2. Cohomology theory of Lie supertriple systems

The idea of this subsection springs from [Reference Zhang29]. We show that the cohomology of Lie supertriple systems defined in [Reference Guo9] can be described by means of the cohomology of the associated Leibniz superalgebra.

We begin with a brief review of basic definitions and properties of Lie supertriple systems. As is customary on superspaces, we denote the degree of ${\mathbb{Z}}_2$ -graded vector spaces $V=V_{\bar {0}}\oplus V_{\bar {1}}$ by

\begin{equation*} |x|=\left \{ \begin{array}{l@{\quad}l} \bar {0},& \textrm {if} x\in V_{\bar {0}} ;\\[3pt] \bar {1},& \textrm {if} x\in V_{\bar {1}} . \end{array} \right . \end{equation*}

We regard $x$ as a ${\mathbb{Z}}_2$ -homogeneous element when $|x|$ occurs in some expressions.

Definition 2.1. [Reference Okubo19] A Lie supertriple system ( $\textrm {LSTS}$ in short) is a ${\mathbb{Z}}_2$ -graded space $\mathfrak{L}=\mathfrak{L}_{\bar {0}}\oplus \mathfrak{L}_{\bar {1}}$ over $\mathbb{F}$ with a trilinear product $[\cdot ,\cdot ,\cdot ]$ satisfying the following conditions:

(2.1) \begin{align} |[x,y,z]|=|x|+|y|+|z|, \\[-30pt] \nonumber \end{align}

(2.2) \begin{align} [y,x,z]=-(\!-\!1)^{|x||y|}[x,y,z], \\[-30pt] \nonumber \end{align}

(2.3) \begin{align} (\!-\!1)^{|x||z|}[x,y,z]+(\!-\!1)^{|y||x|}[y,z,x]+(\!-\!1)^{|z||y|}[z,x,y]=0, \\[-30pt] \nonumber \end{align}

(2.4) \begin{align} [x,y,[z,w,u]]=[[x,y,z],w,u]+(\!-\!1)^{(|x|+|y|)|z|}[z,[x,y,w],u]+(\!-\!1)^{(|x|+|y|)(|z|+|w|)}[z,w,[x,y,u]], \\[5pt] \nonumber \end{align}

for all homogeneous elements $x,y,z,w,u\in \mathfrak{L}$ . Denote it by $(\mathfrak{L},[\cdot ,\cdot ,\cdot ])$ . Obviously, the even part to itself is a Lie triple system.

A homomorphism $f:(\mathfrak{L},[\cdot ,\cdot ,\cdot ])\rightarrow (\mathfrak{L}', [\cdot ,\cdot ,\cdot ]')$ of $\textrm {LSTS}$ is a homogeneous linear map satisfying

\begin{equation*}f[x,y,z]=[f(x),f(y),f(z)]', \, \forall x,y,z\in \mathfrak{L}. \end{equation*}

Definition 2.2. [Reference Guo9] A representation of a $\textrm {LSTS}$ $(\mathfrak{L},[\cdot ,\cdot ,\cdot ])$ on a vector space $V$ is an even bilinear map $\theta \;:\; \mathfrak{L}\times \mathfrak{L} \rightarrow \textrm {End}(V)$ such that for all homogeneous elements $x,y,z,w\in \mathfrak{L}$ ,

(2.5) \begin{align} (\!-\!1)^{(|x|+|y|)(|z|+|w|)}\theta (z,w)\theta (x,y)-(\!-\!1)^{|x||y|+|w|(|x|+|z|)}\theta (y,w)\theta (x,z)\notag \\[3pt] -\theta (x,[y,z,w])+(\!-\!1)^{|x|(|y|+|z|)}D(y,z)\theta (x,w)=0, \\[-30pt] \nonumber \end{align}

(2.6) \begin{align} (\!-\!1)^{(|x|+|y|)(|z|+|w|)}\theta (z,w)D(x,y)-D(x,y)\theta (z,w)+\theta ([x,y,z],w)+(\!-\!1)^{|z|(|x|+|y|)}\theta (z,[x,y,w])=0, \\[5pt] \nonumber \end{align}

where $D(x,y)=(\!-\!1)^{|x||y|}\theta (y,x)-\theta (x,y)$ . Also $V$ is called a $\mathfrak{L}$ -module. In the sequel, a representation of $\textrm {LSTS}$ is denoted by $(V;\;\theta ).$

In particular, let $V=\mathfrak{L}$ , and $\theta (x,y)(z)=(\!-\!1)^{|z|(|x|+|y|)}[z,x,y]$ . Then it is easy to see that $D(x,y)(z)=[x,y,z]$ and $\mathfrak{L}$ is showed to be the adjoint $\mathfrak{L}$ -module. In this case, $(\mathfrak{L};\;\theta )$ is called the adjoint representation of $(\mathfrak{L},[\cdot ,\cdot ,\cdot ])$ .

Remark 2.3. Using (2.6) and $D(x,y)=(\!-\!1)^{|x||y|}\theta (y,x)-\theta (x,y)$ , we immediately get

(2.7) \begin{equation} D(x,y)D(z,w)-(\!-\!1)^{(|x|+|y|)(|z|+|w|)}D(z,w)D(x,y)=D([x,y,z],w)+(\!-\!1)^{|z|(|x|+|y|)}D(z,[x,y,w]) \end{equation}

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

Proposition 2.4. [Reference Guo9] Let $(\mathfrak{L},[\cdot ,\cdot ,\cdot ])$ be a $\textrm {LSTS}$ . $(V;\;\theta )$ is a representation of $(\mathfrak{L},[\cdot ,\cdot ,\cdot ])$ if and only if there is a $\textrm {LSTS}$ structure on the direct sum $\mathfrak{L} \oplus V$ of vector spaces (called the semidirect product ) defined by

\begin{equation*}[x+a,y+b,z+c]_V=[x,y,z]+(\!-\!1)^{|x|(|y|+|z|)}\theta (y,z)a-(\!-\!1)^{|y||z|}\theta (x,z)b+D(x,y)c,\end{equation*}

where $x,y,z\in \mathfrak{L}, a,b,c \in V$ are homogeneous elements and $|x+a|=|x|=|a|$ . A semidirect product of $\textrm {LSTS}$ is denoted by $\mathfrak{L} \ltimes V$ .

In what follows, we recall some basic definitions of Leibniz superalgebras.

Definition 2.5. [Reference Liu and Hu14] A Leibniz superalgebra is a ${\mathbb{Z}}_2$ -graded vector space $\mathfrak{g}=\mathfrak{g}_{\bar {0}} \oplus \mathfrak{g}_{\bar {1}}$ over a field $\mathbb{F}$ equipped with a $\mathbb{F}$ -bilinear map $\ast \;:\; \mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g}$ satisfying $\mathfrak{g}_\alpha \ast \mathfrak{g}_{\beta } \subset \mathfrak{g}_{\alpha +\beta }, \forall \alpha , \beta \in \mathbb{Z}_2$ and the Leibniz identity

\begin{equation*} (a\ast b)\ast c=a\ast (b\ast c)-(\!-\!1)^{|a||b|}b\ast (a\ast c), \end{equation*}

for all homogeneous elements $ a, b, c \in \mathfrak{g}$ .

Definition 2.6. [Reference Liu and Hu14] Let $(\mathfrak{g},\ast )$ be a Leibniz superalgebra. A ${\mathbb{Z}}_2$ -graded space $M=M_{\bar {0}}\oplus M_{\bar {1}}$ together with two bilinear maps: $[\cdot ,\cdot ]_L\;:\;\mathfrak{g}\times M\rightarrow M$ and $[\cdot ,\cdot ]_R\;:\;M\times \mathfrak{g}\rightarrow M$ is said to be a representation over $(\mathfrak{g},\ast )$ if the following three axioms are satisfied:

(2.8) \begin{aligned}[x\ast y,m]_L=[x,[y,m]_L]_L-(\!-\!1)^{|x||y|}[y,[x,m]_L]_L,\end{aligned}

(2.9) \begin{aligned}[[x,m]_L,y]_R=[x,[m,y]_R]_L-(\!-\!1)^{|x||m|}[m,x\ast y]_R,\end{aligned}

(2.10) \begin{aligned}[[m,x]_R,y]_R=[m, x\ast y]_R-(\!-\!1)^{|x||m|}[x,[m,y]_R]_L,\end{aligned}

for any homogeneous elements $m\in M$ and $x,y\in \mathfrak{g}$ .

Remark 2.7. By Eqs. (2.9) and (2.10), we also get

(2.11) \begin{equation} [[m,x]_R,y]_R+(\!-\!1)^{|x||m|}[[x,m]_L,y]_R=0. \end{equation}

In fact, assume that Eq. (2.8) holds, then it is easy to check that any of Eqs. (2.9), (2.10) and (2.11) can be derived from the other two.

It was shown in [Reference Zhang29] that one can construct a Leibniz algebra from a Lie triple system. Now we generalize this idea to $\textrm {LSTS}$ .

Let $(\mathfrak{L},[\cdot , \cdot , \cdot ])$ be a $\textrm {LSTS}$ and $\mathcal{L}\;:\!=\;\otimes ^2 \mathfrak{L}$ . Denote by $\mathfrak{X}=(x_1,x_2)$ any homogeneous element of $\mathcal{L}$ , which is called the fundamental object of the $\textrm {LSTS}$ $(\mathfrak{L},[\cdot , \cdot , \cdot ])$ . Obviously, $|\mathfrak{X}|=|x_1|+|x_2|$ . Now we define a bilinear operation $\ast$ on $\mathcal{L}$ by

(2.12) \begin{equation} \mathfrak{X}\ast \mathfrak{Y}=([x_1,x_2,y_1],y_2)+(\!-\!1)^{|y_1|(|x_1|+|x_2|)}(y_1,[x_1,x_2,y_2]), \end{equation}

for any homogeneous elements $\mathfrak{X}=(x_1,x_2), \mathfrak{Y}=(y_1,y_2)\in \mathcal{L}$ . By straightforward tedious calculations, one can prove that $(\mathcal{L},\ast )$ is a Leibniz superalgebra, called the associated Leibniz superalgebra of $(\mathfrak{L},[\cdot , \cdot , \cdot ])$ .

Remark 2.8. $(1)$ For any homogeneous elements $\mathfrak{X}=(x_1,x_2), \mathfrak{Y}=(y_1,y_2)\in \mathcal{L}$ , by (2.12), we can reformulate (2.7) as the following form:

(2.13) \begin{align} D(\mathfrak{X}\ast \mathfrak{Y})=[D(\mathfrak{X}),D(\mathfrak{Y})]. \end{align}

$(2)$ Define a map $\textrm {ad}_{\mathcal{L}}\;:\; \mathcal{L}\rightarrow \mathfrak {gl}(\mathfrak{L})$ by

\begin{equation*}\textrm {ad}_{\mathcal{L}}(\mathfrak{X})(y)=[x_1,x_2,y],\, \forall \mathfrak{X}=(x_1,x_2)\in \mathcal{L}, y\in \mathfrak{L}.\end{equation*}

It directly follows from (2.4) that $\textrm {ad}_{\mathcal{L}}$ is a homomorphism of Leibniz superalgebra, i.e.,

(2.14) \begin{align} \textrm {ad}_{\mathcal{L}}(\mathfrak{X}\ast \mathfrak{Y})=\textrm {ad}_{\mathcal{L}}(\mathfrak{X})\textrm {ad}_{\mathcal{L}}(\mathfrak{Y})-(\!-\!1)^{|\mathfrak{X}||\mathfrak{Y}|}\textrm {ad}_{\mathcal{L}}(\mathfrak{Y})\textrm {ad}_{\mathcal{L}}(\mathfrak{X}). \end{align}

Given a $\textrm {LSTS}$ $(\mathfrak{L},[\cdot , \cdot , \cdot ])$ and its representation $(V;\;\theta )$ , notice that $\mathrm{Hom}(\mathfrak{L},V)=\mathrm{Hom}(\mathfrak{L},V)_{\bar {0}}\oplus \mathrm{Hom}(\mathfrak{L},V)_{\bar {1}},$ where $\mathrm{Hom}(\mathfrak{L},V)_{\alpha }=\{f\in \mathrm{Hom}(\mathfrak{L},V)\,|\, f(\mathfrak{L}_{\beta })\subseteq V_{\alpha +\beta }, \,\forall \alpha , \beta \in {\mathbb{Z}}_2\}$ , then we define two even bilinear maps

\begin{equation*}[\cdot ,\cdot ]_L\;:\; \mathcal{L} \times \mathrm{Hom}(\mathfrak{L},V)\rightarrow \mathrm{Hom}(\mathfrak{L},V) \,\textrm {and}\, [\cdot , \cdot ]_R\;:\; \mathrm{Hom}(\mathfrak{L},V)\times \mathcal{L}\rightarrow \mathrm{Hom}(\mathfrak{L},V)\end{equation*}

by

(2.15) \begin{equation} [(x_1,x_2), f]_L(x_3)=D(x_1,x_2)f(x_3)-(\!-\!1)^{|f|(|x_1|+|x_2|)}f([x_1,x_2,x_3]) \end{equation}

and

(2.16) \begin{align} [\,f,(x_1,x_2)]_R(x_3)&=f([x_1,x_2,x_3])-(\!-\!1)^{|f|(|x_1|+|x_2|)}D(x_1,x_2)f(x_3)+ (\!-\!1)^{|f|(|x_1|+|x_3|)+|x_2||x_3|}\theta (x_1,x_3)f(x_2)\notag \\ &\quad -(\!-\!1)^{(|f|+|x_1|)(|x_2|+|x_3|)}\theta (x_2,x_3)f(x_1), \end{align}

for any homogeneous map $f\in \mathrm{Hom}(\mathfrak{L},V)$ and homogeneous elements $x_i\in \mathfrak{L}$ $(i=1,2,3)$ .

With these notations, we have the following proposition, which gives the relationship between representations of $\textrm {LSTS}$ and representations of the associated Leibniz superalgebra.

Proof. It suffices to verify that Eqs. (2.8)–(2.10) hold for $(\mathrm{Hom}(\mathfrak{L},V);[\cdot ,\cdot ]_L,[\cdot ,\cdot ]_R)$ . Let $\mathfrak{X}=(x_1,x_2), \mathfrak{Y}=(y_1,y_2)$ and $y_3\in \mathfrak{L}$ . First, we verify the equality

(2.17) \begin{align} &[\mathfrak{X}\ast \mathfrak{Y}, f]_{L}(y_3)=[\mathfrak{X},[\mathfrak{Y},f]_L]_L(y_3)-(\!-\!1)^{|\mathfrak{X}||\mathfrak{Y}|}[\mathfrak{Y},[\mathfrak{X},f]_L]_L(y_3). \end{align}

By (2.15) and the definition of $\textrm {ad}_{\mathcal{L}}$ , the left-hand side of (2.17) is

\begin{align*} \rm {LHS\,of \,Eq. (2.17)}=D(\mathfrak{X}\ast \mathfrak{Y})f(y_3)-(\!-\!1)^{|f|(|\mathfrak{X}|+|\mathfrak{Y}|)}f(\rm {ad}_{\mathcal{L}}(\mathfrak{X}\ast \mathfrak{Y})(y_3)), \end{align*}

and the right-hand side of (2.17) is

\begin{align*} & \rm {RHS \,of \,Eq. (2.17)}\\&\quad =D(\mathfrak{X})D(\mathfrak{Y})f(y_3)\underline {-(\!-\!1)^{|f||\mathfrak{Y}|}D(\mathfrak{X})f(\textrm {ad}_{\mathcal{L}}(\mathfrak{Y})(y_3))}- \underbrace {(\!-\!1)^{|\mathfrak{X}|(|f|+|\mathfrak{Y}|)}D(\mathfrak{Y})f(\textrm {ad}_{\mathcal{L}}(\mathfrak{X})(y_3))} \\[3pt] \nonumber &\qquad +(\!-\!1)^{|\mathfrak{X}|(|f|+|\mathfrak{Y}|)+|\mathfrak{Y}||f|}f(\textrm {ad}_{\mathcal{L}}(\mathfrak{Y})\textrm {ad}_{\mathcal{L}}(\mathfrak{X})(y_3))-(\!-\!1)^{|\mathfrak{X}||\mathfrak{Y}|}D(\mathfrak{Y})D(\mathfrak{X}) \\[3pt] \nonumber &\qquad \underbrace {+(\!-\!1)^{|\mathfrak{X}|(|f|+|\mathfrak{Y}|)}D(\mathfrak{Y})f(\textrm {ad}_{\mathcal{L}}(\mathfrak{X})(y_3))}+\underline {(\!-\!1)^{|f||\mathfrak{Y}|}D(\mathfrak{X})f(\textrm {ad}_{\mathcal{L}}(\mathfrak{Y})(y_3))} \\[3pt] \nonumber &\qquad -(\!-\!1)^{|f|(|\mathfrak{X}|+|\mathfrak{Y}|)}f(\textrm {ad}_{\mathcal{L}}(\mathfrak{X})\textrm {ad}_{\mathcal{L}}(\mathfrak{Y})(y_3)) \\[3pt] \nonumber &\quad =D(\mathfrak{X})D(\mathfrak{Y})f(y_3)-(\!-\!1)^{|\mathfrak{X}||\mathfrak{Y}|}D(\mathfrak{Y})D(\mathfrak{X})-(\!-\!1)^{|f|(|\mathfrak{X}|+|\mathfrak{Y}|)}f(\textrm {ad}_{\mathcal{L}}(\mathfrak{X})\textrm {ad}_{\mathcal{L}}(\mathfrak{Y})(y_3)) \\[3pt] \nonumber &\qquad +(\!-\!1)^{|\mathfrak{X}|(|f|+|\mathfrak{Y}|)+|\mathfrak{Y}||f|}f(\textrm {ad}_{\mathcal{L}}(\mathfrak{Y})\textrm {ad}_{\mathcal{L}}(\mathfrak{X})(y_3))\\[3pt] \nonumber &\quad =[D(\mathfrak{X}),D(\mathfrak{Y})]f(y_3)-(\!-\!1)^{|f|(|\mathfrak{X}|+|\mathfrak{Y}|)}f\Big ((\textrm {ad}_{\mathcal{L}}(\mathfrak{X})\textrm {ad}_{\mathcal{L}}(\mathfrak{Y})-(\!-\!1)^{|\mathfrak{X}||\mathfrak{Y}|} \textrm {ad}_{\mathcal{L}}(\mathfrak{Y})\textrm {ad}_{\mathcal{L}}(\mathfrak{X}))(y_3)\Big ). \end{align*}

Thus, we conclude that (2.17) is valid due to (2.13) and (2.14).

