Super-biderivations of the contact Lie superalgebra $K(m,n;\underline{t})$

TL;DR

This paper investigates the structure of super-biderivations in the contact Lie superalgebra $K(m,n; ext{ extunderscore}t)$ over a field with characteristic greater than 3, showing they are all inner using weight space decomposition.

Contribution

It demonstrates that all skew-symmetric super-biderivations of $K(m,n; ext{ extunderscore}t)$ are inner, providing a detailed analysis based on the algebra's weight space decomposition.

Findings

All super-biderivations are inner.

Utilizes weight space decomposition for analysis.

Provides structure results for contact Lie superalgebras.

Abstract

Let $K$ denote the contact Lie superalgebra $K(m,n;\underline{t})$ over a field of characteristic $p>3$, which has a finite $\mathbb{Z}$-graded structure. Let $T_K$ be the canonical torus of $K$, which is an abelian subalgebra of $K_{0}$ and operates on $K_{-1}$ by semisimple endomorphisms. Utilizing the weight space decomposition of $K$ with respect to $T_K$, %we show the action of the skew-symmetric super-biderivation on the elements of $T$ and the contact of $K$. %Moreover, we prove that each skew-symmetric super-biderivation of $K$ is inner.

Equations

$$D([x,y])=[D(x),y]+[x,D(y)],$$

$$\psi (x,[y,z])$$

$$\psi ([x,y],z)$$

$$[x,y]$$

$$[x,[y,z]]$$

$$D([x,y])=[D(x),y]+(-1{)}^{\mathrm{p}(D)\mathrm{p}(x)}[x,D(y)],$$

$$skew-symmetry:\varphi (x,y)=-(-1{)}^{\mathrm{p}(x)\mathrm{p}(y)}\varphi (y,x),$$

$$\varphi ([x,y],z)=(-1{)}^{\mathrm{p}(\varphi )\mathrm{p}(x)}[x,\varphi (y,z)]+(-1{)}^{\mathrm{p}(y)\mathrm{p}(z)}[\varphi (x,z),y],$$

$$\varphi (x,[y,z])=[\varphi (x,y),z]+(-1{)}^{(\mathrm{p}(\varphi )+\mathrm{p}(x))\mathrm{p}(y)}[y,\varphi (x,z)],$$

$$\mathrm{BDer}(L)={\mathrm{BDer}}_{\overline{0}}(L)\oplus {\mathrm{BDer}}_{\overline{1}}(L).$$

$$x^{(\alpha )}{x}^{(\beta )}=\left(\genfrac{}{}{0ex}{}{\alpha +\beta }{\alpha }\right)x^{(\alpha +\beta )},\forall \alpha ,\beta \in {\mathbb{N}}^m.$$

$$x_k{x}_l=-x_l{x}_k,\forall k,l\in \{m+1,\dots ,s\};$$

$$x^{(\alpha )}{x}_k=x_k{x}^{(\alpha )},\forall \alpha \in {\mathbb{N}}^m,k\in \{m+1,\dots ,s\}.$$

$${\mathbb{B}}_k:=\{⟨i_1,i_2,\dots ,i_k⟩\mid m+1\le i_1<i_2<\dots <i_k\le s\}$$

$$W(m,n):=\left\{\sum f_r{\partial }_r|f_r\in \mathcal{O}(m,n),r\in \mathrm{I}\right\}.$$

$$[f{\partial }_i,g{\partial }_j]=f{\partial }_i(g){\partial }_j-(-1{)}^{\mathrm{p}(f{\partial }_i)\mathrm{p}(g{\partial }_j)}g{\partial }_j(f){\partial }_i,$$

$$O(m,n;\underline{t})=\{x^{(\alpha )}{x}^u|0\le \alpha \le \pi ,u\in \mathbb{B}\}$$

$$W(m,n;\underline{t})={\mathrm{span}}_{\mathbb{F}}\{x^{(\alpha )}{x}^u{\partial }_r|0\le \alpha \le \pi ,u\in \mathbb{B},r\in \mathrm{I}\}$$

$$W(m,n;\underline{t})=\underset{r=-1}{\overset{\xi -1}{⨁}}W(m,n;\underline{t}{)}_r,$$

$$i^{\prime }:=\{\begin{array}{ll}i+r,& 1\le i\le r,\\ i-r,& r<i\le 2r,\\ i,& i=m,\\ i+t,& m<i\le m+t,\\ i-t,& m+t<i\le s;\end{array}\sigma (i):=\{\begin{array}{ll}1,& 1\le i\le r\\ -1,& r<i\le 2r\\ 1,& 2r<i\le s.\end{array}$$

