[x_1,x_4,x_5]+{(-1)}^{{\overline{x}}_2({\overline{a}}_1+{\overline{a}}_4+{\overline{a}}_5+{\overline{x}}_3)}\rho (x_3,x_1)(a_4{a}_5)[x_2,x_4,x_5]\hfill \\ \hfill & +{(-1)}^{{\overline{x}}_1({\overline{a}}_4+{\overline{a}}_5+{\overline{x}}_2+{\overline{x}}_3)}\rho (x_1,x_2)(a_4{a}_5)[x_3,x_4,x_5]\hfill \\ \hfill & +{(-1)}^{({\overline{a}}_4+{\overline{x}}_5)({\overline{x}}_2+{\overline{x}}_3)+({\overline{x}}_1+{\overline{x}}_4)({\overline{x}}_2+{\overline{x}}_3+{\overline{a}}_5)}{a}_4\rho (x_1,x_4)a_5[x_5,x_2,x_3]\hfill \\ \hfill & +{(-1)}^{{\overline{a}}_4({\overline{x}}_2+{\overline{x}}_3)+({\overline{x}}_4+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_3)+{\overline{x}}_4{\overline{a}}_5}{a}_4\rho (x_2,x_4)a_5[x_5,x_3,x_1]\hfill \\ \hfill & +{(-1)}^{{\overline{a}}_4({\overline{x}}_2+{\overline{x}}_3)+({\overline{x}}_4+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_2)+{\overline{x}}_4{\overline{a}}_5}{a}_4\rho (x_3,x_4)a_5[x_5,x_1,x_2]\hfill \\ \hfill & -{(-1)}^{({\overline{x}}_1+{\overline{a}}_5+{\overline{x}}_5)({\overline{a}}_4+{\overline{x}}_2+{\overline{x}}_3)+{\overline{x}}_4({\overline{x}}_2+{\overline{x}}_3)}{a}_5\rho (x_1,x_5)a_4[x_4,x_2,x_3]\hfill \\ \hfill & -{(-1)}^{({\overline{a}}_5+{\overline{x}}_5)({\overline{x}}_2+{\overline{x}}_3)+({\overline{a}}_4+{\overline{x}}_5)({\overline{x}}_3+{\overline{x}}_5)}{a}_5\rho (x_2,x_5)a_4[x_4,x_3,x_1]\hfill \\ \hfill & -{(-1)}^{{\overline{a}}_5({\overline{x}}_2+{\overline{x}}_3+{\overline{a}}_4)+({\overline{x}}_2+{\overline{x}}_5)({\overline{x}}_1+{\overline{x}}_4+{\overline{a}}_4)+{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_5)+{\overline{x}}_1{\overline{x}}_4}{a}_5\rho (x_3,x_5)a_4[x_4,x_1,x_2])=0,\hfill \end{array}$$(2.3) , we can deduce that $(B,[·,·,·]\prime )$ is a 3-Lie superalgebra and B is an A-module. Thanks to (1.10)(1.10) $$\begin{array}{c}\rho (x_1,x_2)\rho (x_3,x_4)-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}\rho (x_3,x_4)\rho (x_1,x_2)\hfill \\ =\rho ([x_1,x_2,x_3],x_4)-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}\rho ([x_1,x_2,x_4],x_3),\hfill \end{array}$$(1.10) and (1.11)(1.11) $$\begin{array}{c}\rho ([x_1,x_2,x_3],x_4)=\rho (x_1,x_2)\rho (x_3,x_4)+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)}\rho (x_2,x_3)\rho (x_1,x_4)\hfill \\ +{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2)}\rho (x_3,x_1)\rho (x_2,x_4),\hfill \end{array}$$(1.11) , for all $x_1,x_2,x_3\in \mathcal{H}(L)$ and $b,a_1,a_2,a_3\in \mathcal{H}(A),$ it is easy to show that $(A,\rho \prime )$ is a representation of B. Indeed, $$\begin{array}{c}2\rho \prime (a_1{x}_1,a_2{x}_2)\rho \prime (a_3{x}_3,a_4{x}_4)-{(-1)}^{({\overline{a}}_1+{\overline{x}}_1+{\overline{a}}_2+{\overline{x}}_2)({\overline{x}}_3+{\overline{a}}_3+{\overline{x}}_4+{\overline{x}}_4)}\rho \prime (a_3{x}_3,a_4{x}_4)\rho \prime (a_1{x}_1,a_2{x}_2)\\ \hspace{1em}=\rho \prime ([a_1{x}_1,a_2{x}_2,a_3{x}_3]\prime ,a_4{x}_4)+{(-1)}^{({\overline{a}}_1+{\overline{x}}_1+{\overline{a}}_2+{\overline{x}}_2)({\overline{a}}_3+{\overline{x}}_3)}\rho \prime (a_3{x}_3,[a_1{x}_1,a_2{x}_2,a_4{x}_4]\prime ),\\ \rho \prime (a_1{x}_1,a_2{x}_2)\rho \prime (a_3{x}_3,a_4{x}_4)+{(-1)}^{({\overline{a}}_1+{\overline{x}}_1)({\overline{a}}_2+{\overline{x}}_2+{\overline{x}}_3+{\overline{x}}_3)}\rho \prime (a_2{x}_2,a_3{x}_3)\rho \prime (a_1{x}_1,a_4{x}_4)\\ \hspace{1em}+{(-1)}^{({\overline{a}}_1+{\overline{x}}_1+{\overline{a}}_2+{\overline{x}}_2)({\overline{a}}_3+{\overline{x}}_3)}\rho \prime (a_3{x}_3,a_1{x}_1)\rho \prime (a_2{x}_2,a_4{x}_4)=\rho \prime ([a_1{x}_1,a_2{x}_2,a_3{x}_3],a_4{x}_4).\end{array}$$

To prove the compatibility condition (1.15)(1.15) $$[x,y,az]={(-1)}^{\overline{a}(\overline{x}+\overline{y})}a[x,y,z]+\rho (x,y)az, \forall x,y,z\in \mathcal{H}(L),a\in \mathcal{H}(A).$$(1.15) , we compute as follows $$\begin{array}{c}[a_1{x}_1,a_2{x}_2,b(a_3{x}_3)]\prime ={(-1)}^{{\overline{a}}_2{\overline{x}}_1+({\overline{a}}_3+\overline{b})({\overline{x}}_1+{\overline{x}}_3)}{a}_1{a}_{2}b{a}_3[x_1,x_2,x_3]\\ \hspace{1em}+{(-1)}^{{\overline{a}}_2{\overline{x}}_1}{a}_1{a}_2\rho (x_1,x_2)(b{a}_3)x_3\\ \hspace{1em}+{(-1)}^{(\overline{b}+{\overline{a}}_3){\overline{x}}_2+(\overline{a}+{\overline{x}}_1)({\overline{a}}_2+{\overline{x}}_2+{\overline{a}}_3+{\overline{x}}_3)}{a}_{2}b{a}_3\rho (x_2,x_3)a_1{x}_1\\ \hspace{1em}+{(-1)}^{(\overline{b}+{\overline{a}}_3){\overline{x}}_1+({\overline{a}}_2+{\overline{x}}_2)({\overline{a}}_3+\overline{b}+{\overline{x}}_3}{a}_{1}b{a}_3\rho (x_1,x_3)a_2{x}_2\\ ={(-1)}^{\overline{b}({\overline{a}}_1+{\overline{x}}_1+{\overline{a}}_2+{\overline{x}}_2)}b(a_1{a}_2{a}_3[x_1,x_2,x_3]+{(-1)}^{{\overline{a}}_2{\overline{x}}_1}{a}_1{a}_2\rho (x_1,x_2)a_3{x}_3\\ \hspace{1em}+{(-1)}^{{\overline{a}}_3{\overline{x}}_2+({\overline{a}}_1+{\overline{x}}_1)({\overline{a}}_2+{\overline{x}}_2+{\overline{a}}_3+{\overline{x}}_3)}{a}_2{a}_3\rho (x_2,x_3)a_1{x}_1\\ \hspace{1em}+{(-1)}^{{\overline{a}}_3{\overline{x}}_1+({\overline{a}}_2+{\overline{x}}_2)({\overline{a}}_3+{\overline{x}}_3)}{a}_1{a}_3\rho (x_1,x_3)a_2{x}_2)+{(-1)}^{{\overline{a}}_2{\overline{x}}_1}{a}_1{a}_2\rho (x_1,x_2)b(a_3{x}_3)\\ ={(-1)}^{\overline{b}({\overline{a}}_1+{\overline{x}}_1+{\overline{a}}_2+{\overline{x}}_2)}b[a_1{x}_1,a_2{x}_2,a_3{x}_3]\prime +\rho \prime (a_1{x}_1,a_2{x}_2)b(a_3{x}_3).\end{array}$$

It is obvious to show that $$\rho \prime (b(a_1{x}_1),a_2{x}_2)=b\rho \prime (a_1{x}_1,a_2{x}_2)={(-1)}^{\overline{b}(\overline{a}+{\overline{x}}_1)}\rho \prime (a_1{x}_1,b(a_2{x}_2)).$$

Therefore, $(B,A,[·,·,·]\prime ,\rho \prime )$ is a 3-Lie-Rinehart superalgebra. □

Theorem 2.7.