Similarly, by (2.6) and (2.7), one can verify that the equality

\begin{align*} [[\mathfrak{X},f]_L, \mathfrak{Y}]_R=[\mathfrak{X},[f,\mathfrak{Y}]_R]_L-(\!-\!1)^{|f||\mathfrak{X}|}[f,\mathfrak{X}\ast \mathfrak{Y}]_R \end{align*}

holds by some straightforward tedious computations. We omit the details here.

At last, we verify that the equality

(2.18) \begin{align} [[f,\mathfrak{X}]_R,\mathfrak{Y}]_R(y_3)+(\!-\!1)^{|f||\mathfrak{X}|}[[\mathfrak{X},f]_L,\mathfrak{Y}]_R(y_3)=0 \end{align}

holds. By Remark 2.7, we deduce that (2.10) is valid for $[\cdot ,\cdot ]_L$ and $[\cdot ,\cdot ]_R$ , which can be derived from (2.17) and (2.18). From (2.15) and (2.16), we have

\begin{equation*}\big ([f,\mathfrak{X}]_R+(\!-\!1)^{|\mathfrak{X}||f|}[\mathfrak{X},f]_L\big )(y_3)=(\!-\!1)^{|f|(|x_1|+|y_3|)+|x_2||y_3|}\theta (x_1,y_3)f(x_2)-(\!-\!1)^{(|f|+|x_1|)(|x_2|+|y_3|)}\theta (x_2,y_3)f(x_1).\end{equation*}

Thus, by (2.5), we have

\begin{align*} &[[f,\mathfrak{X}]_R,\mathfrak{Y}]_R(y_3)+(\!-\!1)^{|f||\mathfrak{X}|}[[\mathfrak{X},f]_L,\mathfrak{Y}]_R(y_3)\\[3pt] \nonumber =&\;[[f,\mathfrak{X}]_R+(\!-\!1)^{|\mathfrak{X}||f|}[\mathfrak{X},f]_L, \mathfrak{Y}]_R(y_3)\\[3pt] \nonumber =&\;([f,\mathfrak{X}]_R+(\!-\!1)^{|\mathfrak{X}||f|}[\mathfrak{X},f]_L)([y_1,y_2,y_3])-(\!-\!1)^{(|f|+|\mathfrak{X}|)|\mathfrak{Y}|}D(\mathfrak{Y})([f,\mathfrak{X}]_R+(\!-\!1)^{|\mathfrak{X}||f|}[\mathfrak{X},f]_L)(y_3)\\[3pt] \nonumber &-(\!-\!1)^{|y_1|(|f|+|\mathfrak{X}|)+|y_3|(|f|+|\mathfrak{X}|+|y_2|)}\theta (y_1,y_3)([f,\mathfrak{X}]_R+(\!-\!1)^{|\mathfrak{X}||f|}[\mathfrak{X},f]_L)(y_2)\\[3pt] \nonumber &-(\!-\!1)^{(|y_2|+|y_3|)(|f|+|\mathfrak{X}|+|y_1|)}\theta (y_2,y_3)([f,\mathfrak{X}]_R+(\!-\!1)^{|\mathfrak{X}||f|}[\mathfrak{X},f]_L)(y_1)\\[3pt] \nonumber =&\;(\!-\!1)^{|f||x_1|+(|x_2|+|f|)(|y_1|+|y_2|+|y_3|)}\theta (x_1,[y_1,y_2,y_3])f(x_2)\\[3pt] \nonumber &-(\!-\!1)^{(|f|+|x_1|)|x_2|+(|f|+|x_1|)(|y_1|+|y_2|+|y_3|)}\theta (x_2,[y_1,y_2,y_3])f(x_1)\\[3pt] \nonumber &-(\!-\!1)^{|f||x_1|+(|x_2|+|f|)(|y_1|+|y_2|+|y_3|)+|x_1|(|y_1|+|y_2|)}D(y_1,y_2)\theta (x_1,y_3)f(x_2)\\[3pt] \nonumber &+(\!-\!1)^{(|f|+|x_1|)|x_2|+(|f|+|x_1|)(|y_1|+|y_2|+|y_3|)+|x_2|(|y_1|+|y_3|)}D(y_1,y_2)\theta (x_2,y_3)f(x_1)\\[3pt] \nonumber &+(\!-\!1)^{|f||x_1|+(|x_2|+|f|)(||y_1|+|y_2||+|y_3|)+|x_1|(|y_1|+|y_3|)+|y_2||y_3|}\theta (y_1,y_3)\theta (x_1,y_2)f(x_2)\\[3pt] \nonumber &-(\!-\!1)^{(|f|+|x_1|)|x_2|+(|f|+|x_1|)(||y_1|+|y_2|+|y_3|)+|x_2|(|y_1|+|y_3|)+|y_3||y_3|}\theta (y_1,y_3)\theta (x_2,y_2)f(x_1)\\[3pt] \nonumber &-(\!-\!1)^{|f||x_1|+(|x_2|+|f|)(|y_1|+|y_2|+|y_3|)+(|x_1|+|y_1|)(|y_2|+|y_3|)}\theta (y_2,y_3)\theta (x_1,y_1)f(x_2)\\[3pt] \nonumber &+(\!-\!1)^{(|f|+|x_1|)|x_2|+(|f|+|x_1|)(|y_1|+|y_2|+|y_3|)+(|y_2|+|y_3|)(|x_2|+|y_1|)}\theta (y_2,y_3)\theta (x_2,y_1)f(x_1)\\[3pt] \nonumber =&\;(\!-\!1)^{|f||x_1|+(|x_2|+|f|)(|y_1|+|y_2|+|y_3|)}\Big (\theta (x_1,[y_1,y_2,y_3])-(\!-\!1)^{|x_1|(|y_1|+|y_2|)}D(y_1,y_2)\theta (x_1,y_3)\\[3pt] \nonumber &+(\!-\!1)^{|x_1|(|y_1|+|y_3|)+|y_2||y_3|}\theta (y_1,y_3)\theta (x_1,y_2)-(\!-\!1)^{(|x_1|+|y_1|)(|y_2|+|y_3|)}\theta (y_2,y_3)\theta (x_1,y_1)\Big )f(x_2)\\[3pt] \nonumber &-(\!-\!1)^{(|f|+|x_1|)|x_2|+(|f|+|x_1|)(|y_1|+|y_2|+|y_3|)}\Big (\theta (x_2,[y_1,y_2,y_3])-(\!-\!1)^{|x_2|(|y_1|+|y_3|)}D(y_1,y_2)\theta (x_2,y_3)\\[3pt] \nonumber &+(\!-\!1)^{|x_2|(|y_1|+|y_3|)+|y_3||y_3|}\theta (y_1,y_3)\theta (x_2,y_2)-(\!-\!1)^{(|y_2|+|y_3|)(|x_2|+|y_1|)}\theta (y_2,y_3)\theta (x_2,y_1)\Big )f(x_1)\\ \overset {(2.5)}{=}&0. \end{align*}

Therefore, we complete the proof.

We complete this section with a description of the cohomology of the $\textrm {LSTS}$ by the cohomology of the associated Leibniz superalgebra.

Definition 2.10. Let $(V;\;\theta )$ be a representation of $(\mathfrak{L},[\cdot ,\cdot ,\cdot ])$ . A homogeneous $(2n+1)$ -linear map $f\;:\; \underbrace {\mathfrak{L}\times \cdots \times \mathfrak{L}}_{(2n+1)\,times}\rightarrow V$ of ${\mathbb{Z}}_2$ -degree $|f|$ is called a $(2n+1)$ -cochain of $\mathfrak{L}$ on $V$ for $n\geq 0$ , if it satisfies

\begin{equation*}|f(x_1,x_2,\cdots ,x_{2n+1})|=|f|+|x_1|+|x_2|+\cdots +|x_{2n+1}|,\end{equation*}

\begin{equation*}f(x_1,x_2,\cdots ,x_{2n-2},x, y, x_{2n+1})=-(\!-\!1)^{|x||y|}f(x_1,x_2,\cdots ,x_{2n-2},y,x, x_{2n+1}),\end{equation*}

and

\begin{equation*} (\!-\!1)^{|x||z|}f(x_1,\cdots , x_{2n-2},x,y,z)+(\!-\!1)^{|y||x|}f(x_1,\cdots , x_{2n-2},y,z,x)+(\!-\!1)^{|y||z|}f(x_1,\cdots , x_{2n-2},z,x,y)=0.\end{equation*}

Denote by $C^{2n+1}(\mathfrak{L},V)_{\alpha }$ the set of all homogeneous $(2n+1)$ -cochains of degree $\alpha \in {\mathbb{Z}}_2$ . Then $C^{2n+1}(\mathfrak{L},V)=C^{2n+1}(\mathfrak{L},V)_{\bar {0}}\oplus C^{2n+1}(\mathfrak{L},V)_{\bar {1}}$ .

The cohomology theory of Leibniz superalgebras was studied in [Reference Dzhumadil’daev5, Reference Liu and Hu14, Reference Yadav27]. We present the cohomology for the associated Leibniz superalgebra $\mathcal{L}$ of $\textrm {LSTS}$ $(\mathfrak{L}, [\cdot ,\cdot ,\cdot ])$ with coefficients in the representation $\mathrm{Hom}(\mathfrak{L},V)$ as follows. Denote the space of $n$ -cochains by

\begin{equation*}C^{n}(\mathcal{L},\mathrm{Hom}(\mathfrak{L},V))\;:\!=\;\mathrm{Hom}(\otimes ^{n}\mathcal{L},\mathrm{Hom}(\mathfrak{L},V)).\end{equation*}

The corresponding coboundary operator $\delta _{\mathcal{L}}\;:\; C^{n-1}(\mathcal{L},\mathrm{Hom}(\mathfrak{L},V)) \to C^{n}(\mathcal{L},\mathrm{Hom}(\mathfrak{L},V))$ is defined by

(2.19) \begin{eqnarray} \,\nonumber &&(\delta _{\mathcal{L}}f)(\mathfrak X_1,\cdots ,\mathfrak X_n)(z)\\ \,\nonumber &=&\sum \limits _{1 \le i \lt j\le n}(- 1)^{i+|\mathfrak X_i|(|\mathfrak X_{i+1}|+\cdots +|\mathfrak X_{j-1}|)}f({\mathfrak X_1},\cdots ,\mathop {\hat {{\mathfrak X_i}} } ,\cdots ,{\mathfrak X_{j - 1}},\mathfrak X_i\ast \mathfrak X_j,\cdots ,\mathfrak X_n)(z) \\ \, \nonumber && + \sum \limits _{i=1}^n(\!-\!1)^{i+1+|\mathfrak X_i|(|f|+|\mathfrak X_{1}|+\cdots +|\mathfrak X_{i-1}|)}[\mathfrak X_i, f(\mathfrak X_1,\cdots , {\hat {\mathfrak X_i}},\cdots ,\mathfrak X_n)]_L(z)\\ \,&&-(\!-\!1)^{n-1}[f(\mathfrak X_1,\cdots ,\mathfrak X_{n-1}),\mathfrak X_n]_R(z) \end{eqnarray}

for all homogeneous cochain $f \in C^{n-1}(\mathcal{L},\mathrm{Hom}(\mathfrak{L},V))$ and homogeneous elements $\mathfrak X_i\in \mathcal{L},\,z \in \mathfrak{L},\,i=1,2,\cdots ,n$ , where the operation $\ast$ is defined by (2.12) and $[\cdot ,\cdot ]_L$ , $[\cdot ,\cdot ]_R$ are given by (2.15) and (2.16), respectively.

Note that

\begin{equation*}C^{n}(\mathcal{L},\mathrm{Hom}(\mathfrak{L},V))=\mathrm{Hom}(\otimes ^{n}(\otimes ^2\mathfrak{L}), \mathrm{Hom}(\mathfrak{L},V)) \cong \mathrm{Hom}(\otimes ^{2n+1}\mathfrak{L}, V)=C^{2n+1}(\mathfrak{L},V),\end{equation*}

then let $\mathfrak X_i=(x_{2i-1}, x_{2i+1})$ , $z=x_{2n+1}$ , we get a coboundary operator

\begin{equation*}\delta =(\!-\!1)^{n+1}\delta _{\mathcal{L}}\;:\;C^{2n-1}(\mathfrak{L},V)\rightarrow C^{2n+1}(\mathfrak{L},V).\end{equation*}

By a direct calculation, we see that

\begin{align*} &\delta f(x_1,\cdots , x_{2n+1})\notag \\ =&\;(\!-\!1)^{n+1}\delta _{\mathcal{L}}f(\mathfrak X_1,\cdots ,\mathfrak X_n,z)\notag \\ =&\;\sum \limits _{1 \le i \lt j\le n}(\!-\!1)^{n+1 +i+(|x_{2i-1}|+|x_{2i}|)(|x_{2i+1}|+|x_{2i+2}|+\cdots +|x_{2j-2}|)}\Big (f(x_1,\cdots ,\widehat {x_{2i-1}},\widehat {x_{2i}},\cdots ,[x_{2i-1},x_{2i}, x_{2j-1}],x_{2j},\cdots , x_{2n+1})\notag \\ &+(\!-\!1)^{|x_{2j-1}|(|x_{2i-1}|+|x_{2i}|)}f(x_1,\cdots ,\widehat {x_{2i-1}},\widehat {x_{2i}},\cdots ,x_{2j-1},[x_{2i-1},x_{2i}, x_{2j}],\cdots , x_{2n+1})\Big )\notag \\ &+\sum \limits _{i=1}^{n-1}(\!-\!1)^{n+i+(|f|+|x_1|+\cdots +|x_{2i-2}|)(|x_{2i-1}|+|x_{2i}|)}[(x_{2i-1},x_{2i}),f(x_1,\cdots ,\widehat {x_{2i-1}},\widehat {x_{2i}},\cdots ,x_{2n})]_L(x_{2n+1})\notag \\ &+(\!-\!1)^{2n+1}[f(x_1,\cdots ,x_{2n-2}),(x_{2n-1},x_{2n})]_R(x_{2n+1})\notag \\ =&\;\sum \limits _{1 \le i \lt j\le n}(\!-\!1)^{n+1 +i+(|x_{2i-1}|+|x_{2i}|)(|x_{2i+1}|+|x_{2i+2}|+\cdots +|x_{2j-2}|)}\Big (f(x_1,\cdots ,\widehat {x_{2i-1}},\widehat {x_{2i}},\cdots ,[x_{2i-1},x_{2i}, x_{2j-1}],x_{2j},\cdots , x_{2n+1})\notag \\ &+(\!-\!1)^{|x_{2j-1}|(|x_{2i-1}|+|x_{2i}|)}f(x_1,\cdots ,\widehat {x_{2i-1}},\widehat {x_{2i}},\cdots ,x_{2j-1},[x_{2i-1},x_{2i}, x_{2j}],\cdots , x_{2n+1})\Big )\notag \\ &+\sum \limits _{i=1}^{n-1}(\!-\!1)^{n+i+(|f|+|x_1|+\cdots +|x_{2i-2}|)(|x_{2i-1}|+|x_{2i}|)}D(x_{2i-1},x_{2i})f(x_1,\cdots ,\widehat {x_{2i-1}},\widehat {x_{2i}},\cdots ,x_{2n},x_{2n+1})\notag \\ \end{align*}