$$D_K(f)=\sum _{i\in \mathrm{J}}(-1{)}^{\mathrm{p}({\partial }_i)\mathrm{p}(f)}(x_i{\partial }_m(f)+\sigma (i^{\prime }){\partial }_{i^{\prime }}(f)){\partial }_i+(2f-\sum _{i\in \mathrm{J}}{x}_i{\partial }_m(f)){\partial }_m.$$

$$D_K:\mathcal{O}(m,n;\underline{t})\to K(m,n;\underline{t}).$$

$$K(m,n;\underline{t})=[\tilde{K}(m,n;\underline{t}),\tilde{K}(m,n;\underline{t})].$$

$$⟨f,g⟩:=D_K(f)(g)-2{\partial }_m(f)(g),$$

$$(K(m,n;\underline{t}),[\cdot ,\cdot ])\cong (\mathcal{O}(m,n;\underline{t}),⟨\cdot ,\cdot ⟩).$$

$$[\varphi (x,y),[u,v]]=(-1{)}^{\mathrm{p}(\varphi )(\mathrm{p}(x)+\mathrm{p}(y))}[[x,y],\varphi (u,v)]$$

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

$$⟨x_i{x}_{i^{\prime }},x^{(\alpha )}{x}^u⟩=({\alpha }_{i^{\prime }}-{\alpha }_i+{\delta }_{(i^{\prime }\in u)}-{\delta }_{(i\in u)})x^{(\alpha )}{x}^u,$$

$$(\alpha +⟨u⟩)(x_i{x}_{i^{\prime }})={\alpha }_{i^{\prime }}-{\alpha }_i+{\delta }_{(i^{\prime }\in u)}-{\delta }_{(i\in u)}.$$

$$K=\underset{(\alpha +⟨u⟩)}{⨁}{K}_{(\alpha +⟨u⟩)}.$$

$$\varphi (x_i{x}_{i^{\prime }},x^{(\alpha )}{x}^u)\in K_{(\alpha +⟨u⟩)},$$

$$⟨x_l{x}_{l^{\prime }},\varphi (x_i{x}_{i^{\prime }},x^{(\alpha )}{x}^u)⟩$$

Full text

Key words: Torus; Weight space decomposition; Super-biderivation.

Mathematics Subject Classification(2010): 17B05; 17B40; 17B50

000 Corresponding author(L. Chen): chenly640@nenu.edu.cn.000Supported by NNSF of China (Nos. 11771069), NSF of Jilin province (No. 20170101048JC) and the project of jilin province department of education (No. JJKH20180005K).

1 Introduction

Let $L$ be a Lie algebra over an arbitrary field $\mathbb{F}$. An $\mathbb{F}$-linear map $D:L\rightarrow L$ is a derivation satisfying

[TABLE]

for all $x,y\in L$. A bilinear map $\psi:L\times L\rightarrow L$ is called a biderivation if it is a derivation with respect to both components, meaning that

[TABLE]

for all $x,~{}y,~{}z\in L$. A biderivation $\psi$ is called skew-symmetric if $\psi(x,y)=-\psi(y,x)$ for all $x,~{}y\in L$. Obviously, if a biderivation $\psi$ is skew-symmetric, we can omit one of the equations (1.1) and (1.2). Meanwhile, we can view $\psi(x,\cdot)$ or $\psi(\cdot,x)$ as a derivation of $L$.

The study of biderivations traces back to the research on the commuting map in the associative ring [1], where the author showed that all biderivations on associative prime rings are inner. The notation of biderivations of Lie algebras was introduced in [2]. In recent years, there exist a lot of interests in studying biderivations and commuting maps on Lie algebras[3, 4, 5, 6, 7, 8, 9]. Moreover, the authors gave the notion of the skew-symmetric super-biderivation in [11]. So the results about the skew-symmetric super-biderivation of Lie superalgebras arise in [11, 12, 13].

The Cartan modular Lie superalgebra is an important branch of the modular Lie superalgebra, which is a Lie superalgebra over an algebraically closed field of characteristic $p>0$. And the contact Lie superalgebra $K(m,n;\underline{t})$ is an important class of Cartan modular Lie superalgebras. There are many research results about the contact Lie superalgebra $K(m,n;\underline{t})$, such as, derivation superalgebras[14, 15, 16], noncontractible filtrations[17], nondegenerate associative bilinear forms[18].

In this paper, we prove that each skew-symmetric super-biderivation of $K(m,n;\underline{t})$ is inner. The paper is organized as follows. In Section 2, we recall the basic notation. In Section 3, we use the weight space decomposition of $K(m,n;\underline{t})$ with respect to the canonical torus $T_{K}$ to prove that all skew-symmetric super-biderivation of $K$ is inner(Theorem 3.14).