Let $(L,A,[·,·,·],\rho )$ be a 3-Lie-Rinehart superalgebra and $$E=L\oplus A=\{(x,a)|x\in L,a\in A\}.$$

Then $(E,A,{[·,·,·]}_1,{\rho }_1)$ is a 3-Lie-Rinehart superalgebra, where for any $a,b,c\in \mathcal{H}(A),$ $x,y,z\in \mathcal{H}(L)$ and $k\in \mathbb{K},$ (2.7) $$k(x,a)=(kx,ka),(x,a)+(y,b)=(x+y,a+b),a(y,b)=(ay,ab),$$(2.7) (2.8) $${[(x,a),(y,b),(z,c)]}_1=([x,y,z],\rho (x,y)c-{(-1)}^{\overline{y}\overline{z}}\rho (x,z)b+{(-1)}^{\overline{x}(\overline{y}+\overline{z})}\rho (y,z)a),$$(2.8) (2.9) $${\rho }_1:E\wedge E\to \mathit{Der}(A),{\rho }_1((x,a),(y,b))=\rho (x,y).$$(2.9)

Note that $E=E_{\overline{0}}\oplus E_{\overline{1}}$, where $E_{\overline{0}}=L_{\overline{0}}\oplus A_{\overline{0}}$ and $E_{\overline{1}}=L_{\overline{1}}\oplus A_{\overline{1}}.$

Proof.

Let $x_i\in \mathcal{H}(L),a_i\in \mathcal{H}(A),i=1,\dots ,5.$ Thanks to (2.7)(1.12) $$\text{ad}:\mathfrak{g}\wedge \mathfrak{g}\to \mathfrak{g}\mathfrak{l}(\mathfrak{g}), {\text{ad}}_{x,y}(z)=[x,y,z].$$(1.12) , E is an $A-$ module, and the 3-ary linear multiplication defined by (2.8)(2.8) $${[(x,a),(y,b),(z,c)]}_1=([x,y,z],\rho (x,y)c-{(-1)}^{\overline{y}\overline{z}}\rho (x,z)b+{(-1)}^{\overline{x}(\overline{y}+\overline{z})}\rho (y,z)a),$$(2.8) is super skew-symmetric. We have $$\begin{array}{c}{[(x_1,a_1),(x_2,a_2),{[(x_3,a_3),(x_4,a_4),(x_5,a_5)]}_1]}_1\\ =[(x_1,a_1),(x_2,a_2),([x_3,x_4,x_5],\rho (x_3,x_4)a_5-{(-1)}^{{\overline{x}}_4{\overline{x}}_5}\rho (x_3,x_5)a_4\\ +{(-1)}^{{\overline{x}}_3({\overline{x}}_4+{\overline{x}}_5)}\rho (x_4,x_5)a_3){]}_1\\ =([x_1,x_2,[x_3,x_4,x_5]],\rho (x_1,x_2)\rho (x_3,x_4)a_5-{(-1)}^{{\overline{x}}_4{\overline{x}}_5}\rho (x_1,x_2)\rho (x_3,x_5)a_4\\ +{(-1)}^{{\overline{x}}_3({\overline{x}}_4+{\overline{x}}_5)}\rho (x_1,x_2)\rho (x_4,x_5)a_3-{(-1)}^{{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_4+{\overline{x}}_5)}\rho (x_1,[x_3,x_4,x_5])a_2\\ -{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3+{\overline{x}}_4+{\overline{x}}_5)}\rho (x_2,[x_3,x_4,x_5])a_1).\end{array}$$

Similarly, we obtain $$\begin{array}{c}{[{[(x_1,a_1),(x_2,a_2),(x_3,a_3)]}_1,(x_4,a_4),(x_5,a_5)]}_1\\ +{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2)}{[(x_3,a_3),{[(x_1,a_1),(x_2,a_2),(x_4,a_4)]}_1,(x_5,a_5)]}_1\\ +{(-1)}^{({\overline{x}}_3+{\overline{x}}_4)({\overline{x}}_1+{\overline{x}}_2)}{[(x_3,a_3),(x_4,a_4),{[(x_1,a_1),(x_2,a_2),(x_5,a_5)]}_1]}_1\\ =[([x_1,x_2,x_3],\rho (x_1,x_2)a_3-{(-1)}^{{\overline{x}}_2{\overline{x}}_3}\rho (x_1,x_3)a_2\\ +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)}\rho (x_2,x_3)a_2),(x_4,a_4),(x_5,a_5){]}_1\\ +{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2)}([(x_3,a_3),([x_1,x_2,x_4],\rho (x_1,x_2)a_4-{(-1)}^{{\overline{x}}_2{\overline{x}}_4}\rho (x_1,x_4)a_2\\ +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_4)}\rho (x_2,x_4)a_2),(x_5,a_5){]}_1)\\ +{(-1)}^{({\overline{x}}_3+{\overline{x}}_4)({\overline{x}}_1+{\overline{x}}_2)}([(x_3,a_3),(x_4,a_5),([x_1,x_2,x_5],\rho (x_1,x_2)a_5-{(-1)}^{{\overline{x}}_2{\overline{x}}_5}\rho (x_1,x_5)a_2\\ +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_5)}\rho (x_2,x_5)a_2){]}_1)\\ =([x_1,x_2,[x_3,x_4,x_5]],\rho (x_1,x_2)\rho (x_3,x_4)a_5-{(-1)}^{{\overline{x}}_4{\overline{x}}_5}\rho (x_1,x_2)\rho (x_3,x_5)a_4\\ +{(-1)}^{{\overline{x}}_4({\overline{x}}_4+{\overline{x}}_5)}\rho (x_1,x_2)\rho (x_4,x_5)a_3-{(-1)}^{{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_4+{\overline{x}}_5)}\rho (x_1,[x_3,x_4,x_5])a_2\\ -{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3+{\overline{x}}_4+{\overline{x}}_5)}\rho (x_2,[x_3,x_4,x_5])a_1)\\ +{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2)}([x_3,[x_1,x_2,x_4],x_5],\rho (x_3,[x_1,x_2,x_4])a_5\\ -{(-1)}^{{\overline{x}}_5({\overline{x}}_1+{\overline{x}}_2+{\overline{x}}_4)}(\rho (x_3,x_5)\rho (x_1,x_2)a_4-{(-1)}^{{\overline{x}}_2{\overline{x}}_4}\rho (x_3,x_5)\rho (x_1,x_4)a_2\\ +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_4)}\rho (x_3,x_5)\rho (x_2,x_4)a_1)+{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2+{\overline{x}}_4+{\overline{x}}_5)}\rho ([x_1,x_2,x_4],x_5)a_3)\\ +{(-1)}^{({\overline{x}}_3+{\overline{x}}_4)({\overline{x}}_1+{\overline{x}}_2)}([x_3,x_4,[x_1,x_2,x_5]],\rho (x_3,x_4)\rho (x_1,x_2)a_5-{(-1)}^{{\overline{x}}_2{\overline{x}}_5}\rho (x_3,x_4)\rho (x_1,x_5)a_2\\ +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_5)}\rho (x_3,x_4)\rho (x_2,x_5)a_1-{(-1)}^{{\overline{x}}_4({\overline{x}}_1+{\overline{x}}_2+{\overline{x}}_5)}\rho (x_3,[x_1,x_2,x_5])a_4\\ +{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2+{\overline{x}}_4+{\overline{x}}_5)}\rho (x_4,[x_1,x_2,x_5])a_3).\end{array}$$

Then (1.6)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) holds thanks to (1.10)(1.13) $${\rho }_{\tau }(x,y)=\tau (x)\mu (y)-{(-1)}^{\overline{x}\overline{y}}\tau (y)\mu (x),\hspace{1em}\forall x,y\in \mathcal{H}(\mathfrak{g}).$$(1.13) . Therefore, $(E,{[·,·,·]}_1)$ is a 3-Lie superalgebra.

Now we prove that ρ₁ is a representation of E over A. By (2.9)(1.7) $$\begin{array}{c}\left[\right[x_1,x_2,x_3],x_4,x_5]-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}\left[\right[x_1,x_2,x_4],x_3,x_5]+{(-1)}^{{\overline{x}}_2({\overline{x}}_3+{\overline{x}}_4)}\left[\right[x_1,x_3,x_4],x_2,x_5]\hfill \\ -{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3+{\overline{x}}_4)}\left[\right[x_2,x_3,x_4],x_1,x_5]=0,\hfill \end{array}$$(1.7) , we have $$\begin{array}{c}{\rho }_1([(x_1,a_1),(x_2,a_2),(x_3,a_3)],(x_4,a_4))-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}{\rho }_1((x_3,a_3),[(x_1,a_1),(x_2,a_2),(x_4,a_4)])\\ ={\rho }_1(([x_1,x_2,x_3],\rho (x_1,x_2)a_3-{(-1)}^{{\overline{x}}_2{\overline{x}}_3}\rho (x_1,x_2)a_3+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)}\rho (x_2,x_3)a_1),(x_4,a_4))\\ -{(-1)}^{{\overline{x}}_3{\overline{x}}_4}({\rho }_1((x_3,a_3),([x_1,x_2,x_4],\rho (x_1,x_2)a_4-{(-1)}^{{\overline{x}}_2{\overline{x}}_4}\rho (x_1,x_2)a_4\\ +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_4)}\rho (x_2,x_4)a_1)))\\ =\rho ([x_1,x_2,x_3],x_4)-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}\rho (x_3,[x_1,x_2,x_4])\\ =\rho (x_1,x_2)\rho (x_3,x_4)-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}\rho (x_3,x_4)\rho (x_1,x_2)\\ ={\rho }_1((x_1,a_1),(x_2,a_2)){\rho }_1((x_3,a_3),(x_4,a_4))\\ \hspace{1em}-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}\rho ((x_3,a_3),(x_4,a_4))\rho ((x_1,a_1),(x_2,a_2)).\end{array}$$