(2.20) \begin{align} &-\sum \limits _{i=1}^{n-1}(\!-\!1)^{n+i+(|x_{2i+1}|+\cdots +x_{2n})(|x_{2i-1}|+|x_{2i}|)}f(x_1,\cdots ,\widehat {x_{2i-1}},\widehat {x_{2i}},\cdots ,[x_{2i-1},x_{2i},x_{2n+1}])\notag \\ &-f(x_1,\cdots ,x_{2n-2},[x_{2n-1},x_{2n},x_{2n+1}])+(\!-\!1)^{(|f|+|x_1|+\cdots +|x_{2n-2}|)(|x_{2n-1}|+|x_{2n}|)}D(x_{2n-1},x_{2n})f(x_1,\cdots ,x_{2n-2},x_{2n+1})\notag \\ &-(\!-\!1)^{|x_{2n-1}|(|f|+|x_1|+\cdots +|x_{2n-2}|)+|x_{2n+1}|(|f|+|x_1|+\cdots +|x_{2n-2}|+|x_{2n}|)}\theta (x_{2n-1},x_{2n+1})f(x_1,\cdots ,x_{2n-2},x_{2n})\notag \\ &+(\!-\!1)^{(|x_{2n}|+|x_{2n+1}|)(|f|+|x_1|+\cdots +|x_{2n-1}|)}\theta (x_{2n},x_{2n+1})f(x_1,\cdots ,x_{2n-1})\notag \\ =&\;(\!-\!1)^{(|x_{2n}|+|x_{2n+1}|)(|f|+|x_1|+\cdots +|x_{2n-1}|)}\theta (x_{2n},x_{2n+1})f(x_1,\cdots ,x_{2n-1})\notag \\ &-(\!-\!1)^{(|x_{2n-1}|+|x_{2n+1}|)(|f|+|x_1|+\cdots +|x_{2n-2}|)+|x_{2n}||x_{2n+1}|}\theta (x_{2n-1},x_{2n+1})f(x_1,\cdots ,x_{2n-2},x_{2n})\notag \\ &+\sum \limits _{i=1}^{n}(\!-\!1)^{n+i+(|f|+|x_1|+\cdots +|x_{2i-2}|)(|x_{2i-1}|+|x_{2i}|)}D(x_{2i-1},x_{2i})f(x_1,\cdots ,\widehat {x_{2i-1}},\widehat {x_{2i}},\cdots ,x_{2n},x_{2n+1})\notag \\ &+\sum \limits _{i=1}^{n}\sum \limits _{j=2i+1}^{2n+1}(\!-\!1)^{n+1 +i+(|x_{2i-1}|+|x_{2i}|)(|x_{2i+1}|+|x_{2i+2}|+\cdots +|x_{j-1}|)}f(x_1,\cdots ,\widehat {x_{2i-1}},\widehat {x_{2i}},\cdots ,[x_{2i-1},x_{2i}, x_{j}],\cdots , x_{2n+1}). \end{align}

It is easy to see that $\delta \circ \delta =0$ for $n\geq 1$ . Then, the corresponding cohomology group is defined by

\begin{equation*}H^{2n+1}(\mathfrak{L},V)=Z^{2n+1}(\mathfrak{L},V)/B^{2n+1}(\mathfrak{L},V),\,\forall \, n\geq 0,\end{equation*}

where $Z^{2n+1}(\mathfrak{L},V)=\{f\in C^{2n+1}(\mathfrak{L},V)\,|\,\delta f=0\}$ is the space spanned by $(2n+1)$ -cocycles of $\mathfrak{L}$ on $V$ and $B^{2n+1}(\mathfrak{L},V)=\{\delta f\,|\,f\in C^{2n-1}(\mathfrak{L},V)\}$ is the space spanned by $(2n+1)$ -coboundaries of $\mathfrak{L}$ on $V$ .

Note that the coboundary operator defined above coincides with the coboundary operator in [Reference Guo9] when a Hom-Lie supertriple system reduces to a Lie supertriple system. Hence, we actually show that the coholomoy theory of Lie supertriple systems can be obtained by the coholomoy theory of Leibniz superalgebras.

In the end, by (2.20), we list two low-dimensional coboundary operators for later use.

For homogeneous cochain $f\in C^1(\mathfrak{L},V)$ , we have

\begin{align*} \begin{split} \delta f(x_1, x_2, x_3)= & (\!-\!1)^{(|f|+|x_1|)(|x_2|+|x_3|)} \theta (x_2, x_3) f(x_1)-(\!-\!1)^{|x_2||x_3|+|f|(|x_1|+|x_3|)} \theta (x_1, x_3) f(x_2)\\ & +(\!-\!1)^{|f|(|x_1|+|x_2|)} D(x_1, x_2) f(x_3)-f([x_1, x_2, x_3]), \end{split} \end{align*}

and for homogeneous cochain $f\in C^3(\mathfrak{L},V)$ , we have

\begin{align*} &\delta f(x_1, x_2, x_3, x_4, x_5)\\ & = (\!-\!1)^{(|f|+|x_1|+|x_2|+|x_3|)(|x_4|+|x_5|)} \theta (x_4, x_5) f(x_1, x_2, x_3) -{(\!-\!1)}^{(|f|+|x_1|+|x_2|)(|x_3|+|x_5|)+|x_4||x_5|} \theta (x_3, x_5) f(x_1, x_2, x_4) \\ &\quad -(\!-\!1)^{|f|(|x_1|+|x_2|)} D(x_1, x_2) f(x_3, x_4, x_5)+{(\!-\!1)}^{(|f|+|x_1|+|x_2|)(|x_3|+|x_4|)} D(x_3, x_4) f(x_1, x_2, x_5)\\ &\quad +f([x_1, x_2, x_3], x_4, x_5)-f(x_1, x_2,[x_3, x_4, x_5])+{(\!-\!1)}^{|x_3|(|x_1|+|x_2|)} f(x_3,[x_1, x_2, x_4], x_5) \\ &\quad +(\!-\!1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)} f(x_3, x_4,[x_1, x_2, x_5]). \end{align*}

3. Representations and cohomologies of LSTSDer pairs

In this section, we give definitions of LSTSDer pairs and their representations. The cohomology of a LSTSDer pair with coefficients in a representation will be defined. We also describe the relationship between the cohomology theory of a LSTSDer pair and the cohomology of the associated LeibSDer pair.

3.1. LSTSDer pairs and their representations

Definition 3.1. [Reference Ma, Chen and Liu17] A homogeneous linear map $\phi _{\mathfrak{L}}\;:\;\mathfrak{L}\to \mathfrak{L}$ is called a superderivation on a $\textrm {LSTS}$ $(\mathfrak{L},[\cdot ,\cdot ,\cdot ])$ of ${\mathbb{Z}}_2$ -degree $|\phi _{\mathfrak{L}}|$ provided that

(3.1) \begin{equation} \phi _{\mathfrak{L}}([x,y,z])=[\phi _{\mathfrak{L}}(x),y,z]+(\!-\!1)^{|\phi _{\mathfrak{L}}||x|}[x,\phi _{\mathfrak{L}}(y),z]+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x|+|y|)}[x,y,\phi _{\mathfrak{L}}(z)], \end{equation}

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

Denote by $\textrm {Der}(\mathfrak{L})_{\bar {0}}$ (resp. $\textrm {Der}(\mathfrak{L})_{\bar {1}}$ ) the set of all superderivations on the $\mathfrak{L}$ of ${\mathbb{Z}}_2$ -degree $\bar {0}$ (resp. $\bar {1}$ ). Clearly, $\textrm {Der}(\mathfrak{L})=\textrm {Der}(\mathfrak{L})_{\bar {0}}\oplus \textrm {Der}(\mathfrak{L})_{\bar {1}}$ .

Definition 3.2. A LSTSDer pair is a $\textrm {LSTS}$ $(\mathfrak{L},[\cdot ,\cdot ,\cdot ])$ with a superderivation $\phi _{\mathfrak{L}}\in \textrm {Der}(\mathfrak{L})$ . We denote it by $(\mathfrak{L},[\cdot ,\cdot ,\cdot ],\phi _{\mathfrak{L}})$ or simply by $(\mathfrak{L},\phi _{\mathfrak{L}})$ .

Definition 3.3. Let $(\mathfrak{L},\phi _{\mathfrak{L}})$ and $(\mathfrak{L}',\phi _{\mathfrak{L}'})$ be two LSTSDer pairs. A LSTSDer pair morphism from $(\mathfrak{L},\phi _{\mathfrak{L}})$ to $(\mathfrak{L}',\phi _{\mathfrak{L}'})$ is a $\textrm {LSTS}$ homomorphism $\eta \;:\;\mathfrak{L}\rightarrow \mathfrak{L}'$ such that $\eta \circ \phi _{\mathfrak{L}}=\phi _{\mathfrak{L}'}\circ \eta$ .

Next we give the definition of representations of LSTSDer pairs.

Definition 3.4. A representation of a LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ on a ${\mathbb{Z}}_2$ -graded vector space $V$ is a pair $(\theta ,\phi _V)$ in which $\theta\; :\;\mathfrak{L}\times \mathfrak{L}\rightarrow \textrm {End}(V)$ is a representation of $\mathfrak{L}$ and $\phi _V\in \textrm {End}(V)$ is a homogeneous linear map with $|\phi _V|=|\phi _{\mathfrak{L}}|$ satisfying

(3.2) \begin{aligned}\phi _V\theta (x,y)-(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x|+|y|)}\theta (x,y)\phi _V=\theta (\phi _{\mathfrak{L}}(x),y)+(\!-\!1)^{|\phi _{\mathfrak{L}}||x|}\theta (x,\phi _{\mathfrak{L}}(y)),\end{aligned}

(3.3) \begin{aligned}\phi _VD(x,y)-(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x|+|y|)}D(x,y)\phi _V=D(\phi _{\mathfrak{L}}(x),y)+(\!-\!1)^{|\phi _{\mathfrak{L}}||x|}D(x,\phi _{\mathfrak{L}}(y))\end{aligned}

for all homogeneous elements $x,y\in \mathfrak{L}$ , where $D(x,y)=(\!-\!1)^{|x||y|}\theta (y,x)-\theta (x,y)$ . We denote a representation of a LSTSDer pair by $(V;\;\theta ,\phi _V)$ .

Example 3.5. Let $(\mathfrak{L},\phi _{\mathfrak{L}})$ be a LSTSDer pair and $(\mathfrak{L};\;\theta )$ be the adjoint representation of $\mathfrak{L}$ . Then $(\mathfrak{L};\;\theta ,\phi _{\mathfrak{L}})$ is a representation of the LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ , which is called the adjoint representation of $(\mathfrak{L},\phi _{\mathfrak{L}})$ . In particular, a representation $(V;\;\theta ,\phi _V)$ of a LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ is said to be trivial if $\theta =0$ .

Proposition 3.6. Let $(V;\;\theta , \phi _V)$ be a representation of a LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ . Define a homogeneous linear map $\phi _{\mathfrak{L}}+\phi _V\;:\; \mathfrak{L}\oplus V\rightarrow \mathfrak{L}\oplus V$ by

(3.4) \begin{align} (\phi _{\mathfrak{L}}+\phi _V)(x+u)=\phi _{\mathfrak{L}}(x)+\phi _V(u), \end{align}

for all homogeneous elements $ x\in \mathfrak{L}, u\in V$ , where $|\phi _{\mathfrak{L}}+\phi _V|=|\phi _{\mathfrak{L}}|=|\phi _V|$ and $|x+u|=|x|=|u|$ . Then $(\mathfrak{L}\oplus V,\phi _{\mathfrak{L}}+\phi _V)$ , with the $\textrm {LSTS}$ structure given in Proposition 2.4 , is a LSTSDer pair, which we call the semi-direct product of the LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ by the representation $(V;\;\theta ,\phi _V)$ and denote it by $ \mathfrak{L}\ltimes _{\phi }V$ .

Proof. It suffices to show that $\phi _{\mathfrak{L}}+\phi _V$ is a superderivation on $\mathfrak{L}\oplus V$ . For any homogeneous elements $x,y,z\in \mathfrak{L}$ and $u,v,w\in V$ , on one hand, we have

\begin{align*} &(\phi _{\mathfrak{L}}+\phi _V)[x+u,y+v,z+w]_V\\[3pt] \nonumber &=(\phi _{\mathfrak{L}}+\phi _V)\big ([x,y,z]+(\!-\!1)^{|x|(|y|+|z|)}\theta (y,z)(u)-(\!-\!1)^{|y||z|}\theta (x,z)(v)+D(x,y)(w)\big )\\[3pt] \nonumber &=\phi _{\mathfrak{L}}([x,y,z])+(\!-\!1)^{|x|(|y|+|z|)}\phi _V\theta (y,z)(u)-(\!-\!1)^{|y||z|}\phi _V\theta (x,z)(v)+\phi _{V}D(x,y)(w). \end{align*}

On the other hand, we obtain

\begin{align*} &[(\phi _{\mathfrak{L}}+\phi _V)(x+u),y+v,z+w]_V+(\!-\!1)^{|\phi _{\mathfrak{L}}||x|}[x+u,(\phi _{\mathfrak{L}}+\phi _V)(y+v),z+w]_V\\[3pt] \nonumber &+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x|+|y|)}[x+u,y+v,(\phi _{\mathfrak{L}}+\phi _V)(z+w)]_V\\[3pt] \nonumber =&\;[\phi _{\mathfrak{L}}(x),y,z]+(\!-\!1)^{(|x|+|\phi _{\mathfrak{L}}|)(|y|+|z|)}\theta (y,z)\phi _V(u)-(\!-\!1)^{|y||z|}\theta (\phi _{\mathfrak{L}}(x),z)(v)+D(\phi _{\mathfrak{L}}(x),y)(w)\\[3pt] \nonumber &+(\!-\!1)^{|\phi _{\mathfrak{L}}||x|}[x,\phi _{\mathfrak{L}}(y),z]+(\!-\!1)^{|x|(|y|+|z|)}\theta (\phi _{\mathfrak{L}}(y),z)(u)-(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x|+|z|)+|y||z|}\theta (x,z)\phi _V(v) \end{align*}

\begin{align*} &+(\!-\!1)^{|\phi _{\mathfrak{L}}||x|}D(x,\phi _{\mathfrak{L}}(y))(w)+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x|+|y|)}[x,y,\phi _{\mathfrak{L}}(z)]+(\!-\!1)^{|\phi _{\mathfrak{L}}||y|+|x|(|y|+|z|)}\theta (y,\phi _{\mathfrak{L}}(z))(u)\\[3pt] \nonumber &-(\!-\!1)^{|\phi _{\mathfrak{L}}||x|+|y||z|}\theta (x,\phi _{\mathfrak{L}}(z))(v)+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x|+|y|)}D(x,y)\phi _V(w). \end{align*}

One concludes by (3.2)–(3.3) and the fact that $\phi _{\mathfrak{L}}$ is a superderivation.

3.2. Cohomologies of LSTSDer pairs

In this subsection, we aim to define the cohomology of a LSTSDer pair.

Let $(V;\;\theta ,\phi _V)$ be a representation of a LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ . Define the set of LSTSDer pair $0$ -cochains to be $0$ . For $n\geq 1$ , we define the set of LSTSDer pair $(2n+1)$ -cochains by

\begin{eqnarray*} C^{2n+1}_{\mathrm{LSTSDer}}(\mathfrak{L},V)\;:\!=\;C^{2n+1}(\mathfrak{L},V)\times C^{2n-1}(\mathfrak{L},V) \end{eqnarray*}

and $C^1_{\mathrm{LSTSDer}}(\mathfrak{L},V)=\textrm {Hom}(\mathfrak{L},V)$ . Clearly, $C^{2n+1}_{\mathrm{LSTSDer}}(\mathfrak{L},V)$ ( $n\geq 0$ ) is ${\mathbb{Z}}_2$ -graded.

For $n\geq 0$ , we define a new linear map $\mathrm{d}\;:\; C^{2n+1}(\mathfrak{L},V)\rightarrow C^{2n+1}(\mathfrak{L},V)$ by

(3.5) \begin{align} &&&(\mathrm{d} f)(x_1,\cdots , x_{2n+1})\notag \\ &&=&\sum \limits _{i=1}^{2n+1}(\!-\!1)^{|\phi _{\mathfrak{L}}|\sum \limits _{j=1}^{i-1}|x_{j}|}f(x_1,\cdots ,x_{i-1},\phi _{\mathfrak{L}}(x_i),x_{i+1},\cdots ,x_{2n+1})-(\!-\!1)^{|\phi _{\mathfrak{L}}||f|}\phi _V\circ f(x_1, \cdots , x_{2n+1}). \end{align}

Obviously, $\mathrm{d}$ is homogeneous with $|\mathrm{d}|=|\phi _{\mathfrak{L}}|=|\phi _V|$ .

Let $\delta$ be the coboundary operator given by (2.20). Define $\partial \;:\; C^{1}_{\mathrm{LSTSDer}}(\mathfrak{L},V)\rightarrow C^{3}_{\mathrm{LSTSDer}}(\mathfrak{L},V)$ by

(3.6) \begin{equation} \partial f_1=(\delta f_1,-\mathrm{d} f_1),\,\forall f_1\in \textrm {Hom}(\mathfrak{L},V). \end{equation}

For $n\geq 1$ , define $\partial \;:\; C^{2n+1}_{\mathrm{LSTSDer}}(\mathfrak{L},V)\rightarrow C^{2n+3}_{\mathrm{LSTSDer}}(\mathfrak{L},V)$ by

(3.7) \begin{eqnarray} \partial (f_{2n+1},g_{2n-1})=(\delta f_{2n+1},\delta g_{2n-1}+(\!-\!1)^{n-1}\mathrm{d} f_{2n+1}),\,\forall (f_{2n+1},g_{2n-1})\in C^{2n+1}_{\mathrm{LSTSDer}}(\mathfrak{L},V). \end{eqnarray}

Then we get the following lemma, which will be used to prove that $\partial$ is a coboundary operator.

Lemma 3.7. The operators $\delta$ and $\mathrm{d}$ are commutative, that is, $\textrm {d}\circ \delta =\delta \circ \textrm {d}$ .

Proof. It follows by straightforward tedious calculations. We omit the details here.

Proposition 3.8. The map $\partial$ is a coboundary operator, that is, $\partial \circ \partial =0$ .

Proof. For any $f_1\in C^1_{\mathrm{LSTSDer}}(\mathfrak{L},V)$ , by Lemma 3.7, we have

\begin{equation*}(\partial \circ \partial )f_1=\partial (\delta f_1,-\delta f_1)=\big ((\delta \circ \delta )f_1,-(\delta \circ \mathrm{d}) f_1+(\mathrm{d} \circ \delta )f_1\big )=(0,0).\end{equation*}

Similarly, for $n\geq 1$ , let $(f_{2n+1},g_{2n-1})\in C^{2n+1}_{\mathrm{LSTSDer}}(\mathfrak{L},V)$ , by Lemma 3.7, we get

\begin{align*} (\partial \circ \partial )(f_{2n+1},g_{2n-1})&=\partial (\delta f_{2n+1}, \delta g_{2n-1}+(\!-\!1)^{n-1}\mathrm{d} f_{2n+1})\\ &=(\delta (\delta f_{2n+1}),\delta (\delta g_{2n-1}+(\!-\!1)^{n-1}\mathrm{d} f_{2n+1})+(\!-\!1)^{n}\mathrm{d}(\delta f_{2n+1}))\\ &=(0,(\!-\!1)^{n-1}(\delta \circ \mathrm{d}-\mathrm{d} \circ \delta )f_{2n+1})\\ &=(0,0). \end{align*}

Hence, the proof is completed.

Definition 3.9. Let $(V;\;\theta ,\phi _V)$ be a representation of a LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ . The cohomology of the cochain complex $(C^{\bullet }_{\mathrm{LSTSDer}}(\mathfrak{L},V),\partial )$ is taken to be the cohomology of the LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ . The corresponding $(2n+1)$ -th cohomology group is defined by

(3.8) \begin{align} H_{\mathrm{LSTSDer}}^{2n+1}(\mathfrak{L},V)=Z_{\mathrm{LSTSDer}}^{2n+1}(\mathfrak{L},V)/B_{\mathrm{LSTSDer}}^{2n+1}(\mathfrak{L},V), \end{align}

where $Z_{\mathrm{LSTSDer}}^{2n+1}(\mathfrak{L},V)=\{f\in C^{2n+1}_{\mathrm{LSTSDer}}(\mathfrak{L},V)\,|\,\partial f=0\}$ is called the space of $(2n+1)$ -cocycles, and $B_{\mathrm{LSTSDer}}^{2n+1}(\mathfrak{L},V)=\{\partial f\,|\, f\in C^{2n-1}_{\mathrm{LSTSDer}}(\mathfrak{L},V)\}$ is called the space of $(2n+1)$ -coboundaries.

3.3. Maurer–Cartan characterizations of LSTSDer pairs

In this subsection, we aim to construct a differential graded Lie algebra whose Maurer–Cartan elements are LSTDer pairs.

First of all, we recall the notions of a differential graded Lie algebra and its Maurer–Cartan elements.

Definition 3.10. [Reference Loday and Vallette16] A differential graded Lie algebra is a triple $(\mathfrak{g}= \oplus _{k\in {\mathbb{Z}}_{\geq 0}} \mathfrak{g}_k, [\cdot ,\cdot ], d)$ consisting of a graded Lie algebra $(\mathfrak{g}= \oplus _{k\in {\mathbb{Z}}_{\geq 0}} \mathfrak{g}_k, [\cdot ,\cdot ])$ and a differential $d \;:\; \mathfrak{g} \rightarrow \mathfrak{g}$ satisfying $d(\mathfrak{g}_k)\subseteq \mathfrak{g}_{k+1}$ and

\begin{equation*}d([x,y])=[dx,y]+(\!-\!1)^k[x,dy],\,\forall x\in \mathfrak{g}_k,y\in \mathfrak{g}_{k'}.\end{equation*}

Moreover, an element $P\in \mathfrak{g}_1$ is called a Maurer–Cartan element of the differential graded Lie algebra $(\mathfrak{g}, [\cdot ,\cdot ], d)$ if it satisfies $dP+\frac {1}{2}[P,P]=0$ .

Remark 3.11. Any graded Lie algebra is a differential graded Lie algebra with the differential $d=0$ .

Let $\mathfrak{L}$ be a ${\mathbb{Z}}_2$ -graded vector space. Consider the ${\mathbb{Z}}\times {\mathbb{Z}}_2$ -graded vector space

\begin{equation*}\mathfrak{C}^*(\mathfrak{L},\mathfrak{L})=\oplus _{(n,\alpha )\in {\mathbb{Z}}_{\geq 0}\times {\mathbb{Z}}_2}\mathfrak{C}^n(\mathfrak{L},\mathfrak{L})_{\alpha }=\oplus _{(n,\alpha )\in {\mathbb{Z}}_{\geq 0}\times {\mathbb{Z}}_2}C^{2n+1}(\mathfrak{L},\mathfrak{L})_{\alpha },\end{equation*}

where $\mathfrak{C}^0(\mathfrak{L},\mathfrak{L})=\mathrm{Hom}(\mathfrak{L},\mathfrak{L})$ and for $n\geq 1$ , $\mathfrak{C}^n(\mathfrak{L},\mathfrak{L})_\alpha =\mathrm{Hom}(\underbrace {(\otimes ^2\mathfrak{L})\otimes \cdots \otimes (\otimes ^2\mathfrak{L})}_{n}\otimes \mathfrak{L}, \mathfrak{L})_{\alpha }$ is the set of homogeneous linear maps satisfying

\begin{equation*}f(\mathfrak X_1,\cdots ,\mathfrak X_{n-1},x,x,y)=0, \end{equation*}

and

\begin{equation*}(\!-\!1)^{|x||z|}f(\mathfrak X_1,\cdots ,\mathfrak X_{n-1},x,y,z)+(\!-\!1)^{|x||y|}f(\mathfrak X_1,\cdots ,\mathfrak X_{n-1},y,z,x)+(\!-\!1)^{|y||z|)}f(\mathfrak X_1,\cdots ,\mathfrak X_{n-1},z,x,y)=0\end{equation*}

for any $f\in \mathfrak{C}^n(\mathfrak{L},\mathfrak{L})_{\alpha }$ , $\mathfrak X_i=x_i\otimes y_i\in \otimes ^2\mathfrak{L}$ with $|\mathfrak X_i|=|x_i|+|y_i|$ , $1\leq i\leq n-1$ , and homogeneous elements $x,y, z\in \mathfrak{L}$ . The $\mathbb{Z}$ -degree and ${\mathbb{Z}}_2$ -degree of elements in $\mathfrak{C}^n(\mathfrak{L},\mathfrak{L})_{\alpha }$ are assumed to be $n$ and $\alpha$ , respectively. For $P\in \mathfrak{C}^p(\mathfrak{L},\mathfrak{L})_\alpha , Q\in \mathfrak{C}^q(\mathfrak{L},\mathfrak{L})_\beta$ $(p,q\geq 1, \alpha ,\beta \in {\mathbb{Z}}_2)$ , we define the operation $P\circ Q\in \mathfrak{C}^{p+q}(\mathfrak{L},\mathfrak{L})_{\alpha +\beta }$ by

\begin{equation*} \begin{aligned} &(P{\circ }Q)(\mathfrak X_1,\cdots ,\mathfrak X_{p+q},x)\\ =&\;(\!-\!1)^{pq}\sum _{\sigma \in \mathbb{S}(p,q)}(\!-\!1)^{\sigma +\mathbb{K}(\sigma ;\;\mathfrak X,p)+\beta (|\mathfrak X_{\sigma (1)}|+\cdots +|\mathfrak X_{\sigma (p)}|)} P\big (\mathfrak X_{\sigma (1)},\cdots ,\mathfrak X_{\sigma (p)}, Q(\mathfrak X_{\sigma (p+1)},\cdots ,\mathfrak X_{\sigma (p+q-1)},\mathfrak X_{\sigma (p+q)},x)\big )\\ &+\sum _{k=1}^{p}(\!-\!1)^{(k-1)q}\sum _{\sigma \in \mathbb{S}(k-1,q)}(\!-\!1)^{\sigma +\mathbb{K}(\sigma ;\;\mathfrak X,k-1)+|x_{k+q}|(|\mathfrak X_{\sigma (k)}|+\cdots +|\mathfrak X_{\sigma (k+q-1)}|+\beta )+\beta (|\mathfrak X_{\sigma (1)}|+\cdots +|\mathfrak X_{\sigma (k-1)}|)} \\ &P\big (\mathfrak X_{\sigma (1)},\cdots ,\mathfrak X_{\sigma (k-1)}, x_{k+q}\otimes Q(\mathfrak X_{\sigma (k)}, \cdots ,\mathfrak X_{\sigma (k+q-1)},y_{k+q}),\mathfrak X_{k+q+1},\cdots ,\mathfrak X_{p+q},x\big )\\ &+\sum _{k=1}^{p}(\!-\!1)^{(k-1)q}\sum _{\sigma \in \mathbb{S}(k-1,q)}(\!-\!1)^{\sigma +\mathbb{K}(\sigma ;\;\mathfrak X,k-1)+\beta (|\mathfrak X_{\sigma (1)}|+\cdots +|\mathfrak X_{\sigma (k-1)}|)} P\big (\mathfrak X_{\sigma (1)},\cdots ,\mathfrak X_{\sigma (k-1)}, \\ &Q(\mathfrak X_{\sigma (k)},\cdots ,\mathfrak X_{\sigma (k+q-1)},x_{k+q})\otimes y_{k+q}, \mathfrak X_{k+q+1},\cdots ,\mathfrak X_{p+q},x\big ),\\ \end{aligned} \end{equation*}

for all homogeneous elements $\mathfrak X_i=x_i\otimes y_i\in \otimes ^2\mathfrak{L}$ with $|\mathfrak X_i|=|x_i|+|y_i|$ and $x\in \mathfrak{L}$ , where $\mathbb{S}_{(i,n-i)}$ denotes the set of $(i,n-i)$ -unshuffles, namely, the permutation $\sigma \in \mathbb{S}_n$ satisfies $\sigma (1)\lt \cdots \lt \sigma (i)$ and $\sigma (i+1)$ $\lt \cdots \lt \sigma (n)$ , and for any $1\leq m\leq p$ ,

\begin{equation*}\mathbb{K}(\sigma ;\;\mathfrak X,m)=\textrm {card}\big \{(i,j)\,|\,i\lt j, \mathfrak X_{\sigma (i)}\in \mathfrak{L}_{\bar {1}}, \mathfrak X_{\sigma (j)}\in \mathfrak{L}_{\bar {1}}, \sigma (i)\lt \sigma (j)\big \}=\sum _{1\leq s\leq m\atop 1\leq t\leq q}\sum _{\sigma (m+t)\lt \sigma (s)}|\mathfrak X_{\sigma (s)}||\mathfrak X_{\sigma (m+t)}|.\end{equation*}

In particular, for $P\in \mathfrak{C}^p(\mathfrak{L},\mathfrak{L})_{\alpha }, f\in \mathfrak{C}^{0}(\mathfrak{L},\mathfrak{L})_{\beta }$ , we define

(3.9) \begin{align} P\circ f(\mathfrak X_1,\cdots , \mathfrak X_p,x)=&\;\sum _{k=1}^p(\!-\!1)^{(|\mathfrak X_1|+\cdots +|\mathfrak X_{k-1}|+|x_k|)\beta }P(\mathfrak X_1,\cdots ,\mathfrak X_{k-1},x_k\otimes f(y_k),\mathfrak X_{k+1},\cdots , \mathfrak X_p,x)\notag \\ &+\sum _{k=1}^p(\!-\!1)^{(|\mathfrak X_1|+\cdots +|\mathfrak X_{k-1}|)\beta }P(\mathfrak X_1,\cdots ,\mathfrak X_{k-1},f(x_k)\otimes y_k,\mathfrak X_{k+1},\cdots , \mathfrak X_p,x)\notag \\ &+(\!-\!1)^{(|\mathfrak X_1|+\cdots +|\mathfrak X_{p}|)\beta }P(\mathfrak X_1,\cdots ,\mathfrak X_p,f(x)), \end{align}

and

(3.10) \begin{equation} f\circ P(\mathfrak X_1,\cdots , \mathfrak X_p,x)=f(P(\mathfrak X_1,\cdots , \mathfrak X_p,x)). \end{equation}

Now we define

\begin{equation*}[P,Q]_{\textrm {LSTS}}=P\circ Q-(\!-\!1)^{pq+\alpha \beta }Q\circ P.\end{equation*}

It was shown in [Reference Chtioui, Hajjaji, Mabrouk and Makhlouf2] that the subspace $\mathfrak{C}^*(\mathfrak{L},\mathfrak{L})_{\bar {0}}=\oplus _{n\geq 0}\mathfrak{C}^{n}(\mathfrak{L},\mathfrak{L})_{\bar {0}}$ equipped with the above graded commutator is a $\mathbb{Z}$ -graded Lie algebra. By similar computations, we get that $(\mathfrak{C}^*(\mathfrak{L},\mathfrak{L}), [\cdot ,\cdot ]_{\textrm {LSTS}})$ forms a ${\mathbb{Z}}\times {\mathbb{Z}}_2$ -graded Lie algebra. In particular, $(\mathfrak{C}^*(\mathfrak{L},\mathfrak{L})_{\bar {0}}, [\cdot ,\cdot ]_{\textrm {LSTS}}, d=0)$ is a differential graded Lie algebra, and $\pi \in \mathfrak{C}^1(\mathfrak{L},\mathfrak{L})_{\bar {0}}$ defines a $\textrm {LSTS}$ structure on $\mathfrak{L}$ if and only if $[\pi ,\pi ]_{\textrm {LSTS}}=0$ , namely, $\pi$ is a Maurer–Cartan element of $(\mathfrak{C}^*(\mathfrak{L},\mathfrak{L})_{\bar {0}},[\cdot ,\cdot ]_{\textrm {LSTS}}, d=0)$ .

For $n\geq 1$ , we denote by $\mathfrak{C}_{\mathrm{LSTSDer}}^n(\mathfrak{L},\mathfrak{L})=\mathfrak{C}^n(\mathfrak{L},\mathfrak{L})\times \mathfrak{C}^{n-1}(\mathfrak{L},\mathfrak{L})$ . Then we have the following result.