Therefore, (1.10)(1.10) $$\begin{array}{c}\rho (x_1,x_2)\rho (x_3,x_4)-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}\rho (x_3,x_4)\rho (x_1,x_2)\hfill \\ =\rho ([x_1,x_2,x_3],x_4)-{(-1)}^{{\overline{x}}_3{\overline{x}}_4}\rho ([x_1,x_2,x_4],x_3),\hfill \end{array}$$(1.10) holds. Similarly, we can prove (1.11)(1.11) $$\begin{array}{c}\rho ([x_1,x_2,x_3],x_4)=\rho (x_1,x_2)\rho (x_3,x_4)+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)}\rho (x_2,x_3)\rho (x_1,x_4)\hfill \\ +{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2)}\rho (x_3,x_1)\rho (x_2,x_4),\hfill \end{array}$$(1.11) . Then ρ₁ is a representation of E over A. For all $b\in \mathcal{H}(A),$ we have $$\begin{array}{c}{[(x_1,a_1),(x_2,a_2),b(x_3,a_3)]}_1={[(x_1,a_1),(x_2,a_2),(b{x}_3,b{a}_3)]}_1\\ =([x_1,x_2,b{x}_3],\rho (x_1,x_2)(b{a}_3)-{(-1)}^{{\overline{x}}_2{\overline{x}}_3}\rho (x_1,b{x}_3)a_2+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)}\rho (x_2,b{x}_3)a_1)\\ =({(-1)}^{\overline{b}({\overline{x}}_1+{\overline{x}}_2)}b[x_1,x_2,x_3]+\rho (x_1,x_2)b{x}_3,\rho (x_1,x_2)(b{a}_3)-{(-1)}^{{\overline{x}}_2{\overline{x}}_3}\rho (x_1,b{x}_3)a_2\\ +{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)}\rho (x_2,b{x}_3)a_1)\\ ={(-1)}^{\overline{b}({\overline{x}}_1+{\overline{x}}_2)}b[(x_1,a_1),(x_2,a_2),(x_3,a_3)]+{\rho }_1(x_1,x_2)b(x_3,a_3).\end{array}$$

Moreover, we have $$\begin{array}{c}{\rho }_1(c(x,a),(y,b))={\rho }_1((cx,ca),(y,b))=\rho (cx,y)\\ ={(-1)}^{\overline{c}\overline{x}}\rho (x,cy)={(-1)}^{\overline{c}\overline{x}}{\rho }_1((x,a),(cy,cb))={(-1)}^{\overline{c}\overline{x}}{\rho }_1((x,a),c(y,b)).\end{array}$$

Similarly, we obtain ${\rho }_1(c(x,a),(y,b))=c{\rho }_1((x,a),(y,b)).$ Then $(E,A,{[·,·,·]}_1,{\rho }_1)$ is a 3-Lie-Rinehart superalgebra. □

4. Cohomology and deformations of 3-Lie-Rinehart superalgebras

In this section, we study the notion of a module for a 3-Lie-Rinehart superalgebras and subsequently we introduce a cochain complex and cohomology of a 3-Lie-Rinehart superalgebras with coefficients in a module, then we study relations between 1 and 2 cocycles of a Lie-Rinehart superalgebra and the induced 3-Lie-Rinehart superalgebra. At the end of this section, we study the deformation of 3-Lie-Rinehart superalgebras

4.1. Representations and cohomology of 3-Lie-Rinehart superalgebras

Definition 3.1.

Let M be an A-module. and $\psi :L\otimes L\to \mathit{End}(M)$ be an even linear map. The pair $(M,\psi )$ is called a left module of the 3-Lie-Rinehart superalgebra $(L,A,[·,·,·],\rho )$ if the following conditions hold:

1. ψ is a representation of $(L,[·,·,·])$ on M,

2. $\psi (a·x,y)={(-1)}^{\overline{a}\overline{x}}\psi (x,a·y)=a·\psi (x,y),$ for all $a\in \mathcal{H}(A)$ and $x,y\in \mathcal{H}(L),$

3. $\psi (x,y)(a·m)={(-1)}^{\overline{a}(\overline{x}+\overline{y})}a·\psi (x,y)(m)+\rho (x,y)(a)m,$ for all $x,y\in \mathcal{H}(L),a\in \mathcal{H}(A)$ and $m\in M.$

Example 3.2.

A is a left module over L since ρ is a representation of $(L,[·,·,·])$ over A and the other conditions are satisfied automatically by definition of the map ρ.

Example 3.3.

The pair (L, ad) is a left module over L, which is called the adjoint representation of $(L,A,[·,·,·],\rho ).$

Proposition 3.4.

Let $(L,A,[·,·,·],\rho )$ be a 3-Lie-Rinehart superalgebra. Then $(M,\psi )$ is a left module over $(L,A,[·,·,·],\rho )$ if and only if $(L\oplus M,A,{[·,·,·]}_{L\oplus M},{\rho }_{L\oplus M})$ is a 3-Lie-Rinehart superalgebra with the multiplication: $$\begin{array}{c}{[x_1+m_1,x_2+m_2,x_3+m_3]}_{L\oplus M}:=[x_1,x_2,x_3]+\psi (x_1,x_2)m_3+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)}\psi (x_2,x_3)m_1\\ +{(-1)}^{{\overline{x}}_2{\overline{x}}_3}\psi (x_3,x_1)m_2,\\ {\rho }_{L\oplus M}:(L\oplus M)\otimes (L\oplus M)\to \mathit{Der}(A),\hspace{1em}{\rho }_{L\oplus M}(x_1+m_1,x_2+m_2):=\rho (x_1,x_2),\end{array}$$ for any $x_1,x_2,x_3\in \mathcal{H}(L)$ and $m_1,m_2,m_3\in M$. Note that ${(L\oplus M)}_{\overline{0}}=L_{\overline{0}}\oplus M_{\overline{0}}$, implying that if $x+m\in \mathcal{H}(L\oplus M)$, then $\overline{x+m}=\overline{x}=\overline{m}.$

Proof.

Since L and M are A-modules, then $L\oplus M$ is an A-module via $$a(x+m)=ax+am,\forall a\in A,x\in L,m\in M.$$

If $(M,\psi )$ is a left module over $(L,A,[·,·,·],\rho ).$ Then $(L\oplus M,{[·,·,·]}_{L\oplus M})$ is a 3-Lie superalgebra. It is obvious that ${\rho }_{L\oplus M}$ is a representation of the 3-Lie superalgebra $(L\oplus M,{[·,·,·]}_{L\oplus M})$ over A.

For any $x_1,x_2,x_3\in \mathcal{H}(L),m_1,m_2,m_3\in \mathcal{H}(M)$ and $a\in \mathcal{H}(A),$ $$\begin{array}{c}{[x_1+m_1,x_2+m_2,a(x_3+m_3)]}_{L\oplus M}={[x_1+m_1,x_2+m_2,a{x}_3+a{m}_3]}_{L\oplus M}\\ =[x_1,x_2,a{x}_3]+\psi (x_1,x_2)a{m}_3+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)}\psi (x_2,a{x}_3)m_1+{(-1)}^{{\overline{x}}_2{\overline{x}}_3}\psi (a{x}_3,x_1)m_2\\ ={(-1)}^{\overline{a}({\overline{x}}_1+{\overline{x}}_2)}a[x_1,x_2,x_3]+\psi (x_1,x_2)a{m}_3+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)}\psi (x_2,a{x}_3)m_1\\ +{(-1)}^{{\overline{x}}_2{\overline{x}}_3}\psi (a{x}_3,x_1)m_2+\rho (x_1,x_2)(a{m}_3)\\ ={(-1)}^{\overline{a}({\overline{x}}_1+{\overline{x}}_2)}a([x_1,x_2,x_3]+\psi (x_1,x_2)m_3+{(-1)}^{{\overline{x}}_1({\overline{x}}_2+{\overline{x}}_3)}\psi (x_2,x_3)m_1+{(-1)}^{{\overline{x}}_2{\overline{x}}_3}\psi (x_3,x_1)m_2)\\ +\rho (x_1,x_2)(a{m}_3)\\ ={(-1)}^{\overline{a}({\overline{x}}_1+{\overline{x}}_2)}a{[x_1+m_1,x_2+m_2,x_3+m_3]}_{L\oplus M}+{\rho }_{L\oplus M}(x_1+m_1,x_2+m_2)a(x_3+m_3).\end{array}$$

Moreover, $$\begin{array}{c}{\rho }_{L\oplus M}(a(x_1+m_1),x_2+m_2)={\rho }_{L\oplus M}(a{x}_1+a{m}_1),x_2+m_2)=\rho (a{x}_1,x_2)={(-1)}^{\overline{a}{\overline{x}}_1}\rho (x_1,a{x}_2)\\ ={(-1)}^{\overline{a} \overline{x_1+m_1}}{\rho }_{L\oplus M}(x_1+m_1,a{x}_2+a{m}_2)={(-1)}^{\overline{a} \overline{x_1+m_1}}{\rho }_{L\oplus M}(x_1+m_1,a(x_2+m_2)).\end{array}$$

Similarly, we obtain ${\rho }_{L\oplus M}(a(x_1+m_1),x_2+m_2)=a{\rho }_{L\oplus M}(x_1+m_1,x_2+m_2).$ Therefore, $(L\oplus M,A,{[·,·,·]}_{L\oplus M},{\rho }_{L\oplus M})$ is a 3-Lie-Rinehart superalgebra. The sufficient condition can be done in the same way. □