Proposition 3.12. Let $\mathfrak{C}^*_{\mathrm{LSTSDer}}(\mathfrak{L},\mathfrak{L})=\bigoplus _{n\in {\mathbb{Z}}_{\geq 0}}\mathfrak{C}^n(\mathfrak{L},\mathfrak{L})\times \mathfrak{C}^{n-1}(\mathfrak{L},\mathfrak{L})$ . Define the bracket $[\cdot ,\cdot ]_{\mathrm{LSTSDer}}\;:$ $\mathfrak{C}^p_{\mathrm{LSTSDer}}(\mathfrak{L},\mathfrak{L}) \times \mathfrak{C}^{q}_{\mathrm{LSTSDer}}(\mathfrak{L},\mathfrak{L})\rightarrow \mathfrak{C}^{p+q}_{\mathrm{LSTSDer}}(\mathfrak{L},\mathfrak{L})$ by

\begin{equation*}[(f_{p}, g_{p-1}), (f_{q}, g_{q-1})]_{\mathrm{LSTSDer}}=[f_p,f_q]_{\textrm {LSTS}}+(\!-\!1)^p[f_p,g_{q-1}]_{\textrm {LSTS}}+[g_{p-1},f_q]_{\textrm {LSTS}}.\end{equation*}

Then, $(\mathfrak{C}^*_{\mathrm{LSTSDer}}(\mathfrak{L},\mathfrak{L}), [\cdot , \cdot ]_{\mathrm{LSTSDer}})$ forms a ${\mathbb{Z}}\times {\mathbb{Z}}_2$ -graded Lie algebra. Moreover, for any $\pi \in \mathfrak{C}^1(\mathfrak{L},\mathfrak{L})_{\bar {0}}$ and $\phi \in \mathrm{Hom}(\mathfrak{L},\mathfrak{L})_{\alpha }$ , $(\mathfrak{L}, \pi ,\phi )$ is a LSTSDer if and only if $(\pi ,\phi )$ is a Maurer–Cartan element of the differential graded algebra $(\bigoplus _{(n,\alpha )\in {\mathbb{Z}}_{\geq 0}\times {\mathbb{Z}}_2}\mathfrak{C}^n(\mathfrak{L},\mathfrak{L})_{\bar {0}}\times \mathfrak{C}^{n-1}(\mathfrak{L},\mathfrak{L})_{\alpha }, [\cdot , \cdot ]_{\mathrm{LSTSDer}}, d=0)$ .

Proof. As $(\mathfrak{C}^*(\mathfrak{L},\mathfrak{L}), [\cdot ,\cdot ]_{\textrm {LSTS}})$ is a ${\mathbb{Z}}\times {\mathbb{Z}}_2$ -graded Lie algebra, we have that $\mathfrak{C}^*_{\mathrm{LSTSDer}}(\mathfrak{L},\mathfrak{L})$ equipped with the bracket $[\cdot , \cdot ]_{\mathrm{LSTSDer}}$ forms a ${\mathbb{Z}}\times {\mathbb{Z}}_2$ -graded Lie algebra. In particular, the subalgebra $(\bigoplus _{(n,\alpha )\in {\mathbb{Z}}_{\geq 0}\times {\mathbb{Z}}_2}\mathfrak{C}^n(\mathfrak{L},\mathfrak{L})_{\bar {0}}\times \mathfrak{C}^{n-1}(\mathfrak{L},\mathfrak{L})_{\alpha }, [\cdot , \cdot ]_{\mathrm{LSTSDer}}, d=0)$ is a differential graded Lie algebra. For $\pi \in \mathfrak{C}^1(\mathfrak{L},\mathfrak{L})_{\bar {0}}$ and $\phi \in \mathrm{Hom}(\mathfrak{L},\mathfrak{L})_{\alpha }$ , we see that

\begin{equation*}[(\pi ,\phi ),(\pi ,\phi )]_{\mathrm{LSTSDer}}=([\pi ,\pi ]_{\textrm {LSTS}}, -2[\pi ,\phi ]_{\textrm {LSTS}}).\end{equation*}

So $(\pi ,\phi )$ is a Maurer–Cartan element of $(\bigoplus _{(n,\alpha )\in {\mathbb{Z}}_{\geq 0}\times {\mathbb{Z}}_2}\mathfrak{C}^n(\mathfrak{L},\mathfrak{L})_{\bar {0}}\times \mathfrak{C}^{n-1}(\mathfrak{L},\mathfrak{L})_{\alpha }, [\cdot , \cdot ]_{\mathrm{LSTSDer}}, d=0)$ if and only if $[(\pi ,\phi ),(\pi ,\phi )]_{\mathrm{LSTSDer}}=0$ , that is, $[\pi ,\pi ]_{\textrm {LSTS}}=0$ and $[\pi , \phi ]_{\textrm {LSTS}}=0$ . Since $[\pi ,\pi ]_{\textrm {LSTS}}=0$ , it follows that $(\mathfrak{L},\pi )$ is a LSTS. Moreover, for any $x,y,z\in \mathfrak{L}$ , from $[\pi , \phi ]_{\textrm {LSTS}}=0$ and (3.9)–(3.10), we have

\begin{align*} [\pi ,\phi ]_{\textrm {LSTS}}(x,y,z)={}&(\pi \circ \phi -\phi \circ \pi )(x,y,z)\\ ={}&\pi (\phi (x),y,z)+(\!-\!1)^{|x|\alpha }\pi (x,\phi (y),z)+(\!-\!1)^{(|x|+|y|)\alpha }\pi (x,y,\phi (x))-\phi (\pi (x,y,z)), \end{align*}

which implies that $\phi$ is a superderivation of $(\mathfrak{L},\pi )$ . Thus, $(\mathfrak{L},\pi ,\phi )$ is a LSTSDer pair. The proof is completed.

3.4. The associated LeibSDer pair

In this subsection, we consider the LeibSDer pair, that is, a Leibniz superalgebra with a superderivation. Recall from Section 2 that we can get a Leibniz superalgebra from a $\textrm {LSTS}$ . Using the same idea, we study the connections between LSTSDer pairs and LeibSDer pairs.

First of all, we give the definitions concerning a LeibSDer pair and its representation, which are ${\mathbb{Z}}_2$ -graded generalizations of LeibDer pair given in [Reference Das3].

Let $(\mathfrak{g},[\cdot , \cdot ]_{\mathfrak{g}})$ be a Leibniz superalgebra. A homogeneous linear map $\phi _{\mathfrak{g}}\;:\;\mathfrak{g}\rightarrow \mathfrak{g}$ is a superderivation on $\mathfrak{g}$ of ${\mathbb{Z}}_2$ -degree $|\phi _{\mathfrak{g}}|$ if it satisfies

\begin{equation*}\phi _{\mathfrak{g}}([x,y]_{\mathfrak{g}})=[\phi _{\mathfrak{g}}(x),y]_{\mathfrak{g}}+(\!-\!1)^{|\phi _{\mathfrak{g}}||x|}[x,\phi _{\mathfrak{g}}(y)]_{\mathfrak{g}},\end{equation*}

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

Definition 3.13. A Leibniz superalgebra $\mathfrak{g}$ with a superderivation $\phi _{\mathfrak{g}}$ is called a LeibSDer pair , denoted by $(\mathfrak{g},\phi _{\mathfrak{g}})$ .

Definition 3.14. Let $(\mathfrak{g},\phi _{\mathfrak{g}})$ be a LeibSDer pair. A representation of it is given by a pair of $(M,\phi _M)$ in which $(M,[\cdot ,\cdot ]_L, [\cdot ,\cdot ]_R)$ is a representation of $\mathfrak{g}$ and $\phi _M\;:\; M\rightarrow M$ is a homogeneous linear map with $|\phi _M|=|\phi _{\mathfrak{g}}|$ , satisfying

(3.11) \begin{aligned}\phi _M[x,m]_L&=[\phi _{\mathfrak{g}}(x), m]_L+(\!-\!1)^{|x||\phi _{\mathfrak{g}}|}[x,\phi _M(m)]_L,\end{aligned}

(3.12) \begin{aligned}\phi _M[m,x]_R&=[\phi _M(m),x]_R+(\!-\!1)^{|m||\phi _{\mathfrak{g}}|}[m,\phi _{\mathfrak{g}}(x)]_R,\end{aligned}

for all homogeneous elements $x\in \mathfrak{g}$ and $m\in M$ .

Proposition 3.15. Let $(\mathfrak{L},\phi _{\mathfrak{L}})$ be a LSTSDer pair and $(\mathcal{L}\;:\!=\;\otimes ^{2}\mathfrak{L},\ast )$ be the associated $\textrm {LSTS}$ of $\mathfrak{L}$ defined by (2.12). Then, $(\mathcal{L},\phi _{\mathfrak{L}}\otimes \textrm {Id}+\textrm {Id}\otimes \phi _{\mathfrak{L}})$ is a LeibSDer pair.

Proof. We only need to show that $\phi _{\mathfrak{L}}\otimes \textrm {Id}+\textrm {Id}\otimes \phi _{\mathfrak{L}}$ is a superderivation of $\mathcal{L}$ .

Notice that

\begin{equation*}(\phi _{\mathfrak{L}}\otimes \textrm {Id}+\textrm {Id}\otimes \phi _{\mathfrak{L}})(x_1,x_2)=(\phi _{\mathfrak{L}}(x_1),x_2)+(\!-\!1)^{|x_1||\phi _{\mathfrak{L}}|}(x_1,\phi _{\mathfrak{L}}(x_2))\end{equation*}

for all homogeneous element $(x_1,x_2)\in \mathcal{L}$ . Then for any homogeneous elements $\mathfrak{X}=(x_1,x_2),\mathfrak{Y}=(y_1,y_2)\in \mathcal{L}$ , on one hand, we have

\begin{align*} &(\phi _{\mathfrak{L}}\otimes \textrm {Id}+\textrm {Id}\otimes \phi _{\mathfrak{L}})(\mathfrak{X}\ast \mathfrak{Y})\\ =&\;(\phi _{\mathfrak{L}}\otimes \textrm {Id}+\textrm {Id}\otimes \phi _{\mathfrak{L}})\Big (([x_1,x_2,y_1],y_2)+(\!-\!1)^{(|x_1|+|x_2|)|y_1|}(y_1,[x_1,x_2,y_2])\Big )\\ =&\;(\phi _{\mathfrak{L}}([x_1,x_2,y_1]),y_2)+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|+|y_1|)}([x_1,x_2,y_1],\phi _{\mathfrak{L}}(y_2))+(\!-\!1)^{(|x_1|+|x_2|)|y_1|}(\phi _{\mathfrak{L}}(y_1),[x_1,x_2,y_1])\\ &+(\!-\!1)^{(|x_1|+|x_2|+|\phi _{\mathfrak{L}}|)|y_1|}(y_1,\phi _{\mathfrak{L}}([x_1,x_2,y_2])). \end{align*}

On the other hand, we obtain that

\begin{align*} &\Big ((\phi _{\mathfrak{L}}\otimes \textrm {Id}+\textrm {Id}\otimes \phi _{\mathfrak{L}})(\mathfrak{X})\Big )\ast \mathfrak{Y}+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}\mathfrak{X}\ast \Big ((\phi _{\mathfrak{L}}\otimes \textrm {Id}+\textrm {Id}\otimes \phi _{\mathfrak{L}})(\mathfrak{Y})\Big )\\[3pt] \nonumber =&\,\Big ((\phi _{\mathfrak{L}}(x_1),x_2)+(\!-\!1)^{|\phi _{\mathfrak{L}}||x_1|}(x_1,\phi _{\mathfrak{L}}(x_2))\Big )\ast \mathfrak{Y}\nonumber \\ &+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}\mathfrak{X}\ast \Big ((\phi _{\mathfrak{L}}(y_1),y_2)+(\!-\!1)^{|\phi _{\mathfrak{L}}||y_1|}(y_1,\phi _{\mathfrak{L}}(y_2))\Big )\\[3pt] \nonumber \overset {(2.12)}{=}&([\phi _{\mathfrak{L}}(x_1),x_2,y_1],y_2)+(\!-\!1)^{(|\phi _{\mathfrak{L}}|+|x_1|+|x_2|)|y_1|}(y_1,[\phi _{\mathfrak{L}}(x_1),x_2,y_2])+(\!-\!1)^{|x_1||\phi _{\mathfrak{L}}|}([x_1,\phi _{\mathfrak{L}}(x_2),y_1],y_2)\\[3pt] \nonumber &+(\!-\!1)^{(|x_1|+|x_2|+|\phi _{\mathfrak{L}}|)|y_1|+|x_1||\phi _{\mathfrak{L}}|}(y_1,[x_1,\phi _{\mathfrak{L}}(x_2),y_2])+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}([x_1,x_2,\phi _{\mathfrak{L}}(y_1)],y_2)\\[3pt] \nonumber &+(\!-\!1)^{(|x_1|+|x_2|)|y_1|}(\phi _{\mathfrak{L}}(y_1),[x_1,x_2,y_2])+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|+|y_1|)}([x_1,x_2,y_1], \phi _{\mathfrak{L}}(y_2))\\[3pt] \nonumber &+(\!-\!1)^{(|x_1|+|x_2|+|\phi _{\mathfrak{L}}|)|y_1|+|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}(y_1,[x_1,x_2,\phi _{\mathfrak{L}}(y_2)]). \end{align*}

From the fact that $\phi _{\mathfrak{L}}$ is a superderivation of $\mathfrak{L}$ , we deduce that

\begin{align*}(\phi _{\mathfrak{L}}\otimes \textrm {Id}+\textrm {Id}\otimes \phi _{\mathfrak{L}})(\mathfrak{X}\ast \mathfrak{Y})&=\Big ((\phi _{\mathfrak{L}}\otimes \textrm {Id}+\textrm {Id}\otimes \phi _{\mathfrak{L}})(\mathfrak{X})\Big )\ast \mathfrak{Y}\nonumber \\&\quad +(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}\mathfrak{X}\ast \Big ((\phi _{\mathfrak{L}}\otimes \textrm {Id}+\textrm {Id}\otimes \phi _{\mathfrak{L}})(\mathfrak{Y})\Big ),\end{align*}

namely, $\phi _{\mathfrak{L}}\otimes \textrm {Id}+\textrm {Id}\otimes \phi _{\mathfrak{L}}$ is a superderication of $\mathcal{L}$ . The proof is finished.

Define a homogeneous linear map $\phi _{\mathcal{L}}\;:\;\mathrm{Hom}(\mathfrak{L},V)\rightarrow \mathrm{Hom}(\mathfrak{L},V)$ with $|\phi _{\mathcal{L}}|=|\phi _{\mathfrak{L}}|=|\phi _V|$ by

\begin{equation*} \phi _{\mathcal{L}}(f)=\phi _V\circ f-(\!-\!1)^{|f||\phi _{\mathfrak{L}}|}f\circ \phi _{\mathfrak{L}}, \,\forall f\in \mathrm{Hom}(\mathfrak{L},V).\end{equation*}

Then we have the following proposition, which provide a connection of the representation between a LSTSDer pair and its associated LeibSDer pair.

Proposition 3.16. Assume that $(V;\;\theta ,\phi _V)$ is a representation of a LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ . Then, $(\mathrm{Hom}(\mathfrak{L},V);[\cdot ,\cdot ]_L,[\cdot ,\cdot ]_R,\phi _{\mathcal{L}})$ is a representation of the associated LeibSDer pair $(\mathcal{L},\phi _{\mathfrak{L}}\otimes \textrm {Id}+\textrm {Id}\otimes \phi _{\mathfrak{L}})$ .

Proof. According to Proposition 2.9, we know that $(\mathrm{Hom}(\mathfrak{L},V);[\cdot ,\cdot ]_L,[\cdot ,\cdot ]_R)$ is a representation of $\mathcal{L}$ . So it is sufficient to check that

(3.13) \begin{aligned}\phi _{\mathcal{L}}[\mathfrak{X},f]_L(x_3)=[(\phi _{\mathfrak{L}}\otimes \textrm {Id}+\textrm {Id}\otimes \phi _{\mathfrak{L}})(\mathfrak{X}),f]_L(x_3)+(\!-\!1)^{|\mathfrak{X}||\phi _{\mathfrak{L}}|}[\mathfrak{X},\phi _{\mathcal{L}}(f)]_L(x_3),\end{aligned}

(3.14) \begin{aligned}\phi _{\mathcal{L}}[f,\mathfrak{X}]_R(x_3)=[\phi _{\mathcal{L}}(f), \mathfrak{X}]_R(x_3)+(\!-\!1)^{|f||\phi _{\mathfrak{L}}|}[f,(\phi _{\mathfrak{L}}\otimes \textrm {Id}+\textrm {Id}\otimes \phi _{\mathfrak{L}})(\mathfrak{X})]_R(x_3),\end{aligned}

for all homogeneous elements $\mathfrak{X}=(x_1,x_2)\in \mathcal{L}$ , $f\in \mathrm{Hom}(\mathfrak{L},V)$ and $x_3\in \mathfrak{L}$ .

Here we only verify that the equality (3.13) holds since the other equality can be treated similarly. In fact, by the definition of $\phi _{\mathcal{L}}$ and Eq. (2.15), the left-hand side of Eq. (3.13) is

\begin{align*} &\rm {LHS \, of \, Eq. \,(3.13)}\\ =&\;\phi _V[(x_1,x_2),f]_L(x_3)-(\!-\!1)^{(|f|+|x_1|+|x_2|)|\phi _{\mathfrak{L}}|}[(x_1,x_2),f]_L(\phi _{\mathfrak{L}}(x_3))\\ =&\;\phi _VD(x_1,x_2)f(x_3)-(\!-\!1)^{|f|(|x_1|+|x_2|)}\phi _V( f([x_1,x_2,x_3]))-(\!-\!1)^{(|f|+|x_1|+|x_2|)|\phi _{\mathfrak{L}}|}D(x_1,x_2)(f( \phi _{\mathfrak{L}}(x_3))\\ &+(\!-\!1)^{|f|(|x_1|+|x_2|+|\phi _{\mathfrak{L}}|)+(|x_1|+|x_2|)|\phi _{\mathfrak{L}}|}f([x_1,x_2,\phi _{\mathfrak{L}}(x_3)]). \end{align*}

Similarly, the right-hand side of Eq. (3.13) is

\begin{align*} &\rm {RHS \, of \, Eq. \,(3.13)}\\[3pt] \nonumber =&\;[(\phi _{\mathfrak{L}}(x_1),x_2),f]_L(x_3)+(\!-\!1)^{|x_1||\phi _{\mathfrak{L}}|}[(x_1,\phi _{\mathfrak{L}}(x_2)),f]_L(x_3) +(\!-\!1)^{(|x_1|+|x_2|)|\phi _{\mathfrak{L}}|}D(x_1,x_2)\phi _V( f(x_3))\\[3pt] \nonumber &-(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_1|+|f|)}D(x_1,x_2)(f(\phi _{\mathfrak{L}}(x_3)))-(\!-\!1)^{|f|(|x_1|+|x_2|)}\phi _V( f([x_1,x_2,x_3]))\\[3pt] \nonumber &+(\!-\!1)^{|f|(|x_1|+|x_2|+|\phi _{\mathfrak{L}}|)}f(\phi _{\mathfrak{L}}([x_1,x_2,x_3]))\\[3pt] \nonumber =&\;\underline {D(\phi _{\mathfrak{L}}(x_1),x_2)f(x_3)}\underbrace {-(\!-\!1)^{|f|(|x_1|+|x_2|+|\phi _{\mathfrak{L}}|)}f([\phi _{\mathfrak{L}}(x_1),x_2,x_3])}+\underline {(\!-\!1)^{|x_1||\phi _{\mathfrak{L}}|}D (x_1,\phi _{\mathfrak{L}}(x_2))f(x_3)}\\[3pt] \nonumber &\underbrace {-(\!-\!1)^{|f|(|x_1|+|x_2|+|\phi _{\mathfrak{L}}|)+|x_1||\phi _{\mathfrak{L}}|}f([x_1,\phi _{\mathfrak{L}}(x_2),x_3])}+\underline {(\!-\!1)^{(|x_1|+|x_2|)|\phi _{\mathfrak{L}}|}D(x_1,x_2)\phi _V( f(x_3))}\\[3pt] \nonumber &-(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_1|+|f|)}D(x_1,x_2)(f(\phi _{\mathfrak{L}}(x_3)))-(\!-\!1)^{|f|(|x_1|+|x_2|)}\phi _V( f([x_1,x_2,x_3]))\\[3pt] \nonumber &\underbrace {+(\!-\!1)^{|f|(|x_1|+|x_2|+|\phi _{\mathfrak{L}}|)}f(\phi _{\mathfrak{L}}([x_1,x_2,x_3]))}. \end{align*}

Thus, one concludes by (3.3) and the fact that $\phi _{\mathfrak{L}}$ is a superderivation.

4. Deformations of LSTSDer pairs

In this section, we study formal deformations and deformations of order $n$ of LSTSDer pairs. Let $\mathbb{F}[[t]]$ be the formal power series over $\mathbb{F}$ and $\mathfrak{L}[[t]]$ be the set of formal power series over $\mathfrak{L}$ . Clearly, $\mathfrak{L}[[t]]$ is the $\mathbb{F}[[t]]$ -module.

4.1. Formal deformations of a LSTSDer pair

In this subsection, we investigate 1-parameter formal deformations of a LSTSDer pair. We prove that the LSTSDer pair is rigid provided that the third cohomology group of a LSTSDer pair with coefficient in the adjoint representation is trivial.

Let $(\mathfrak{L},\phi _{\mathfrak{L}})$ be a LSTSDer pair. Denote by $\mu$ the $\textrm {LSTS}$ bracket on $\mathfrak{L}$ , that is, $\mu (x,y,z)=[x,y,z]$ for all homogeneous elements $x,y,z\in \mathfrak{L}$ . Consider a $t$ -parametrized family of homogeneous linear operators

\begin{align*} \mu _t&=\sum \limits _{i\geq 0}\mu _it^i, \, \mu _i\in C^{3}(\mathfrak{L},\mathfrak{L})_{\bar {0}}, \,\textrm {with}\, \mu _0=\mu ,\\ \phi _t&=\sum \limits _{i\geq 0}\phi _it^i,\, \phi _i\,\textrm {is homogeneous in}\, C^{1}(\mathfrak{L},\mathfrak{L}) \,\textrm {with}\, \phi _0=\phi _{\mathfrak{L}}, |\phi _t|=|\phi _i|=|\phi _{\mathfrak{L}}|. \end{align*}

Definition 4.1. A 1-parameter formal deformation of a LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ is a pair of formal power series $(\mu _t,\phi _t)$ , where $(\mu _0,\phi _0)=(\mu ,\phi _{\mathfrak{L}})$ such that $(\mathfrak{L}[[t]],\mu _t,\phi _t)$ is a LSTSDer pair over $\mathbb{F}[[t]]$ .

Given a pair $(\mu _t, \phi _t)$ as above, it is a 1-parameter formal deformation of the LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ if and only if for all homogeneous elements $x_k\in \mathfrak{L}$ , $1\leq k\leq 5$ , the following equalities hold:

(4.1) \begin{aligned}|\mu _t(x_1, x_2, x_3)|=&\,|x_1|+|x_2|+|x_3|,\end{aligned}

(4.2) \begin{aligned}\mu _t(x_2, x_1, x_3)=&\,-(\!-\!1)^{|x_1||x_2|} \mu _t(x_1, x_2, x_3),\end{aligned}

(4.3) \begin{aligned}(\!-\!1)^{|x_1||x_3|} \mu _t(x_1, x_2, x_3) & +(\!-\!1)^{|x_2||x_1|} \mu _t(x_2, x_3, x_1)+(\!-\!1)^{|x_3||x_2|} \mu _t(x_3, x_1, x_2)=0, \end{aligned}

(4.4) \begin{aligned}\mu _t(x_1, x_2, \mu _t(x_3, x_4, x_5))= &\mu _t(\mu _t(x_1, x_2, x_3), x_4, x_5)+ (\!-\!1)^{|x_3|(|x_1|+|x_2|)} \mu _t(x_3, \mu _t(x_1, x_2, x_4), x_5)\notag \\ & +(\!-\!1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)} \mu _t(x_3, x_4, \mu _t(x_1, x_2, x_5)),\end{aligned}

(4.5) \begin{aligned}\phi _t(\mu _t(x_1,x_2,x_3))=&\,\mu _t(\phi _t(x),y,z)+(\!-\!1)^{|\phi _{\mathfrak{L}}||x|}u_t(x,\phi _t(y),z)+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x|+|y|)}\mu _t(x,y,\phi _t(z)).\end{aligned}

Expanding the above equalities and collecting coefficients of $t^{n}$ in (4.4)–(4.5), we have that (4.1)–(4.5) are equivalent to the following equations:

(4.6) \begin{align} |\mu _i(x_1, x_2, x_3)|=|x_1|+|x_2|+|x_3|, \\[-30pt] \nonumber \end{align}

(4.7) \begin{align} \mu _i(x_2, x_1, x_3)=-(\!-\!1)^{|x_1||x_2|} \mu _i(x_1, x_2, x_3), \\[-30pt] \nonumber \end{align}

(4.8) \begin{align} (\!-\!1)^{|x_1||x_3|} \mu _i(x_1, x_2, x_3) +(\!-\!1)^{|x_2||x_1|} \mu _i(x_2, x_3, x_1)+(\!-\!1)^{|x_3||x_2|} \mu _i(x_3, x_1, x_2)=0, \\[-30pt] \nonumber \end{align}

(4.9) \begin{aligned}\sum _{i+j=n\atop i,j\geq 0} &\Big (\mu _i(\mu _j(x_1,x_2,x_3),x_4,x_5) +(\!-\!1)^{|x_3|(|x_1|+|x_2|)}\mu _i(x_3,\mu _j(x_1,x_2,x_4),x_5)\\ &+(\!-\!1)^{(|x_{3}|+|x_4|)(|x_1|+|x_2|)}\mu _i(x_3,x_4,\mu _j(x_1,x_2,x_5))-\mu _i(x_1,x_2,\mu _j(x_3,x_4,x_5))\Big )=0,\end{aligned}

(4.10) \begin{aligned} \sum _{i+j=n\atop i,j\geq 0} &\Big (\phi _i(\mu _j(x_1,x_2,x_3))-\mu _j(\phi _i(x_1),x_2,x_3) -(\!-\!1)^{|\phi _{\mathfrak{L}}||x_1|}\mu _j(x_1,\phi _i(x_2),x_3)\\ &-(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}\mu _j(x_1,x_2,\phi _i(x_3))\Big )=0. \end{aligned}

To characterize $(\mu _1, \phi _1)$ , we need the following low dimensional coboundary operators: for $f\in C^1(\mathfrak{L},\mathfrak{L})$ ,

(4.11) \begin{equation} \begin{split} \delta f(x_1, x_2, x_3)= & [f(x_1), x_2, x_3]+(\!-\!1)^{|f||x_1|}[x_1, f(x_2), x_3] +(\!-\!1)^{|f|(|x_1|+|x_2|)} [x_1, x_2, f(x_3)]\\ &-f([x_1, x_2, x_3]), \end{split} \end{equation}

and for $f\in C^3(\mathfrak{L},\mathfrak{L})$ ,

(4.12) \begin{align} &\delta f(x_1, x_2, x_3, x_4, x_5)\notag \\[5pt] = &[f(x_1, x_2, x_3),x_4, x_5] +(\!-\!1)^{(|f|+|x_1|+|x_2|)|x_3|} [x_3,f(x_1, x_2, x_4),x_5]-(\!-\!1)^{|f|(|x_1|+|x_2|)} [x_1, x_2, f(x_3, x_4, x_5)]\notag \\[5pt] &+(\!-\!1)^{(|f|+|x_1|+|x_2|)(|x_3|+|x_4|)} [x_3, x_4, f(x_1, x_2, x_5)]+f([x_1, x_2, x_3], x_4, x_5)-f(x_1, x_2,[x_3, x_4, x_5])\notag \\[5pt] &+(\!-\!1)^{|x_3|(|x_1|+|x_2|)} f(x_3,[x_1, x_2, x_4], x_5)+(\!-\!1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)} f(x_3, x_4,[x_1, x_2, x_5]). \end{align}

Proposition 4.2. Let $(\mu _t,\phi _t)$ be a 1-parameter formal deformation of the LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ . Then $(\mu _1,\phi _1) \in C^{3}_{\mathrm{LSTSDer}}(\mathfrak{L},\mathfrak{L})$ is a $3$ -cocycle of the LSTSDer pair $(\mathfrak{L},\phi )$ with the coefficient in the adjoint representation.

Proof. Take $n=1$ in (4.9) and (4.10), respectively, we have

(4.13) \begin{align} &\mu _1([x_1,x_2,x_3],x_4,x_5)+(\!-\!1)^{|x_3|(|x_1|+|x_2|)}\mu _1(x_3,[x_1,x_2,x_4],x_5)+(\!-\!1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}\mu _1(x_3,x_4,[x_1,x_2,x_5])\notag \\[3pt] &-\mu _1(x_1,x_2,[x_3,x_4,x_5])+[\mu _1(x_1,x_2,x_3),x_4,x_5]+(\!-\!1)^{|x_3|(|x_1|+|x_2|)}[x_3,\mu _1(x_1,x_2,x_4),x_5]\notag \\[3pt] &+(\!-\!1)^{(|x_3|+|x_4|)(|x_1|+|x_2|)}[x_3,x_4,\mu _1(x_1,x_2,x_5)]-[x_1,x_2,\mu _1(x_3,x_4,x_5)]=0, \end{align}

and

(4.14) \begin{align} & \phi _1([x_1,x_2,x_3])-[\phi _1(x_1),x_2,x_3]-(\!-\!1)^{|\phi _{\mathfrak{L}}||x_1|}[x_1,\phi _1(x_2),x_3]\notag \\[3pt] &-(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}[x_1,x_2,\phi _1(x_3)]+\phi _{\mathfrak{L}}(\mu _1(x_1,x_2,x_3))-\mu _1(\phi _{\mathfrak{L}}(x_1),x_2,x_3)\notag \\[3pt] &-(\!-\!1)^{|x_1||\phi _{\mathfrak{L}}|}\mu _1(x_1,\phi _{\mathfrak{L}}(x_2),x_3)-(\!-\!1)^{(|x_1|+|x_2|)|\phi _{\mathfrak{L}}|}\mu _1(x_1,x_2, \phi _{\mathfrak{L}}(x_3))=0. \end{align}

According to (4.11) and (4.12), we note that (4.13) implies that $\delta \mu _1=0$ and (4.14) implies that $\delta \phi _1+\mathrm{d} \mu _1=0$ . Thus, we have

\begin{equation*}\partial (\mu _1,\phi _1)=(\delta \mu _1,\delta \phi _1+\mathrm{d} \mu _1)=(0,0).\end{equation*}

We complete the proof.

Definition 4.3. The $3$ -cocycle $(\mu _1, \phi _1)$ is called the infinitesimal of the 1-parameter formal deformation $(\mu _t, \phi _t)$ of the LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ .

Definition 4.4. Suppose that $(\mu _t,\phi _t)$ and $(\mu _{t}',\phi _{t}')$ are 1-parameter formal deformations of a LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ . $(\mu _t,\phi _t)$ and $(\mu _{t}',\phi _{t}')$ are called equivalent (i.e., $(\mu _t,\phi _t)\sim (\mu _{t}',\phi _{t}')$ ), if there exists a formal isomorphism $\Psi _t=\sum _{i\geq 0}\psi _i t^i:(\mathfrak{L}[[t]],\mu '_t,\phi '_t)\rightarrow (\mathfrak{L}[[t]],\mu _t,\phi _t)$ , where $\psi _i\in C^1(\mathfrak{L},\mathfrak{L})_{\bar {0}}$ and $\psi _0=\textrm {Id}_{\mathfrak{L}}$ , satisfying that

\begin{align*} \Psi _t\circ u_{t}'&=u_t\circ (\Psi _t\otimes \Psi _t\otimes \Psi _t),\\ \Psi _t\circ \phi _{t}'&=\phi _t\circ \Psi _t. \end{align*}

Definition 4.5. Let $(\mu _{t},\phi _{t})$ be a 1-parameter formal deformation of a LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ . $(\mu _t,\phi _t)$ is said to be trivial if $(\mu _t,\phi _t)\sim (\mu ,\phi _{\mathfrak{L}})$ . Moreover, $(\mathfrak{L},\phi _{\mathfrak{L}})$ is called rigid if every 1-parameter formal deformation of $(\mathfrak{L}, \phi _{\mathfrak{L}})$ is trivial.

The proofs of the following results are essentially the same as in the case of Lie triple systems with derivations. We will not present them here. More details refer to [Reference Guo10].

Proposition 4.6. The infinitesimals of two equivalent 1-parameter formal deformations of a LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ belong to the same cohomology class.

Theorem 4.7. Let $(\mathfrak{L},\phi _{\mathfrak{L}})$ be a LSTSDer pair. If $H_{\mathrm{LSTSDer}}^3(\mathfrak{L},\mathfrak{L})=0$ , then $(\mathfrak{L},\phi _{\mathfrak{L}})$ is rigid.

4.2. Deformations of order $n$ of a LSTSDer pair

In this subsection, we introduce a cohomology class associated to any deformation of order $n$ of a LSTSDer pair. Moreover, a deformation of order $n$ of a LSTSDer pair is showed to be extensible if and only if this cohomology class is trivial.