Let $(M,\psi )$ be a left module of the 3-Lie-Rinehart superalgebra $(L,A,[·,·,·],\rho )$ and $C^n(L,M)$ the space of all linear maps $f:{\wedge }^{2}L\otimes \cdots \otimes {\wedge }^{2}L\wedge L\to M$ satisfying the following conditions:

1. $f(x_1,\dots ,x_i,x_{i+1},\dots ,x_{2n},x_{2n+1})=-{(-1)}^{{\overline{x}}_i{\overline{x}}_{i+1}}f(x_1,\dots ,x_{i+1},x_i,\dots ,x_{2n+1}),$ for all $x_i\in \mathcal{H}(L),1\le i\le 2n+1,$

2. $f(x_1,···,a·x_i,···,x_{2n+1})={(-1)}^{\overline{a}({\overline{x}}_1+\dots +{\overline{x}}_{i-1}+\overline{f})}af(x_1,···,x_i,···,x_{2n+1}),$ for all $x_i\in \mathcal{H}(L),1\le i\le 2n+1$ and $a\in \mathcal{H}(A).$

Next we consider the ${\mathbb{Z}}_{+}$-graded space of $\mathbb{K}$-modules $$C^{\ast }(L,M):={\oplus }_{n\ge 0}{C}^n(L,M).$$

Define the $\mathbb{K}$-linear maps ${\delta }_{3LR}:C^{n-1}(L,M)\to C^n(L,M)$ by $$\begin{array}{c}{\delta }_{3LR}f(x_1,···,x_{2n+1})\\ ={(-1)}^{n+(\overline{f}+{\overline{x}}_1+{\overline{x}}_2+\dots +{\overline{x}}_{2n-2})({\overline{x}}_{2n-1}+{\overline{x}}_{2n+1})+{\overline{x}}_{2n+1} {\overline{x}}_{2n}}\psi (x_{2n-1},x_{2n+1})f(x_1,···,x_{2n-2},x_{2n})\\ +{(-1)}^{n+(\overline{f}+{\overline{x}}_1+{\overline{x}}_2+\dots +{\overline{x}}_{2n-1})({\overline{x}}_{2n}+{\overline{x}}_{2n+1})}\psi (x_{2n},x_{2n+1})f(x_1,···,x_{2n-1})\\ +\sum _{k=1}^n{(-1)}^{k+(\overline{f}+{\overline{x}}_1+{\overline{x}}_2+\dots +{\overline{x}}_{2k-2})({\overline{x}}_{2k-1}+{\overline{x}}_{2k})}\psi (x_{2k-1},x_{2k})f(x_1,···,{\widehat{x}}_{2k-1},{\widehat{x}}_{2k},···,x_{2n+1})\\ +\sum _{k=1}^n\sum _{j=2k+1}^{2n+1}{(-1)}^{k+({\overline{x}}_{2k+1}+\dots +{\overline{x}}_{j-1})({\overline{x}}_{2k-1}+{\overline{x}}_{2k})}f(x_1,···,{\widehat{x}}_{2k-1},{\widehat{x}}_{2k},···,[x_{2k-1},x_{2k},x_j],···,x_{2n+1}).\end{array}$$

Proposition 3.5.

If $f\in C^{n-1}(L,M)$, then ${\delta }_{3LR}f\in C^n(L,M)$ and ${\delta }_{3LR}^2=0.$

Proof.

Let f be a homogenous element in $C^{n-1}(L,M),$ it is obvious that ${\delta }_{3LR}f$ is skew-symmetric. For all $x_1,x_2,\dots ,x_{2n+1}\in \mathcal{H}(L),a\in \mathcal{H}(A)$ and for $i<2n-1,$ $$\begin{array}{c}{\delta }_{3LR}f(x_1,···,a{x}_i,···,x_{2n+1})=\\ {(-1)}^{n+(\overline{f}+{\overline{x}}_1+{\overline{x}}_2+\dots +{\overline{x}}_{2n-2})({\overline{x}}_{2n-1}+{\overline{x}}_{2n+1})+{\overline{x}}_{2n+1} {\overline{x}}_{2n}}\psi (x_{2n-1},x_{2n+1})f(x_1,···,a{x}_i,···,x_{2n-2},x_{2n})\\ +{(-1)}^{n+(\overline{f}+{\overline{x}}_1+{\overline{x}}_2+\dots +{\overline{x}}_{2n-1})({\overline{x}}_{2n}+{\overline{x}}_{2n+1})}\psi (x_{2n},x_{2n+1})f(x_1,···,a{x}_i,···,x_{2n-1})\\ +\sum _{\underset{i<2k-1}{k=1}}^n{(-1)}^{k+(\overline{f}+\overline{a}+{\overline{x}}_1+{\overline{x}}_2+\dots +{\overline{x}}_{2k-2})({\overline{x}}_{2k-1}+{\overline{x}}_{2k})}\psi (x_{2k-1},x_{2k})f(x_1,···,a{x}_i,···,{\widehat{x}}_{2k-1},{\widehat{x}}_{2k},···,x_{2n+1})\\ +\sum _{\underset{i>2k}{k=1}}^n{(-1)}^{k+(\overline{f}+{\overline{x}}_1+{\overline{x}}_2+\dots +{\overline{x}}_{2k-2})({\overline{x}}_{2k-1}+{\overline{x}}_{2k})}\psi (x_{2k-1},x_{2k})f(x_1,···,{\widehat{x}}_{2k-1},{\widehat{x}}_{2k},···,a{x}_i,···,x_{2n+1})\\ +\sum _{k=1}^n{(-1)}^{k+(\overline{f}+{\overline{x}}_1+{\overline{x}}_2+\dots +{\overline{x}}_{2k-2})({\overline{x}}_{2k-1}+\overline{a}+{\overline{x}}_{2k})}\psi (a{x}_{2k-1},x_{2k})f(x_1,···,{\widehat{x}}_{2k-1},{\widehat{x}}_{2k},···,x_{2n+1})\\ +\sum _{k=1}^n{(-1)}^{k+(\overline{f}+{\overline{x}}_1+{\overline{x}}_2+\dots +{\overline{x}}_{2k-2})({\overline{x}}_{2k-1}+\overline{a}+{\overline{x}}_{2k})}\psi (x_{2k-1},a{x}_{2k})f(x_1,···,{\widehat{x}}_{2k-1},{\widehat{x}}_{2k},···,x_{2n+1})\\ +\sum _{\underset{i<2k}{k=1}}^n\sum _{j=2k+1}^{2n+1}{(-1)}^{k+({\overline{x}}_{2k+1}+\dots +{\overline{x}}_{j-1})({\overline{x}}_{2k-1}+{\overline{x}}_{2k})}\\ f(x_1,···,a{x}_i,···,{\widehat{x}}_{2k-1},{\widehat{x}}_{2k},···,[x_{2k-1},x_{2k},x_j],···,x_{2n+1})\\ +\sum _{k=1}^n\sum _{\underset{2k-1<i<j}{j=2k+1}}^{2n+1}{(-1)}^{k+(\overline{a}+{\overline{x}}_{2k+1}+\dots +{\overline{x}}_{j-1})({\overline{x}}_{2k-1}+{\overline{x}}_{2k})}\\ f(x_1,···,{\widehat{x}}_{2k-1},{\widehat{x}}_{2k},···,a{x}_i,···,[x_{2k-1},x_{2k},x_j],···,x_{2n+1})\\ +\sum _{k=1}^n\sum _{\underset{j<i}{j=2k+1}}^{2n+1}{(-1)}^{k+({\overline{x}}_{2k+1}+\dots +{\overline{x}}_{j-1})({\overline{x}}_{2k-1}+{\overline{x}}_{2k})}\\ f(x_1,···,{\widehat{x}}_{2k-1},{\widehat{x}}_{2k},···,[x_{2k-1},x_{2k},x_j],···,a{x}_i,···,x_{2n+1})\\ +\sum _{k=1}^n\sum _{j=2k+1}^{2n+1}{(-1)}^{k+({\overline{x}}_{2k+1}+\dots +{\overline{x}}_{j-1})({\overline{x}}_{2k-1}+\overline{a}+{\overline{x}}_{2k})}f(x_1,···,{\widehat{x}}_{2k-1},{\widehat{x}}_{2k},···,[a{x}_{2k-1},x_{2k},x_j],···,x_{2n+1})\\ +\sum _{k=1}^n\sum _{j=2k+1}^{2n+1}{(-1)}^{k+({\overline{x}}_{2k+1}+\dots +{\overline{x}}_{j-1})({\overline{x}}_{2k-1}+\overline{a}+{\overline{x}}_{2k})}f(x_1,···,{\widehat{x}}_{2k-1},{\widehat{x}}_{2k},···,[x_{2k-1},a{x}_{2k},x_j],···,x_{2n+1})\\ +\sum _{k=1}^n\sum _{j=2k+1}^{2n+1}{(-1)}^{k+({\overline{x}}_{2k+1}+\dots +{\overline{x}}_{j-1})({\overline{x}}_{2k-1}+{\overline{x}}_{2k})}f(x_1,···,{\widehat{x}}_{2k-1},{\widehat{x}}_{2k},···,[x_{2k-1},x_{2k},a{x}_j],···,x_{2n+1}).\end{array}$$

Using Definition 3.1 and (1.15)(1.5) $$[x,y,z]=-{(-1)}^{\overline{x}\overline{y}}[y,x,z],\hspace{1em}[x,y,z]=-{(-1)}^{\overline{y}\overline{z}}[x,z,y],$$(1.5) , we obtain $${\delta }_{3LR}f(x_1,···,a{x}_i,···,x_{2n+1})={(-1)}^{\overline{a}({\overline{x}}_1+\dots +{\overline{x}}_{i-1}+\overline{f})}a{\delta }_{3LR}f(x_1,···,x_i,···,x_{2n+1}).$$

Similarly, we can proof the same result if $i=2n-1,2n,2n+1.$ Then ${\delta }_{3LR}$ is well-defined. Further, ${\delta }_{3LR}^2=0$ follows from the direct but a long calculation. □

By the above proposition, $(C^{\ast }(L,M),{\delta }_{3LR})$ is a cochain complex. The resulting cohomology of the cochain complex can be defined as the cohomology space of 3-Lie-Rinehart superalgebra $(L,A,[·,·,·],\rho )$ with coefficients in $(M,\psi ),$ and we denote this cohomology as $H^{\ast }(L,M).$

Definition 3.6.

Let $(L,A,[·,·,·],\rho )$ be a 3-Lie-Rinehart superalgebra and $(M,\psi )$ be a left module over L. If $\nu \in H_{3LR}^1(L,M)$ satisfies $$\begin{array}{c}{(-1)}^{\overline{\nu }({\overline{x}}_1+{\overline{x}}_2)}\psi (x_1,x_2)\nu (x_3)+{(-1)}^{{\overline{x}}_2{\overline{x}}_3+\overline{\nu }({\overline{x}}_2+{\overline{x}}_3)}\psi (x_1,x_3)\nu (x_2)\\ +{(-1)}^{({\overline{x}}_1+\overline{\nu })({\overline{x}}_2+{\overline{x}}_3)}\psi (x_2,x_3)\nu (x_1)+\nu ([x_1,x_2,x_3])=0,\end{array}$$ for any $x_1,x_2,x_3\in \mathcal{H}(L),$ then ν is called a 1-cocycle associated with ψ.

Definition 3.7.

Let $(L,A,[·,·,·],\rho )$ be a 3-Lie-Rinehart superalgebra and $(M,\psi )$ be a left module over L. If $\omega \in H_{3LR}^2(L,M)$ satisfies $$\begin{array}{c}{(-1)}^{(\overline{\omega }+{\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_5)+{\overline{x}}_4{\overline{x}}_5}\psi (x_3,x_5)\omega (x_1,x_2,x_4)+{(-1)}^{(\overline{\omega }+{\overline{x}}_1+{\overline{x}}_2+{\overline{x}}_3)({\overline{x}}_4+{\overline{x}}_5)}\psi (x_4,x_5)\omega (x_1,x_2,x_3)\\ -{(-1)}^{\overline{\omega }({\overline{x}}_1+{\overline{x}}_2)}\psi (x_1,x_2)\omega (x_3,x_4,x_5)+{(-1)}^{(\overline{\omega }+{\overline{x}}_1+{\overline{x}}_2)({\overline{x}}_3+{\overline{x}}_4)}\psi (x_3,x_4)\omega (x_1,x_2,x_5)\\ -\omega ([x_1,x_2,x_3],x_4,x_5)-{(-1)}^{{\overline{x}}_3({\overline{x}}_1+{\overline{x}}_2)}\omega (x_3,[x_1,x_2,x_4],x_5)\\ -{(-1)}^{({\overline{x}}_3+{\overline{x}}_4)({\overline{x}}_1+{\overline{x}}_2)}\omega (x_3,x_4,[x_1,x_2,x_5])+\omega (x_1,x_2,[x_3,x_4,x_5])=0,\end{array}$$ for any $x_1,x_2,x_3,x_4,x_5\in \mathcal{H}(L),$ then ω is called a 2-cocycle associated with ψ.

Theorem 3.8.

Let $(L,A,[·,·],\mu )$ be a Lie-Rinehart superalgebra, τ be a supertrace and $\phi \in Z_{LR}^2(L,L)$ such that $$\begin{array}{c}\forall x,y,z\in \mathcal{H}(L):\\ \tau \left(x\right)\tau \left(\phi \left(y,z\right)\right)-{(-1)}^{\overline{x}\overline{y}}\tau \left(y\right)\tau \left(\phi \left(x,z\right)\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\tau \left(z\right)\tau \left(\phi \left(x,y\right)\right)=0.\end{array}$$

Define the linear map $\varphi :{\otimes }^{3}L\to L$ by $$\varphi \left(x,y,z\right)=\tau \left(x\right)\phi \left(y,z\right)-{(-1)}^{\overline{x}\overline{y}}\tau \left(y\right)\phi \left(x,z\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\tau \left(z\right)\phi \left(x,y\right).$$

Then $\varphi$ is a 2-cocycle of the induced 3-Lie-Rinehart superalgebra $(L,A,{[·,·,·]}_{\tau },{\rho }_{\tau }).$

Proof.

Let $\phi \in Z_{LR}^2(L,L).$ It is obvious that $\varphi$ is skew-symmetric and $\overline{\varphi }=\overline{\phi }.$ If $x,y,z\in \mathcal{H}(L),$ then $$\begin{array}{c}\varphi \left(ax,y,z\right)=\tau \left(ax\right)\phi \left(y,z\right)-{(-1)}^{\overline{x}\overline{y}}\tau \left(y\right)\phi \left(ax,z\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\tau \left(z\right)\phi \left(ax,y\right)\\ =a\tau \left(x\right)\phi \left(y,z\right)-{(-1)}^{\overline{x}\overline{y}+\overline{a}\overline{\phi }}\tau \left(y\right)a\phi \left(x,z\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})+\overline{a}\overline{\phi }}\tau \left(z\right)a\phi \left(x,y\right))\\ ={(-1)}^{\overline{a}\overline{\phi }}a(\tau \left(x\right)\phi \left(y,z\right)-{(-1)}^{\overline{x}\overline{y}}\tau \left(y\right)\phi \left(x,z\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\tau \left(z\right)\phi \left(x,y\right))\\ ={(-1)}^{\overline{a}\overline{\varphi }}a\varphi \left(x,y,z\right).\end{array}$$