Definition 4.8. A deformation of order $n$ of a LSTSDer pair $(\mathfrak{L}, \phi _{\mathfrak{L}})$ is a pair $(\mu _t,\phi _t)$ such that $\mu _t=\sum _{i=0}^{n}\mu _it^i$ and $\phi _t=\sum _{i=0}^{n}\phi _it^i$ endow the $\mathbb{F}[[t]]/(t^{n+1})$ -module $\mathfrak{L}[[t]]/(t^{n+1})$ the LSTSDer pair structure with $(\mu _0,\phi _0)=(\mu ,\phi _{\mathfrak{L}})$ . Furthermore, if there exists a 3-cochain $(\mu _{n+1},\phi _{n+1}) \in C_{\mathrm{LSTSDer}}^3(\mathfrak{L},\mathfrak{L})$ , such that $(\mu _{t}',\phi _{t}')$ defined by

(4.15) \begin{align} \mu '_{t}=\mu _t+\mu _{n+1}t^{n+1},\phi _{t}'=\phi _t+\phi _{n+1}t^{n+1} \end{align}

is a deformation of order $(n+1)$ of $(\mathfrak{L},\phi _{\mathfrak{L}})$ , we say that $(\mu _t,\phi _t)$ is extensible .

Let $(\mu _t,\phi _t)$ be a deformation of order $n$ of a LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ . For any homogeneous elements $x_k\in \mathfrak{L}$ , $1\leq k\leq 5$ , define a $5$ -cochain $(\Omega _{n+1}^5,\Omega _{n+1}^3)\in C_{\mathrm{LSTSDer}}^5(\mathfrak{L},\mathfrak{L})$ as

\begin{align*} \Omega _{n+1}^5(x_1,x_2,x_3,x_4,x_5)=&\,\sum _{i+j=n+1\atop i,j\geq 1}\Big (\mu _i(\mu _j(x_1,x_2,x_3),x_4,x_5)+(\!-\!1)^{|x_3|(|x_1|+|x_2|)}\mu _i(x_3,\mu _j(x_1,x_2,x_4),x_5)\\ & \ +(\!-\!1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}\mu _i(x_3,x_4,\mu _j(x_1,x_2,x_5))-\mu _i(x_1,x_2,\mu _j(x_3,x_4,x_5))\Big ),\\ \Omega _{n+1}^3(x_1,x_2,x_3)=&\, \sum _{i+j=n+1\atop i,j\geq 1}\Big (\phi _i(\mu _j(x_1,x_2,x_3))-\mu _j(\phi _i(x_1),x_2,x_3)-(\!-\!1)^{|x_1||\phi _{\mathfrak{L}}|}\mu _j(x_1,\phi _i(x_2),x_3)\\ & \ -(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}\mu _j(x_1,x_2,\phi _i(x_3))\Big ). \end{align*}

Proposition 4.9. Keep notations as above. If $(\mu _t,\phi _t)$ is extensible, then the $5$ -cochain $(\Omega _{n+1}^5,\Omega _{n+1}^3)$ is a 5-cocycle of the LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ with the coefficient in the adjoint representation, that is, $\partial (\Omega _{n+1}^5,\Omega _{n+1}^3)=0.$

Proof. Since $(\mu _t,\phi _t)$ is a deformation of order $n$ of the LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ , for all $0\leq i \leq n$ , we see that

(4.16) \begin{aligned} \sum _{k+l=i\atop k,l\geq 0}&\Big (\mu _k(\mu _l(x_1,x_2,x_3),x_4,x_5)+(\!-\!1)^{|x_3|(|x_1|+|x_2|)}\mu _k(x_3,\mu _l(x_1,x_2,x_4),x_5)\\ &+(\!-\!1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)}\mu _k(x_3,x_4,\mu _l(x_1,x_2,x_5))-\mu _k(x_1,x_2,\mu _l(x_3,x_4,x_5))\Big )=0. \end{aligned}

(4.17) \begin{aligned} \sum _{k+l=i\atop k,l\geq 0}&\Big (\phi _k(\mu _l(x_1,x_2,x_3))-\mu _l(\phi _k(x_1),x_2,x_3)-(\!-\!1)^{|x_1||\phi _{\mathfrak{L}}|}\mu _l(x_1,\phi _k(x_2),x_3)\\ &-(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}\mu _l(x_1,x_2,\phi _k(x_3))\Big )=0. \end{aligned}

Due to the extensibility of $(\mu _t,\phi _t)$ , we see that (4.16) and (4.17) hold for $i=n+1$ . It is straightforward to check that $\Omega _{n+1}^5=\delta \mu _{n+1}$ and $\Omega _{n+1}^3=\delta \phi _{n+1}+\mathrm{d} \mu _{n+1}$ . Then, $\delta \Omega _{n+1}^5=\delta (\delta \mu _{n+1})=0$ . By Lemma 3.7, we also have

\begin{align*} \delta \Omega _{n+1}^3-\mathrm{d}\Omega _{n+1}^5=\delta (\delta \phi _{n+1}+\mathrm{d} \mu _{n+1})-\mathrm{d}(\delta \mu _{n+1})=(\delta \circ \mathrm{d}-\mathrm{d}\circ \delta )\mu _{n+1}=0. \end{align*}

Thus, $\partial (\Omega _{n+1}^5,\Omega _{n+1}^3)=(\delta \Omega _{n+1}^5,\delta \Omega _{n+1}^3-\mathrm{d}\Omega _{n+1}^5)=(0,0)$ . The proof is completed.

Definition 4.10. Let $(\mu _t,\phi _t)$ be a deformation of order $n$ of a LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ . We call the cohomology class $[(\Omega _{n+1}^5,\Omega _{n+1}^3)] \in H_{\mathrm{LSTSDer}}^5(\mathfrak{L},\mathfrak{L})$ the obstruction class of $(\mu _t,\phi _t)$ being extensible .

Now, we give the main result of this subsection.

Theorem 4.11. Let $(\mu _t,\phi _t)$ be a deformation of order $n$ of a LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ . Then $(\mu _t,\phi _t)$ is extensible if and only if the obstruction class $[(\Omega _{n+1}^5,\Omega _{n+1}^3)]$ is trivial.

Proof. Suppose that $(\mu _t,\phi _t)$ extends to a deformation $(\mu '_t,\phi _t')$ of order $(n+1)$ , where

\begin{align*} \mu _{t}'=\mu _t+\mu _{n+1}t^{n+1},\, \phi _{t}'=\phi _t+\phi _{n+1}t^{n+1}. \end{align*}

One can easily check that $\Omega _{n+1}^5=\delta \mu _{n+1}, \Omega _{n+1}^3=\delta \phi _{n+1}+\mathrm{d} \mu _{n+1}$ , which implies that

\begin{align*} (\Omega _{n+1}^5,\Omega _{n+1}^3)=(\delta \mu _{n+1},\delta \phi _{n+1}+\mathrm{d} \mu _{n+1})=\partial (\mu _{n+1},\phi _{n+1}). \end{align*}

Thus, by Proposition 4.9, we know that the cohomology class $[(\Omega _{n+1}^5,\Omega _{n+1}^3)]$ is trivial.

Conversely, suppose that $[(\Omega _{n+1}^5,\Omega _{n+1}^3)]$ is trivial. Then, $(\Omega _{n+1}^5,\Omega _{n+1}^3)=\partial (\mu _{n+1},\phi _{n+1})$ , for some 3-cochain $(\mu _{n+1},\phi _{n+1})\in C_{\mathrm{LSTSDer}}^3(\mathfrak{L},\mathfrak{L})$ . Set

\begin{equation*}\mu _{t}'=\mu _t+\mu _{n+1}t^{n+1},\ \phi _{t}'=\phi _t+\phi _{n+1}t^{n+1}.\end{equation*}

It is easy to see that $(\mu _{t}',\phi _{t}')$ satisfies (4.16)–(4.17) for $0\leq i\leq n+1$ , and hence, $(\mu _{t}',\phi _{t}')$ is a deformation of order $(n+1)$ . Namely, $(\mu _t,\phi _t)$ is extensible. We complete the proof.

5. Abelian extensions of LSTSDer pairs

In this section, we study abelian extensions of LSTSDer pairs and show that equivalent abelian extensions of LSTSDer pairs are classified by the third cohomology groups with the coefficient in a suitable representation.

Definition 5.1. Let $(A,\phi _A)$ and $(\mathfrak{L},\phi _{\mathfrak{L}})$ be two LSTSDer pairs. An abelian extension of $(\mathfrak{L},\phi _{\mathfrak{L}})$ by $(A,\phi _A)$ is an exact sequence of LSTSDer pair morphisms

(5.1)

such that $A$ is an abelian ideal of $\widetilde {\mathfrak{L}}$ , i.e., $[x,y,t]_{\widetilde {\mathfrak{L}}}=[x,t,y]_{\widetilde {\mathfrak{L}}}=[t,x,y]_{\widetilde {\mathfrak{L}}}=0$ , $\forall \ x,y\in A, t\in \widetilde {\mathfrak{L}}$ . Denote this abelian extension by $(\widetilde {\mathfrak{L}},\phi _{\widetilde {\mathfrak{L}}})$ . A section of an abelian extension of $(\mathfrak{L},\theta _{\mathfrak{L}})$ by $(A,\theta _A)$ is an even linear map $s\;:\;\mathfrak{L}\rightarrow \widetilde {\mathfrak{L}}$ such that $p\circ s=\textrm {Id}_{\mathfrak{L}}$ .

Let $(\mathfrak{L},\phi _{\mathfrak{L}})$ be a LSTSDer pair and $(A;\;\theta ,\phi _A)$ a representation of $(\mathfrak{L},\phi _{\mathfrak{L}})$ . Suppose that $(\varphi ,\lambda )\in C_{\mathrm{LSTSDer}}^3(\mathfrak{L},A)$ . Define $[\cdot ,\cdot ,\cdot ]_{\varphi }:(\mathfrak{L}\oplus A)\otimes (\mathfrak{L}\oplus A)\otimes (\mathfrak{L}\oplus A)\rightarrow \mathfrak{L}\oplus A$ and $\phi _{\lambda }\;:\;\mathfrak{L}\oplus A\rightarrow \mathfrak{L}\oplus A$ as

(5.2) \begin{aligned}[x_1+a_1,x_2+a_2,x_3+a_3]_{\varphi }=&\, [x_1,x_2,x_3]+\varphi (x_1,x_2,x_3)+(\!-\!1)^{|x_1|(|x_2|+|x_3|)}\theta (x_2,x_3)(a_1)\notag \\ & -(\!-\!1)^{|x_2||x_3|}\theta (x_1,x_3)(a_2)+D(x_1,x_2)(a_3) ,\end{aligned}

(5.3) \begin{aligned}\phi _{\lambda }(x+a)=&\, \phi _{\mathfrak{L}}(x)+\lambda (x)+\phi _A(a)\end{aligned}

for all homogeneous elements $x,x_i\in \mathfrak{L}, a,a_i\in A,$ where $\lambda$ is homogeneous with $|\lambda |=|\phi _{\mathfrak{L}}|=|\phi _A|$ . Denote $\mathfrak{L}\oplus A$ with the bracket operation (5.2) by $\mathfrak{L}\ltimes _{\varphi } A$ .

Proof. Suppose that $(\mathfrak{L}\oplus A,\phi _{\lambda })$ is a LSTSDer pair. By Proposition 2.4 and a direct caculation, we see that $\mathfrak{L}\oplus A$ equipped with the bracket given in (5.2) is a $\textrm {LSTS}$ if and only if

\begin{align*} 0=&\,(\!-\!1)^{(|x_1|+|x_2|+|x_3|)(|x_4|+|x_5|)} \theta (x_4, x_5) \varphi (x_1, x_2, x_3) -(\!-\!1)^{(|x_1|+|x_2|)(|x_3|+|x_5|)+|x_4||x_5|} \theta (x_3, x_5) \varphi (x_1, x_2, x_4) \\[3pt] & - D(x_1, x_2) \varphi (x_3, x_4, x_5)+(\!-\!1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)} D(x_3, x_4) \varphi (x_1, x_2, x_5)+\varphi ([x_1, x_2, x_3], x_4, x_5)\\[3pt] &-\varphi (x_1, x_2,[x_3, x_4, x_5])+(\!-\!1)^{|x_3|(|x_1|+|x_2|)} \varphi (x_3,[x_1, x_2, x_4], x_5) +(\!-\!1)^{(|x_1|+|x_2|)(|x_3|+|x_4|)} \varphi (x_3, x_4,[x_1, x_2, x_5]), \end{align*}

which implies that $\delta \varphi =0$ .

Moreover, by the fact that $\phi _\lambda$ is a superderivation on $\mathfrak{L}\ltimes _{\varphi } A$ , $\phi _{\mathfrak{L}}$ is a superderivation on $\mathfrak{L}$ , and $(A;\;\theta ,\phi _A)$ is a representation of $(\mathfrak{L},\phi _{\mathfrak{L}})$ , for any homogeneous elements $x_i\in \mathfrak{L}, a_i\in A$ , we have

\begin{eqnarray*} &&[\phi _\lambda (x_1+a_1),x_2+a_2,x_3+a_3]_{\varphi }+(\!-\!1)^{|\phi _{\mathfrak{L}}||x_1|}[x_1+a_1,\phi _\lambda (x_2+a_2),x_3+a_3]_{\varphi }\\[3pt] && +(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}[x_1+a_1,x_2+a_2,\phi _\lambda (x_3+a_3)]_{\varphi }-\phi _\lambda ([x_1+a_1,x_2+a_2,x_3+a_3]_{\varphi })\\[3pt] &=&[\phi _{\mathfrak{L}}(x_1),x_2,x_3]+(\!-\!1)^{|\phi _{\mathfrak{L}}||x_1|}[x_1,\phi _{\mathfrak{L}}(x_2),x_3]+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}[x_1,x_2,\phi _{\mathfrak{L}}(x_3)]-\phi _{\mathfrak{L}}([x_1,x_2,x_3])\\[3pt] &&+\varphi (\phi _{\mathfrak{L}}(x_1),x_2,x_3)+(\!-\!1)^{|\phi _{\mathfrak{L}}||x_1|}\varphi (x_1,\phi _{\mathfrak{L}}(x_2),x_3)+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}\varphi (x_1,x_2,\phi _{\mathfrak{L}}(x_3))-\phi _A\circ \varphi (x_1,x_2,x_3)\\[3pt] &&+(\!-\!1)^{(|x_1|+|\phi _{\mathfrak{L}}|)(|x_2|+|x_3|)}\theta (x_2,x_3)\lambda (x_1)-(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_3|)+|x_2||x_3|}\theta (x_1,x_3)\lambda (x_2)\\ &&+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}D(x_1,x_2)\lambda (x_3)\\[3pt] &&-\lambda ([x_1,x_2,x_3])+(\!-\!1)^{|x_1|(|x_2|+|x_3|)}\Big ((\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_2|+|x_3|)}\theta (x_2,x_3)\phi _A(a_1)-\phi _A\theta (x_2,x_3)(a_1)+\theta (\phi _{\mathfrak{L}}(x_2,x_3)(a_1)\\[3pt] &&+(\!-\!1)^{|x_1||\phi _{\mathfrak{L}}|}\theta (x_2,\phi _{\mathfrak{L}}(x_3))(a_1)\Big )-(\!-\!1)^{|x_2||x_3|}\Big ((\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_3|)}\theta (x_1,x_3)\phi _A(a_2)-\phi _A\theta (x_1,x_3)(a_2)\\[3pt] &&+\theta (\phi _{\mathfrak{L}}(x_1),x_3)(a_2)+(\!-\!1)^{|x_2||\phi _{\mathfrak{L}}|}\theta (x_1,\phi _{\mathfrak{L}}(x_3))(a_2)\Big )+(\!-\!1)^{|\phi _t|(|x_1|+|x_2|)}D(x_1,x_2)\phi _A(a_3)-\phi _AD(x_1,x_2)(a_3)\\[3pt] &&+D(\phi _{\mathfrak{L}}(x_1),x_2)(a_3)+(\!-\!1)^{|x_1||\phi _{\mathfrak{L}}|}D(x_1,\phi _{\mathfrak{L}}(x_2))(a_3)\\[3pt] &=&\varphi (\phi _{\mathfrak{L}}(x_1),x_2,x_3)+(\!-\!1)^{|\phi _{\mathfrak{L}}||x_1|}\varphi (x_1,\phi _{\mathfrak{L}}(x_2),x_3)+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}\varphi (x_1,x_2,\phi _{\mathfrak{L}}(x_3))-\phi _A\circ \varphi (x_1,x_2,x_3)\\[3pt] &&+(\!-\!1)^{(|x_1|+|\phi _{\mathfrak{L}}|)(|x_2|+|x_3|)}\theta (x_2,x_3)\lambda (x_1)-(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_3|)+|x_2||x_3|}\theta (x_1,x_3)\lambda (x_2)\\[3pt] &&+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x_1|+|x_2|)}D(x_1,x_2)\lambda (x_3)-\lambda ([x_1,x_2,x_3])\\ &=&(\delta \lambda +\mathrm{d}\varphi )(x_1,x_2,x_3)=0. \end{eqnarray*}

Thus, we have

\begin{align*} \partial (\varphi ,\lambda )=(\delta \varphi ,\delta \lambda +\mathrm{d}\varphi )=(0,0), \end{align*}

which forces that $(\varphi ,\lambda )\in Z_{\mathrm{LSTSDer}}^3(\mathfrak{L},A)$ .