Similarly, $\varphi (x,ay,z)={(-1)}^{\overline{a}(\overline{x}+\overline{\varphi })}a\varphi (x,y,z)$ and $\varphi (x,y,az)={(-1)}^{\overline{a}(\overline{x}+\overline{y}+\overline{\varphi })}a\varphi (x,y,z).$ On the other hand, if $x_1,x_2,y_1,y_2,z\in \mathcal{H}(L),$ then $$\begin{array}{c}{\delta }_{3LR}\varphi \left(x_1,x_2,y_1,y_2,z\right)=\varphi \left(x_1,x_2,{\left[y_1,y_2,z\right]}_{\tau }\right)-\varphi \left({\left[x_1,x_2,y_1\right]}_{\tau },y_2,z\right)\\ -{(-1)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2)}\varphi \left(y_1,{\left[x_1,x_2,y_2\right]}_{\tau },z\right)-{(-1)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2)}\varphi \left(y_1,y_2,{\left[x_1,x_2,z\right]}_{\tau }\right)\\ +{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)\overline{\varphi }}{\left[x_1,x_2,\varphi \left(y_1,y_2,z\right)\right]}_{\tau }-{\left[\varphi \left(x_1,x_2,y_1\right),y_2,z\right]}_{\tau }\\ -{(-1)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })}{\left[y_1,\varphi \left(x_1,x_2,y_2\right),z\right]}_{\tau }-{(-1)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })}{\left[y_1,y_2,\varphi \left(x_1,x_2,z\right)\right]}_{\tau }\\ =\tau \left(y_1\right)\varphi \left(x_1,x_2,\left[y_2,z\right]\right)-{\left(-1\right)}^{{\overline{y}}_1{\overline{y}}_2}\tau \left(y_2\right)\varphi \left(x_1,x_2,\left[z,y_1\right]\right)\\ +{\left(-1\right)}^{\overline{z}({\overline{y}}_1+{\overline{y}}_2)}\tau \left(z\right)\varphi \left(x_1,x_2,\left[y_1,y_2\right]\right)-\tau \left(x_1\right)\varphi \left(\left[x_2,y_1\right],y_2,z\right)\\ +{\left(-1\right)}^{{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\varphi \left(\left[x_1,y_1\right],y_2,z\right)-{\left(-1\right)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2)}\tau \left(y_1\right)\varphi \left(\left[x_1,x_2\right],y_2,z\right)\\ -{\left(-1\right)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2)}\tau \left(x_1\right)\varphi \left(y_1,\left[x_2,y_2\right],z\right)+{\left(-1\right)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2)+{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\varphi \left(y_1,\left[x_1,y_2\right],z\right)\\ -{\left(-1\right)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2)}\tau \left(y_2\right)\varphi \left(y_1,\left[x_1,x_2\right],z\right)-{\left(-1\right)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2)}\tau \left(x_1\right)\varphi \left(y_1,y_2,\left[x_2,z\right]\right)\\ +{\left(-1\right)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2)+{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\varphi \left(y_1,y_2,\left[x_1,z\right]\right)-{\left(-1\right)}^{(\overline{z}+{\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2)}\tau \left(z\right)\varphi \left(y_1,y_2,\left[x_1,x_2\right]\right)\\ +{\left(-1\right)}^{\overline{\varphi }({\overline{x}}_1+{\overline{x}}_2)}\tau \left(x_1\right)\left[x_2,\varphi \left(y_1,y_2,z\right)\right]-{\left(-1\right)}^{\overline{\varphi }({\overline{x}}_1+{\overline{x}}_2)+{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\left[x_1,\varphi \left(y_1,y_2,z\right)\right]\\ +{\left(-1\right)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{y}}_1+{\overline{y}}_2+\overline{z})}\tau \left(\varphi \left(y_1,y_2,z\right)\right)\left[x_1,x_2\right]-\tau \left(\varphi \left(x_1,x_2,y_1\right)\right)\left[y_2,z\right]\\ +{\left(-1\right)}^{{\overline{y}}_2({\overline{x}}_1+{\overline{x}}_2+{\overline{y}}_1+\overline{\varphi })}\tau \left(y_2\right)\left[\varphi \left(x_1,x_2,y_1\right),z\right]-{\left(-1\right)}^{\overline{z}({\overline{x}}_1+{\overline{x}}_2+{\overline{y}}_1+{\overline{y}}_2+\overline{\varphi })}\tau \left(z\right)\left[\varphi \left(x_1,x_2,y_1\right),y_2\right]\\ -{\left(-1\right)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })}\tau \left(y_1\right)\left[\varphi \left(x_1,x_2,y_2\right),z\right]+{\left(-1\right)}^{{\overline{y}}_1{\overline{y}}_2}\tau \left(\varphi \left(x_1,x_2,y_2\right)\right)\left[y_1,z\right]\end{array}$$ $$\begin{array}{c}-{\left(-1\right)}^{({\overline{y}}_1+\overline{z})({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })+\overline{z}({\overline{y}}_1+{\overline{y}}_2)}\tau \left(z\right)\left[y_1,\varphi \left(x_1,x_2,y_2\right)\right]\\ -{\left(-1\right)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })}\tau \left(y_1\right)\left[y_2,\varphi \left(x_1,x_2,z\right)\right]\\ +{\left(-1\right)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })+{\overline{y}}_1{\overline{y}}_2}\tau \left(y_2\right)\left[y_1,\varphi \left(x_1,x_2,z\right)\right]\\ -{\left(-1\right)}^{\overline{z}({\overline{y}}_1+{\overline{y}}_2)}\tau \left(\varphi \left(x_1,x_2,z\right)\right)\left[y_1,y_2\right]\\ =\tau \left(y_1\right)(\tau \left(x_1\right)\phi \left(x_2,\left[y_2,z\right]\right)-{(-1)}^{{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\phi \left(x_1,\left[y_2,z\right]\right)\\ +{(-1)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2)}\tau \left(y_2\right)\phi \left(\left[x_1,x_2\right],z\right).-{(-1)}^{({\overline{y}}_1+\overline{z})({\overline{x}}_1+{\overline{x}}_2)+\overline{z}{\overline{y}}_2}\tau \left(z\right)\phi \left(\left[x_1,x_2\right],y_2\right)\\ -{(-1)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })}\tau \left(x_1\right)\left[\phi \left(x_2,y_2\right),z\right]+{(-1)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })+{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\left[\phi \left(x_1,y_2\right),z\right]\\ -{(-1)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })+{\overline{y}}_2({\overline{x}}_1+{\overline{x}}_2)}\tau \left(y_2\right)\left[\phi \left(x_1,x_2\right),z\right]-{(-1)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })}\tau \left(x_1\right)\left[y_2,\phi \left(x_2,z\right)\right]\\ +{(-1)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })}\tau \left(x_1\right)\left[y_2,\phi \left(x_2,z\right)\right]-{(-1)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })+\overline{z}({\overline{x}}_1+{\overline{x}}_2)}\tau \left(z\right)\left[y_2,\phi \left(x_1,x_2\right)\right])\\ -{(-1)}^{{\overline{y}}_1{\overline{y}}_2}\tau \left(y_2\right)(\tau \left(x_1\right)\phi \left(x_2,\left[y_1,z\right]\right)-{(-1)}^{{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\phi \left(x_1,\left[y_1,z\right]\right)\\ +{(-1)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2)+{\overline{y}}_1{\overline{y}}_2}\tau \left(y_1\right)\phi \left(\left[x_1,x_2\right],z\right)+{(-1)}^{({\overline{y}}_1+{\overline{y}}_2+\overline{z})({\overline{x}}_1+{\overline{x}}_2)+{\overline{y}}_2({\overline{y}}_1+\overline{z})}\tau \left(z\right)\phi \left(y_1,\left[x_1,x_2\right]\right)\end{array}$$ $$\begin{array}{c}-{(-1)}^{{\overline{y}}_2({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })}\tau \left(x_1\right)\left[\phi \left(x_2,y_1\right),z\right]+{(-1)}^{{\overline{y}}_2({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })+{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\left[\phi \left(x_1,y_1\right),z\right]\\ -{(-1)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2)+{\overline{y}}_2\overline{\varphi }}\tau \left(y_1\right)\left[\phi \left(x_1,x_2\right),z\right]-{(-1)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })}\tau \left(x_1\right)\left[y_1,\phi \left(x_2,z\right)\right]\\ +{(-1)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })+{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\left[y_1,\phi \left(x_1,z\right)\right]\\ -{(-1)}^{({\overline{y}}_1+{\overline{y}}_2+\overline{z})({\overline{x}}_1+{\overline{x}}_2)+({\overline{y}}_1+{\overline{y}}_2)\overline{\varphi }}\tau \left(z\right)\left[y_1,\phi \left(x_1,x_2\right)\right])\\ +{(-1)}^{\overline{z}({\overline{y}}_1+{\overline{y}}_2)}\tau \left(z\right)(\tau \left(x_1\right)\phi \left(x_2,\left[y_1,y_2\right]\right)-{(-1)}^{{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\phi \left(x_1,\left[y_1,y_2\right]\right)\\ -{(-1)}^{(\overline{z}+{\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2+\overline{z})}\tau \left(y_1\right)\phi \left(y_2,\left[x_1,x_2\right]\right)+{(-1)}^{(\overline{z}+{\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2+\overline{z})+{\overline{y}}_1{\overline{y}}_2}\tau \left(y_2\right)\phi \left(y_1,\left[x_1,x_2\right]\right)\\ -{(-1)}^{\overline{z}({\overline{x}}_1+{\overline{x}}_2+{\overline{y}}_1+{\overline{y}}_2+\overline{\varphi })}\tau \left(x_1\right)\left[\phi \left(x_2,y_1\right),y_2\right]+{(-1)}^{\overline{z}({\overline{x}}_1+{\overline{x}}_2+{\overline{y}}_1+{\overline{y}}_2+\overline{\varphi })+{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\left[\phi \left(x_1,y_1\right),y_2\right]\\ -{(-1)}^{\overline{z}({\overline{x}}_1+{\overline{x}}_2+{\overline{y}}_1+{\overline{y}}_2+\overline{\varphi })+{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2)}\tau \left(y_1\right)\left[\phi \left(x_1,x_2\right),y_2\right]\\ +{(-1)}^{({\overline{y}}_1+\overline{z})({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })+\overline{z}({\overline{y}}_1+{\overline{y}}_2)}\tau \left(x_1\right)\left[y_1,\phi \left(x_2,y_2\right)\right]\\ -{(-1)}^{({\overline{y}}_1+\overline{z})({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })+\overline{z}({\overline{y}}_1+{\overline{y}}_2)+{\overline{x}}_1+{\overline{x}}_2}\tau \left(x_2\right)\left[y_1,\phi \left(x_1,y_2\right)\right]\\ +{(-1)}^{({\overline{y}}_1+{\overline{y}}_2+\overline{z})({\overline{x}}_1+{\overline{x}}_2+\overline{\varphi })+\overline{z}({\overline{y}}_1+{\overline{y}}_2)+{\overline{y}}_2\overline{\varphi }}\tau \left(y_2\right)\left[y_1,\phi \left(x_1,x_2\right)\right])\\ +\tau \left(x_1\right)({(-1)}^{{\overline{y}}_2({\overline{x}}_2+{\overline{y}}_1)}\tau \left(y_2\right)\phi \left(\left[x_2,y_1\right],z\right)-{(-1)}^{\overline{z}({\overline{y}}_1+{\overline{y}}_2+{\overline{x}}_2)}\tau \left(z\right)\phi \left(\left[x_2,y_1\right],y_2\right)\\ -{(-1)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2)}\tau \left(y_1\right)\phi \left(\left[x_2,y_2\right],z\right)-{(-1)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2)+\overline{z}({\overline{y}}_1+{\overline{y}}_2+{\overline{x}}_2)}\tau \left(z\right)\phi \left(y_1,\left[x_2,y_2\right]\right)\end{array}$$ $$\begin{array}{c}-{(-1)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2)}\tau \left(y_1\right)\phi \left(y_2,\left[x_2,z\right]\right)+{(-1)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2)+{\overline{y}}_1{\overline{y}}_2}\tau \left(x_2\right)\phi \left(y_1,\left[x_2,z\right]\right)\\ +{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)\overline{\varphi }}\tau \left(y_1\right)\left[x_2,\phi \left(y_2,z\right)\right]-{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)\overline{\varphi }+{\overline{y}}_1{\overline{y}}_2}\tau \left(y_2\right)\left[x_2,\phi \left(y_1,z\right)\right]\\ +{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)\overline{\varphi }+\overline{z}({\overline{y}}_2+{\overline{y}}_2)}\tau \left(z\right)\left[x_2,\phi \left(y_1,y_2\right)\right])\\ +{(-1)}^{{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)(-{(-1)}^{{\overline{y}}_2({\overline{x}}_1+{\overline{x}}_2)}\tau \left(y_2\right)\phi \left(\left[x_1,y_1\right],z\right)+{(-1)}^{\overline{z}({\overline{x}}_1+{\overline{y}}_1+{\overline{y}}_2)}\tau \left(z\right)\phi \left(\left[x_1,y_1\right],y_2\right)\\ +{(-1)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2)}\tau \left(y_1\right)\phi \left(\left[x_1,y_1\right],z\right)+{(-1)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2)+\overline{z}({\overline{x}}_1+{\overline{y}}_1+{\overline{y}}_2)}\tau \left(z\right)\phi \left(y_1,\left[x_1,y_2\right]\right)\\ +{(-1)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2)}\tau \left(y_1\right)\phi \left(y_2,\left[x_1,z\right]\right)-{(-1)}^{({\overline{y}}_1+{\overline{y}}_2)({\overline{x}}_1+{\overline{x}}_2)+{\overline{y}}_1{\overline{y}}_2}\tau \left(y_2\right)\phi \left(y_1,\left[x_1,z\right]\right)\\ -{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)\overline{\varphi }}\tau \left(y_1\right)\left[x_1,\phi \left(y_1,z\right)\right]+{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)\overline{\varphi }+{\overline{y}}_1{\overline{y}}_2}\tau \left(y_2\right)\left[x_1,\phi \left(y_1,z\right)\right]\\ -{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)\overline{\varphi }+\overline{z}({\overline{y}}_1+{\overline{y}}_2)}\tau \left(z\right)\left[x_1,\phi \left(y_1,y_2\right)\right])\end{array}$$ $$\begin{array}{c}+{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{y}}_1+{\overline{y}}_2+\overline{z})}(\tau \left(y_1\right)\tau \left(\phi \left(y_2,z\right)\right)-{(-1)}^{{\overline{y}}_1{\overline{y}}_2}\tau \left(y_2\right)\tau \left(\phi \left(y_1,z\right)\right)\\ +{(-1)}^{\overline{z}({\overline{y}}_1+{\overline{y}}_2)}\tau \left(z\right)\tau \left(\phi \left(y_1,y_2\right)\right))\left[x_1,x_2\right]\\ -(\tau \left(x_1\right)\tau \left(\phi \left(x_2,y_1\right)\right)-{(-1)}^{{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\tau \left(\phi \left(x_1,y_1\right)\right)\\ +{(-1)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2)}\tau \left(y_1\right)\tau \left(\phi \left(x_1,x_2\right)\right))\left[y_2,z\right]\\ +{(-1)}^{{\overline{y}}_1{\overline{y}}_2}(\tau \left(x_1\right)\tau \left(\phi \left(x_2,y_2\right)\right)-{(-1)}^{{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\tau \left(\phi \left(x_1,y_2\right)\right)\\ +{(-1)}^{{\overline{y}}_2({\overline{x}}_1+{\overline{x}}_2)}\tau \left(y_2\right)\tau \left(\phi \left(x_1,x_2\right)\right))\left[y_1,z\right]\\ -{(-1)}^{\overline{z}({\overline{y}}_1+{\overline{y}}_2)}(\tau \left(x_1\right)\tau \left(\phi \left(x_2,z\right)\right)-{(-1)}^{{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\tau \left(\phi \left(x_1,z\right)\right)\\ +{(-1)}^{\overline{z}({\overline{x}}_1+{\overline{x}}_2)}\tau \left(z\right)\tau \left(\phi \left(x_1,x_2\right)\right))\left[y_1,y_2\right]\\ =-\tau \left(y_1\right)\tau \left(x_1\right){\delta }_{LR}\phi \left(z,y_2,x_2\right)-\tau \left(y_1\right)\tau \left(x_2\right){\delta }_{LR}\phi \left(y_2,z,x_1\right)\\ -\tau \left(y_2\right)\tau \left(x_1\right){\delta }_{LR}\phi \left(y_1,z,x_2\right)-\tau \left(y_2\right)\tau \left(x_2\right){\delta }_{LR}\phi \left(z,y_1,x_1\right)\\ -\tau \left(z\right)\tau \left(x_1\right){\delta }_{LR}\phi \left(y_2,y_1,x_2\right)-\tau \left(z\right)\tau \left(x_2\right){\delta }_{LR}\phi \left(y_1,y_2,x_1\right)\\ +{(-1)}^{({\overline{x}}_1+{\overline{x}}_2)({\overline{y}}_1+{\overline{y}}_2+\overline{z})}(\tau \left(y_1\right)\tau \left(\phi \left(y_2,z\right)\right)-{(-1)}^{{\overline{y}}_1{\overline{y}}_2}\tau \left(y_2\right)\tau \left(\phi \left(y_1,z\right)\right)\\ +{(-1)}^{\overline{z}({\overline{y}}_1+{\overline{y}}_2)}\tau \left(z\right)\tau \left(\phi \left(y_1,y_2\right)\right))\left[x_1,x_2\right]\\ -(\tau \left(x_1\right)\tau \left(\phi \left(x_2,y_1\right)\right)-{(-1)}^{{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\tau \left(\phi \left(x_1,y_1\right)\right)\\ +{(-1)}^{{\overline{y}}_1({\overline{x}}_1+{\overline{x}}_2)}\tau \left(y_1\right)\tau \left(\phi \left(x_1,x_2\right)\right))\left[y_2,z\right]\\ +{(-1)}^{{\overline{y}}_1{\overline{y}}_2}(\tau \left(x_1\right)\tau \left(\phi \left(x_2,y_2\right)\right)-{(-1)}^{{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\tau \left(\phi \left(x_1,y_2\right)\right)\\ +{(-1)}^{{\overline{y}}_2({\overline{x}}_1+{\overline{x}}_2)}\tau \left(y_2\right)\tau \left(\phi \left(x_1,x_2\right)\right))\left[y_1,z\right]\\ -{(-1)}^{\overline{z}({\overline{y}}_1+{\overline{y}}_2)}(\tau \left(x_1\right)\tau \left(\phi \left(x_2,z\right)\right)-{(-1)}^{{\overline{x}}_1{\overline{x}}_2}\tau \left(x_2\right)\tau \left(\phi \left(x_1,z\right)\right)\\ +{(-1)}^{\overline{z}({\overline{x}}_1+{\overline{x}}_2)}\tau \left(z\right)\tau \left(\phi \left(x_1,x_2\right)\right))\left[y_1,y_2\right].\end{array}$$

Since for all $x,y,z\in \mathcal{H}(L),$ $$\tau \left(x\right)\tau \left(\phi \left(y,z\right)\right)-{(-1)}^{\overline{x}\overline{y}}\tau \left(y\right)\tau \left(\phi \left(x,z\right)\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\tau \left(z\right)\tau \left(\phi \left(x,y\right)\right)=0,$$ we get ${\delta }_{3LR}\varphi =0.$ □

Corollary 3.9.

Let $\phi \in Z_{LR}^2(L,\mathbb{K})$. Then $$\psi \left(x,y,z\right)=\tau \left(x\right)\phi \left(y,z\right)-{(-1)}^{\overline{x}\overline{y}}\tau \left(y\right)\phi \left(x,z\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\tau \left(z\right)\phi \left(x,y\right)$$ is a 2-cocycle of the induced 3-Lie-Rinehart superalgebra.

Theorem 3.10.

Every 1-cocycle for the scalar cohomology of the Lie-Rinehart superalgebra $(L,A,\left[·,·\right],\mu )$ is a 1-cocycle for the scalar cohomology of the induced 3-Lie-Rinehart superalgebra $(L,A,{\left[·,·,·\right]}_{\tau },{\rho }_{\tau }).$

Proof.

Let $\omega \in Z_{LR}^1(L,\mathbb{K}).$ Then, $$\forall x,y\in L:{\delta }_{LR}\omega (x,y)=\omega \left(\left[x,y\right]\right)=0,$$ which is equivalent to $\left[L,L\right]\subset \ker \omega .$ It is obvious to prove that ${\left[L,L,L\right]}_{\tau }\subset \left[L,L\right]$ and then ${\left[L,L,L\right]}_{\tau }\subset \ker \omega ,$ that is $$\forall x,y,z\in L:\omega \left({\left[x,y,z\right]}_{\tau }\right)={\delta }_{3LR}\omega \left(x,y,z\right)=0,$$ which means that ω is a 1-cocycle for the scalar cohomology of $(L,A,{\left[·,·,·\right]}_{\tau },{\rho }_{\tau }).$ □

Lemma 3.11.

Let $\phi \in C^1(L,\mathbb{K})$. Then, for all $x,y,z\in \mathcal{H}(L),$ $${\delta }_{3LR}\phi \left(x,y,z\right)=\tau \left(x\right){\delta }_{LR}\phi \left(y,z\right)-{(-1)}^{\overline{x}\overline{y}}\tau \left(y\right){\delta }_{LR}\phi \left(x,z\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\tau \left(z\right){\delta }_{LR}\phi \left(x,y\right).$$

Proof.

Let $\phi \in C^1(L,\mathbb{K}),x,y,z\in \mathcal{H}(L).$ Then, $$\begin{array}{c}{\delta }_{3LR}\phi \left(x,y,z\right)=\phi \left({\left[x,y,z\right]}_{\tau }\right)\\ =\tau \left(x\right)\phi \left(\left[y,z\right]\right)-{(-1)}^{\overline{x}\overline{y}}\tau \left(y\right)\phi \left(\left[x,z\right]\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\tau \left(z\right)\phi \left(\left[x,y\right]\right)\\ =\tau \left(x\right){\delta }_{LR}\phi \left(y,z\right)-{(-1)}^{\overline{x}\overline{y}}\tau \left(y\right){\delta }_{LR}\phi \left(x,z\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\tau \left(z\right){\delta }_{LR}\phi \left(x,y\right)\end{array}$$ completes the proof. □

Proposition 3.12.