The converse can be proved similarly. We omit the details.

Let $(\widetilde {\mathfrak{L}},\phi _{\widetilde {\mathfrak{L}}})$ be an abelian extension of $(\mathfrak{L},\phi _{\mathfrak{L}})$ by $(A,\phi _A)$ and $s\;:\;\mathfrak{L}\rightarrow \widetilde {\mathfrak{L}}$ a section. Define even linear maps $\theta _A\;:\;\mathfrak{L}\otimes \mathfrak{L}\rightarrow \textrm {End}(A)$ , $D_A\;:\;\mathfrak{L}\otimes \mathfrak{L}\rightarrow \textrm {End}(A)$ , $\omega \;:\;\mathfrak{L}\otimes \mathfrak{L}\otimes \mathfrak{L}\rightarrow A$ and homogeneous linear map $\chi \;:\;\mathfrak{L}\rightarrow A$ with $|\chi |=|\phi _{\mathfrak{L}}|$ by

(5.4) \begin{aligned}\theta _A(x,y)(a)&=(\!-\!1)^{|a|(|x|+|y|)}[a,s(x),s(y)]_{\widetilde {\mathfrak{L}}},\end{aligned}

(5.5) \begin{aligned}D_A(x,y)(a)&=[s(x),s(y),a]_{\widetilde {\mathfrak{L}}},\end{aligned}

(5.6) \begin{aligned}\omega (x,y,z)&=[s(x),s(y),s(z)]_{\widetilde {\mathfrak{L}}}-s([x,y,z]),\end{aligned}

(5.7) \begin{aligned}\chi (x)&=\phi _{\widetilde {\mathfrak{L}}}(s(x))-s(\phi _{\mathfrak{L}}(x))\end{aligned}

for all homogenous elements $x,y,z\in \mathfrak{L}, a\in A$ . Clearly, $D_A(x,y)(a)=(\!-\!1)^{|x||y|}\theta _A(y,x)(a)-\theta _A(x,y)(a)$ . With these notations, note that $\widetilde {\mathfrak{L}}$ is actually an abelian extension of the $\textrm {LSTS}$ $\mathfrak{L}$ by $A$ , then we recall the following proposition.

Proof. By Proposition 5.3, we know that $(A;\;\theta _A)$ is a representation of the $\textrm {LSTS}$ algebra $\mathfrak{L}$ on $A$ . We only need to prove the following equalities

(5.8) \begin{aligned}\phi _A\theta _A(x,y)(a)&=\theta _A(\phi _{\mathfrak{L}}(x),y)(a)+(\!-\!1)^{|x||\phi _{\mathfrak{L}}|}\theta _A(x,\phi _{\mathfrak{L}}(y))(a)+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x|+|y|)}\theta _A(x,y)\phi _A(a),\end{aligned}

(5.9) \begin{aligned}\phi _AD_A(x,y)(a)&=D_A(\phi _{\mathfrak{L}}(x),y)(a)+(\!-\!1)^{|x||\phi _{\mathfrak{L}}|}D_A(x,\phi _{\mathfrak{L}}(y))(a)+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x|+|y|)}D_A(x,y)\phi _A(a),\end{aligned}

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

Notice that we can identify $A$ with the corresponding subalgebra of $\widetilde {\mathfrak{L}}$ and $\phi _A=\phi _{\widetilde {\mathfrak{L}}}|_{A}$ , then by a direct calculation, we have

\begin{eqnarray*} &&\theta _A(\phi _{\mathfrak{L}}(x),y)(a)+(\!-\!1)^{|x||\phi _{\mathfrak{L}}|}\theta _A(x,\phi _{\mathfrak{L}}(y))(a)+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x|+|y|)}\theta _A(x,y)\phi _A(a)\\[3pt] &=&(\!-\!1)^{|a|(|\phi _{\mathfrak{L}}|+|x|+|y|)}[a,s(\phi _{\mathfrak{L}}(x)),s(y)]_{\widetilde {\mathfrak{L}}}+(\!-\!1)^{|x||\phi _{\mathfrak{L}}|+|a|(|\phi _{\mathfrak{L}}|+|x|+|y|)}[a,s(x),s(\phi _{\mathfrak{L}}(y))]_{\widetilde {\mathfrak{L}}}\\[3pt] &&+(\!-\!1)^{|\phi _{\mathfrak{L}}|(|x|+|y|)+(|a|+|\phi _{\mathfrak{L}}|)(|x|+|y|)}[\phi _A(a),s(x),s(y)]_{\widetilde {\mathfrak{L}}}\\[3pt] &=&(\!-\!1)^{|a|(|\phi _{\mathfrak{L}}|+|x|+|y|)}[a,\phi _{\widetilde {\mathfrak{L}}}(s(x))-\chi (x),s(y)]_{\widetilde {\mathfrak{L}}}+(\!-\!1)^{|x||\phi _{\mathfrak{L}}|+|a|(|\phi _{\mathfrak{L}}|+|x|+|y|)}[a,s(x),\phi _{\widetilde {\mathfrak{L}}}(s(y))-\chi (y)]_{\widetilde {\mathfrak{L}}}\\[3pt] &&+(\!-\!1)^{|a|(|x|+|y|)}[\phi _A(a),s(x),s(y)]_{\widetilde {\mathfrak{L}}}\\[3pt] &=&(\!-\!1)^{|a|(|\phi _{\mathfrak{L}}|+|x|+|y|)}[a,\phi _{\widetilde {\mathfrak{L}}}(s(x)),s(y)]_{\widetilde {\mathfrak{L}}}+(\!-\!1)^{|x||\phi _{\mathfrak{L}}|+|a|(|\phi _{\mathfrak{L}}|+|x|+|y|)}[a,s(x),\phi _{\widetilde {\mathfrak{L}}}(s(y))]_{\widetilde {\mathfrak{L}}}\\[3pt] &&+(\!-\!1)^{|a|(|x|+|y|)}[\phi _A(a),s(x),s(y)]_{\widetilde {\mathfrak{L}}}\\[3pt] &=&\phi _{\widetilde {\mathfrak{L}}}((\!-\!1)^{|a|(|x|+|y|)}[a,s(x),s(y)]_{\widetilde {\mathfrak{L}}})\\[3pt] &=&\phi _A(\theta _A(x,y)(a)). \end{eqnarray*}

Similarly, one can show that (5.9) holds. Thus, $(A;\;\theta _A,\phi _A)$ is a representation of the LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ .

Since $(\widetilde {\mathfrak{L}},\phi _{\widetilde {\mathfrak{L}}})$ is an abelian extension of $(\mathfrak{L},\phi _{\mathfrak{L}})$ by $(A,\phi _A)$ and $(\omega ,\chi )$ defined in (5.6)–(5.7) belongs to $C^3_{\mathrm{LSTSDer}}(\mathfrak{L},A)$ , by Proposition 5.2, we see that $(\omega ,\chi )\in Z_{\mathrm{LSTSDer}}^3(\mathfrak{L},A)$ .

Suppose that $s_1$ and $s_2$ are two sections of $p$ . Set $\nu (x)=s_1(x)-s_2(x)$ for any homogeneous element $x\in \mathfrak{L}$ . Note that $\nu \in \textrm {Hom}(\mathfrak{L},A)_{\bar {0}}$ , then

\begin{align*} \omega _1(x,y,z)=&\;[s_1(x),s_1(y),s_1(z)]_{\widetilde {\mathfrak{L}}}-s_1([x,y,z])\\[3pt] =&\;[s_2(x)+\nu (x),s_2(y)+\nu (y),s_2(z)+\nu (z)]_{\widetilde {\mathfrak{L}}}-s_2([x,y,z])-\nu ([x,y,z])\\[3pt] =&\;[s_2(x),s_2(y),s_2(z)]_{\widetilde {\mathfrak{L}}}-s_2([x,y,z])+[s_2(x),s_2(y),\nu (z)]_{\widetilde {\mathfrak{L}}}+[\nu (x),s_2(y),s_2(z)]_{\widetilde {\mathfrak{L}}}\\[3pt] &+[s_2(x),\nu (y),s_2(z)]_{\widetilde {\mathfrak{L}}}-\nu ([x,y,z])\\[3pt] =&\;\omega _2(x,y,z)+[s_2(x),s_2(y),\nu (z)]_{\widetilde {\mathfrak{L}}}+[\nu (x),s_2(y),s_2(z)]_{\widetilde {\mathfrak{L}}}+[s_2(x),\nu (y),s_2(z)]_{\widetilde {\mathfrak{L}}}-\nu ([x,y,z]), \end{align*}

which implies that $\omega _1-\omega _2=\delta \nu$ . Besides,

\begin{align*} \chi _1(x)-\chi _2(x)=&\; \phi _{\widetilde {\mathfrak{L}}}(s_1(x))-s_1(\phi _{\mathfrak{L}}(x))-(\phi _{\widetilde {\mathfrak{L}}}(s_2(x))-s_2(\phi _{\mathfrak{L}}(x)))\\[3pt] =&\; \phi _{\widetilde {\mathfrak{L}}}(s_1(x)-s_2(x))-(s_1(\phi _{\mathfrak{L}}(x))-s_2(\phi _{\mathfrak{L}}(x)))\\[3pt] =&\; \phi _{\widetilde {\mathfrak{L}}}(\nu (x))-\nu (\phi _{\mathfrak{L}}(x))\\[3pt] =&\; -(\mathrm{d}\nu )(x). \end{align*}

Then, we deduce $(\omega _1,\chi _1)-(\omega _2,\chi _2)=(\delta \nu ,-\mathrm{d}\nu )=\partial \nu$ . Thus, the cohomology class $(\omega _1,\chi _1)$ and $(\omega _2,\chi _2)$ are in the same cohomological class.

Definition 5.5. Two abelian extensions $(\widetilde {\mathfrak{L}}_1,\phi _{\widetilde {\mathfrak{L}}_1}), (\widetilde {\mathfrak{L}}_2,\phi _{\widetilde {\mathfrak{L}}_2})$ of $(\mathfrak{L},\phi _{\mathfrak{L}})$ by $(A,\phi _A)$ are called equivalent , if there exists a LSTSDer pair morphism $\eta \;:\;(\widetilde {\mathfrak{L}}_1,\phi _{\widetilde {\mathfrak{L}}_1})\rightarrow (\widetilde {\mathfrak{L}}_2,\phi _{\widetilde {\mathfrak{L}}_2})$ such that we have the following commutative diagram:

(5.10)

Since $\theta _A$ defined by (5.4) does not depend on the choice of sections, we know that two equivalent abelian extensions of LSTSDer pairs give the same representation of the $\textrm {LSTS}$ $\mathfrak{L}$ on $A$ . Thus, $(A;\;\theta _A,\phi _A)$ is also a representation of the LSTSDer pair $(\mathfrak{L},\phi _{\mathfrak{L}})$ , which does not depend on equivalence classes of abelian extensions of $(\mathfrak{L},\phi _{\mathfrak{L}})$ by $(A,\theta _A)$ . Furthermore, we have

Proof. Let $(\widetilde {\mathfrak{L}}_1,\phi _{\widetilde {\mathfrak{L}}_1}), (\widetilde {\mathfrak{L}}_2,\phi _{\widetilde {\mathfrak{L}}_2})$ be two equivalent abelian extensions of $(\mathfrak{L},\phi _{\mathfrak{L}})$ by $(A,\phi _A)$ , and $\eta \;:\;(\widetilde {\mathfrak{L}}_1,\phi _{\widetilde {\mathfrak{L}}_1})\rightarrow (\widetilde {\mathfrak{L}}_2,\phi _{\widetilde {\mathfrak{L}}_2})$ is a LSTSDer pair morphism such that the diagram in Definition 5.5 is commutative. Choose a section $s_1\;:\;\mathfrak{L}\rightarrow \widetilde {\mathfrak{L}}_1$ of $p_1$ . Since

\begin{align*} p_2\circ (\eta \circ s_1)=(p_2\circ \eta )\circ s_1=p_1\circ s_1=\textrm {Id}_{\mathfrak{L}}, \end{align*}

we obtain that $s_2\;:\!=\;\eta \circ s_1$ is a section of $p_2$ . Then we get 3-cocycles $(\omega _i,\chi _i)$ given by (5.6)–(5.7) corresponding to $s_i$ , $i=1,2$ . By Proposition 5.4, we see that the cohomology class $[(\omega _i,\chi _i)]$ of the abelian extension $(\widetilde {\mathfrak{L}}_i,\phi _{\widetilde {\mathfrak{L}}_i})$ does not depend on the choice of sections.

Now we go on to show that equivalent abelian extensions give rise to the same element in $H_{\mathrm{LSTSDer}}^3(\mathfrak{L},A)$ . As $\eta$ is a LSTS pair morphism and $\eta |_A=\textrm {Id}_A$ , for any homogeneous elements $x,y,z\in \mathfrak{L}$ , we have

\begin{align*} \omega _2(x,y,z)=&\; [s_2(x),s_2(y),s_2(z)]_{\widetilde {\mathfrak{L}}_2}-s_2([x,y,z])\\[3pt] =&\; [\eta (s_1(x)),\eta (s_1(y)),\eta (s_1(z))]_{\widetilde {\mathfrak{L}}_2}-\eta (s_1([x,y,z]))\\[3pt] =&\; \eta ([s_1(x),s_1(y),s_1(z)]_{{\widetilde {\mathfrak{L}}_1}}-s_1([x,y,z]))\\[3pt] =&\; \eta (\omega _1(x,y,z))=\omega _1(x,y,z). \end{align*}

Similarly, by $\eta \circ \phi _{\widetilde {\mathfrak{L}}_1}=\phi _{\widetilde {\mathfrak{L}}_2}\circ \eta$ , we have

\begin{align*} \chi _2(x)=&\;\phi _{\widetilde {\mathfrak{L}}_2}(s_2(x))-s_2(\phi _{\mathfrak{L}}(x))\\[3pt] =&\;\phi _{\widetilde {\mathfrak{L}}_2}(\eta (s_1(x)))-\eta (s_1(\phi _{\mathfrak{L}}(x)))\\[3pt] =&\;\eta (\phi _{\widetilde {\mathfrak{L}}_1}(s_1(x)))-\eta (s_1(\phi _{\mathfrak{L}}(x)))\\[3pt] =&\;\eta (\phi _{\widetilde {\mathfrak{L}}_1}(s_1(x))-s_1(\phi _{\mathfrak{L}}(x)))\\[3pt] =&\;\eta (\chi _1(x))=\chi _1(x). \end{align*}

Hence, we have $(\omega _1,\chi _1)=(\omega _2,\chi _2)$ . Thus, equivalence abelian extensions of $(\mathfrak{L},\phi _{\mathfrak{L}})$ by $(A,\theta _A)$ give the same element in $H_{\mathrm{LSTSDer}}^3(\mathfrak{L},A)$ .

Conversely, let $[(\omega _1,\chi _1)]=[(\omega _2,\chi _2)]\in H_{\mathrm{LSTSDer}}^3(\mathfrak{L},A)$ . Then there exists a map $\xi \in \textrm {Hom}(\mathfrak{L},A)_{\bar {0}}$ such that

\begin{equation*}(\omega _1,\chi _1)-(\omega _2,\chi _2)=\partial \xi =(\delta \xi ,-\mathrm{d}\xi ).\end{equation*}

By Proposition 5.2, we can construct two abelian extensions $(\mathfrak{L}\ltimes _{\omega _1} A, \phi _{\chi _1})$ and $(\mathfrak{L}\ltimes _{\omega _2} A, \phi _{\chi _2})$ as in (5.2) and (5.3). Define $\eta \;:\;\mathfrak{L}\ltimes _{\omega _1} A \rightarrow \mathfrak{L}\ltimes _{\omega _2} A$ by

\begin{align*} \eta (x+a)=x+\xi (x)+a,\ \forall \ x\in \mathfrak{L},\ a\in A. \end{align*}

We can deduce that $\eta$ is a LSTSDer pair morphism such that (5.10) commutes. Thus, these two abelian extensions are equivalent.