Let ${\phi }_1,{\phi }_2\in Z_{LR}^2(L,\mathbb{K})$. If ${\phi }_1,{\phi }_2$ are in the same cohomology class then ${\psi }_1,{\psi }_2$ defined by $${\psi }_i\left(x,y,z\right)=\tau \left(x\right){\phi }_i\left(y,z\right)-{(-1)}^{\overline{x}\overline{y}}\tau \left(y\right){\phi }_i\left(x,z\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\tau \left(z\right){\phi }_i\left(x,y\right),i=1,2$$ are in the same cohomology class.

Proof.

If ${\phi }_1,{\phi }_2\in Z_{LR}^2(L,\mathbb{K})$ are two cocycles in the same cohomology class, that is ${\phi }_2-{\phi }_1={\delta }_{LR}\nu ,\nu \in C^1(L,\mathbb{K}),$ then $$\begin{array}{c}{\psi }_2\left(x,y,z\right)-{\psi }_1\left(x,y,z\right)=\tau \left(x\right){\phi }_2\left(y,z\right)-{(-1)}^{\overline{x}\overline{y}}\tau \left(y\right){\phi }_2\left(x,z\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\tau \left(z\right){\phi }_2\left(x,y\right)\\ \hspace{1em}-\tau \left(x\right){\phi }_1\left(y,z\right)-{(-1)}^{\overline{x}\overline{y}}\tau \left(y\right){\phi }_1\left(x,z\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\tau \left(z\right){\phi }_1\left(x,y\right)\\ =\tau \left(x\right)({\phi }_2-{\phi }_1)\left(y,z\right)-{(-1)}^{\overline{x}\overline{y}}\tau \left(y\right)({\phi }_2-{\phi }_1)\left(x,z\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\tau \left(z\right)({\phi }_2-{\phi }_1)\left(x,y\right)\\ =\tau \left(x\right){\delta }_{LR}\nu \left(y,z\right)-{(-1)}^{\overline{x}\overline{y}}\tau \left(y\right){\delta }_{LR}\nu \left(x,z\right)+{(-1)}^{\overline{z}(\overline{x}+\overline{y})}\tau \left(z\right){\delta }_{LR}\nu \left(x,y\right)={\delta }_{3LR}\nu \left(x,y,z\right),\end{array}$$ which means that ψ₁ and ψ₂ are in the same cohomology class. □

4.2. Deformations of 3-Lie-Rinehart superalgebras

Let $(L,A,[·,·,·],\rho )$ be a 3-Lie-Rinehart superalgebra. Denote by $\mathbb{K}[[t]]$ the space of formal power series ring with parameter t.

Definition 3.13.

A deformation of a 3-Lie-Rinehart superalgebra $(L,A,[·,·,·],\rho )$ is a $\mathbb{K}[[t]]$-trilinear map $${[·,·,·]}_t:L\left[\right[t\left]\right]\times L\left[\right[t\left]\right]\times L\left[\right[t\left]\right]\to L\left[\right[t\left]\right],\hspace{1em}{[x,y,z]}_t=\sum _{i\ge 0}{t}^i{m}_i(x,y,z),$$ where $m_0=[·,·,·]$ and m_i are even trilinear maps for $i\ge 0,$ satisfying (1.5)(1.5) $$[x,y,z]=-{(-1)}^{\overline{x}\overline{y}}[y,x,z],\hspace{1em}[x,y,z]=-{(-1)}^{\overline{y}\overline{z}}[x,z,y],$$(1.5) , (1.6)(1.6) $$\begin{array}{cc}[x,y,[z,u,v\left]\right]=\left[\right[x,y,z],u,v]\hfill & +{(-1)}^{\overline{z}(\overline{x}+\overline{y})}[z,[x,y,u],v]\hfill \\ \hfill & +{(-1)}^{(\overline{z}+\overline{u})(\overline{x}+\overline{y})}[z,u,[x,y,v\left]\right],\hfill \end{array}$$(1.6) , (1.14)(1.14) $$\rho (ax,y)={(-1)}^{\overline{a}\overline{x}}\rho (x,ay)=a\rho (x,y).$$(1.14) and (1.15)(1.15) $$[x,y,az]={(-1)}^{\overline{a}(\overline{x}+\overline{y})}a[x,y,z]+\rho (x,y)az, \forall x,y,z\in \mathcal{H}(L),a\in \mathcal{H}(A).$$(1.15) .

Let ${[·,·,·]}_t$ be a deformation of $[·,·,·].$ Then $$\begin{array}{c}{\mathrm{m}}_t(x,y,{\mathrm{m}}_t(z,u,v))-{\mathrm{m}}_t({\mathrm{m}}_t(x,y,u),v,w)-{(-1)}^{\overline{u}(\overline{x}+\overline{y})}{\mathrm{m}}_t(u,{\mathrm{m}}_t(x,y,v),w)\\ -{(-1)}^{(\overline{u}+\overline{v})(\overline{x}+\overline{y})}{\mathrm{m}}_t(u,v,{\mathrm{m}}_t(x,y,w))=0.\end{array}$$

Comparing the coefficients of tⁿ, $n\ge 0,$ we get the following equation: $$\begin{array}{c}\sum _{i,j=0}^n({\mathrm{m}}_i(x,y,{\mathrm{m}}_j(z,u,v))-{\mathrm{m}}_i({\mathrm{m}}_j(x,y,u),v,w)-{(-1)}^{\overline{u}(\overline{x}+\overline{y})}{\mathrm{m}}_i(u,{\mathrm{m}}_j(x,y,v),w)\\ -{(-1)}^{(\overline{u}+\overline{v})(\overline{x}+\overline{y})}{\mathrm{m}}_i(u,v,{\mathrm{m}}_j(x,y,w)))=0.\end{array}$$

Definition 3.14.

The 3-cochain m₁ is called the infinitesimal of the deformation ${[·,·,·]}_t.$ More generally, if m_i = 0 for $1\le i\le (n-1)$ and m_n is non zero cochain, then m_n is called the n-infinitesimal of the deformation ${[·,·,·]}_t.$

Definition 3.15.

Two deformations ${[·,·,·]}_t$ and $[·,·,·]{\prime }_t$ are said to be equivalent if there exists a formal automorphism $${\Phi }_t:L\left[\right[t\left]\right]\to L\left[\right[t\left]\right], {\Phi }_t=i{d}_L+\sum _{i\ge 1}{t}^i{\varphi }_i,$$ where ${\varphi }_i:L\to L$ is an even $\mathbb{K}$-linear maps such that $$[x,y,z]{\prime }_t={\Phi }_t^{-1}{[{\Phi }_t(x),{\Phi }_t(y),{\Phi }_t(z)]}_t.$$

Definition 3.16.

A deformation is called trivial if it is equivalent to the deformation $m_0=[·,·,·].$

Definition 3.17.

A 3-Lie-Rinehart superalgebra is said to be rigid if every deformation of it is trivial.

Theorem 3.18.

The cohomology class of the infinitesimal of a deformation ${[·,·,·]}_t$ is determined by the equivalence class of ${[·,·,·]}_t.$

Proof.

Let ${\Phi }_t$ represents an equivalence of deformation given by $m_t$ and $\widetilde{m_t}.$ Then $$\widetilde{m_t}(x,y,z)={\Phi }_t^{-1}{m}_t({\Phi }_t(x),{\Phi }_t(y),{\Phi }_t(z)).$$

Comparing the coefficients of t from both sides of the above equation we have $$\begin{array}{c}\widetilde{m_1}(x,y,z)+{\Phi }_1([x,y,z])=m_1(x,y,z)+[{\Phi }_1(x),y,z]+{(-1)}^{\overline{x}\overline{\Phi }}[x,{\Phi }_1(y),z]\\ +{(-1)}^{(\overline{x}+\overline{y})\overline{\Phi }}[x,y,{\Phi }_1(z)],\end{array}$$ or equivalently, $m_1-\widetilde{m_1}={\delta }_{3LR}({\varphi }_1).$ So, cohomology class of infinitesimal of the deformation is determined by the equivalence class of deformation of $m_t.$ □

Theorem 3.19.

A non-trivial deformation of a 3-Lie-Rinehart superalgebra is equivalent to a deformation whose n-infinitesimal cochain is not a coboundary for some $n\ge 1.$

Proof.

Let ${\mathrm{m}}_t$ be a deformation of 3-Lie-Rinehart superalgebra with n-infinitesimal m_n for some $n\ge 1.$ Assume that there exists a 3-cochain $\varphi \in C^2(L,L)$ with $\delta (\varphi )=m_n.$ Set ${\Phi }_t=i{d}_L+\varphi t^n$ and $\tilde{m}={\Phi }_t^{-1}°{\mathrm{m}}_t°{\Phi }_t.$ Comparing the coefficients of tⁿ, we get the following equation: $${\tilde{m}}_n-m_n=-{\delta }_{3LR}(\varphi ).$$

So ${\tilde{m}}_n=0.$ Then, one has the deformation whose n-infinitesimal is not a coboundary for some $n\ge 1.$ □

Acknowlegments

Dr. Sami Mabrouk is grateful to the research environment in Mathematics and Applied Mathematics MAM, Division of Applied Mathematics and Physics, School of Education, Culture and Communication, Mälardalen University for hospitality and excellent and inspiring environment for research cooperation in Mathematics during his visit in 2019. Partial support from Swedish Royal Academy of Sciences foundations is also gratefully acknowledged.