2 Preliminaries

Let $\mathbb{F}$ denote the prime field of the characteristic $p>2$ and $\mathbb{Z}_{2}=\{\overline{0},\overline{1}\}$ the additive group of two elements. For a vector superspace $V=V_{\overline{0}}\oplus V_{\overline{1}}$, we use $\mathrm{p}(x)$ for the parity of $x\in V_{\alpha}$, $\alpha\in\mathbb{Z}_{2}$. If $V=\oplus_{i\in\mathbb{Z}}V_{i}$ is a $\mathbb{Z}$-graded vector space and $x\in V$ is a $\mathbb{Z}$-homogeneous element, write $|x|$ for the $\mathbb{Z}$-degree of $x$. Once the symbol $\mathrm{p}(x)$ or $|x|$ appears in this paper, it implies that $x$ is a $\mathbb{Z}_{2}$-homogeneous element or that $x$ is a $\mathbb{Z}$-homogeneous element. Throughout this paper all vector spaces or algebras are over $\mathbb{F}$.

2.1 Skew-symmetric super-biderivations of a Lie superalgebra

Let us recall some facts related to the superderivation and skew-symmetric super-biderivation of Lie superalgebras. A Lie superalgebra is a vector superspace $L=L_{\overline{0}}\oplus L_{\overline{1}}$ with an even bilinear mapping $[\cdot,\cdot]:L\times L\rightarrow L$ satisfying the following axioms:

[TABLE]

for all $x,y,z\in L$. We call a linear mapping $D:L\times L\rightarrow L$ a superderivation of $L$ if it satisfies the following axiom:

[TABLE]

for all $x,y\in L$, where $\mathrm{p}(D)$ denotes the $\mathbb{Z}_{2}$-degree of $D$. Write $\mathrm{Der}_{\overline{0}}(L)$ (resp. $\mathrm{Der}_{\overline{1}}(L)$) for the set of all superderivations of $\mathbb{Z}_{2}$-degree $\overline{0}$ (resp. $\overline{1}$) of $L$.

We call a bilinear mapping $\phi:L\times L\rightarrow L$ a skew-symmetric super-biderivation of $L$ if it satisfies the following axioms:

[TABLE]

for all $\mathbb{Z}_{2}$-homogeneous elements $x,y,z\in L$. A super-biderivation $\phi$ of $\mathbb{Z}_{2}$-degree $\gamma$ of $L$ is a super-biderivation such that $\phi(L_{\alpha},L_{\beta})\subset L_{\alpha+\beta+\gamma}$ for any $\alpha,~{}\beta\in\mathbb{Z}_{2}$. Denote by $\mathrm{BDer}_{\gamma}(L)$ the set of all skew-symmetric super-biderivations of $\mathbb{Z}_{2}$-degree $\gamma$. Obviously,

[TABLE]

Specially, if the bilinear map $\phi_{\lambda}:L\times L\rightarrow L$ is defined by $\phi_{\lambda}(x,y)=\lambda[x,y]$ for all $x,y\in L$, where $\lambda\in\mathbb{F}$, then it is easy to check that $\phi_{\lambda}$ is a super-biderivation of $L$. This class of super-biderivations is called inner. Denote by $\mathrm{IBDer}(L)$ the set of all inner super-biderivations.

2.2 Contact Lie superalgebras $K(m,n;\underline{t})$

We propose to construct a $\mathbb{Z}_{2}$-gradation tensor algebra via a divided power algebra and a exterior superalgebra. In the follow, we introduce the divided power algebra $\mathcal{O}(m)$ and the exterior superalgebra $\Lambda(n)$. Fix two positive integers $m>1$ and $n>1$. For $\alpha=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{N}^{m}$, where $\mathbb{N}$ denote the set of natural numbers, put $|\alpha|=\Sigma_{i=1}^{m}\alpha_{i}$. For two $m$-tuples $\alpha=(\alpha_{1},\ldots,\alpha_{m})$ and $\beta=(\beta_{1},\ldots,\beta_{m})\in\mathbb{N}^{m}$, we write ${\alpha\choose\beta}=\prod_{i=1}^{m}{\alpha_{i}\choose\beta_{i}}$ and define $\beta\leq\alpha\Longleftrightarrow\beta_{i}\leq\alpha_{i}$, $1\leq i\leq m$. Let $\mathcal{O}(m)$ denote the $\mathbb{F}$-algebra of divided power series in the variable $x_{1},\ldots,x_{m}$, which is called a $divided~{}power~{}algebra$. For convenience, we replace $x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\cdots x_{m}^{\alpha_{m}}$ by $x^{(\alpha)}$, $\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{m})$. Obviously, $\mathcal{O}(m)$ has an $\mathbb{F}$-basis $\{x^{(\alpha)}|\alpha\in\mathbb{N}^{m}\}$ and satisfies the formula:

[TABLE]

Let $\Lambda(n)$ denote the $exterior~{}superalgebra$ over $\mathbb{F}$ with $n$ variables $x_{m+1},\ldots,x_{s}$, where $s=m+n$. The tensor product $\mathcal{O}(m,n)=\mathcal{O}(m)\otimes_{\mathbb{F}}\Lambda(n)$ is an associative superalgebra with a $\mathbb{Z}_{2}$-gradation induced by the trivial $\mathbb{Z}_{2}$-gradation of $\mathcal{O}(m)$ and the natural $\mathbb{Z}_{2}$-gradation of $\Lambda(n)$. Obviously, $\mathcal{O}(m,n)$ is super-commutative. For $g\in\mathcal{O}(m)$, $f\in\Lambda(n)$, it is customary to write $gf$ instead of $g\otimes f$. Including the formula (2.1), the following formulas also hold in $\mathcal{O}(m,n)$:

[TABLE]

[TABLE]

For $k=1,\ldots,n$, set

[TABLE]

and $\mathbb{B}:=\cup_{k=0}^{n}\mathbb{B}_{k}$, where $\mathbb{B}_{0}=\emptyset$. For $u=\langle i_{1},i_{2},\ldots,i_{k}\rangle\in\mathbb{B}_{k}$, set $|u|:=k$, $x^{u}=x_{i_{1}}\cdots x_{i_{k}}$. Specially, we define $|\emptyset|=0$, $x^{\emptyset}=1$, $|\omega|=n$ and $x^{\omega}=x_{m+1}\cdots x_{m+n}$. Clearly, the set $\{x^{(\alpha)}x^{u}|~{}\alpha\in\mathbb{N}^{m},u\in\mathbb{B}\}$ constitutes an $\mathbb{F}$-basis of $\mathcal{O}(m,n)$.

Put $\mathrm{I}_{0}:=\{1,\ldots,m\}$, $\mathrm{I}_{1}:=\{m+1,\ldots,m+n\}$ and $\mathrm{I}:=\mathrm{I}_{0}\cup\mathrm{I}_{1}$. Let $\partial_{1},\partial_{2},\ldots,\partial_{s}$ be the linear transformations of $\mathcal{O}(m,n)$ such that $\partial_{i}(x^{(\alpha)})=x^{(\alpha-\varepsilon_{i})}$ for $i\in\mathrm{I}_{0}$, and $\partial_{i}(x_{k})=\delta_{ik}$, $k\in\mathrm{I}_{1}$, for $i\in\mathrm{I}_{1}$, where $\delta_{ij}$ is denoted the Kronecker symbol. Obviously, $\mathrm{p}(\partial_{i})=\overline{0}$ if $i\in\mathrm{I}_{0}$ and $\mathrm{p}(\partial_{i})=\overline{1}$ if $i\in\mathrm{I}_{1}$. Then $\partial_{1},\partial_{2},\ldots,\partial_{s}$ are superderivations of the superalgebra $\mathcal{O}(m,n)$. Let

[TABLE]

Then $W(m,n)$ is an infinite-dimensional Lie superalgebra contained in $\mathrm{Der}(\mathcal{O}(m,n))$. One can verify that

[TABLE]

for all $f,g\in\mathcal{O}(m,n)$ and $i,j\in\mathrm{I}$.

Fix two $m$-tuples of positive integers $\underline{t}=(t_{1},t_{2},\ldots,t_{m})$ and $\pi=(\pi_{1},\pi_{2},\ldots,\pi_{m})$, where $\pi_{i}=p^{t_{i}}-1$ for all $i\in\mathrm{I}_{0}$ and $p$ is denoted the characteristic of the basic field $\mathbb{F}$. For two $m$-tuples $\alpha=(\alpha_{1},\ldots,\alpha_{m})$ and $\beta=(\beta_{1},\ldots,\beta_{m})\in\mathbb{N}^{m}$, we have ${\alpha+\beta\choose\alpha}=0$ if there is some $i\in\{1,\ldots,m\}$ satisfying $\alpha_{i}+\beta_{i}\geq p^{t_{i}}$. Thence the set

[TABLE]

is a subalgebra of $O(m,n)$ and the set

[TABLE]

is a finite-dimensional simple subalgebra of $W(m,n)$, which is called the generalized Witt Lie superalgebra. $W(m,n;\underline{t})$ possesses a $\mathbb{Z}$-graded structure:

[TABLE]

where $W(m,n;\underline{t})_{r}:=\mathrm{span}_{\mathbb{F}}\{x^{(\alpha)}x^{u}\partial_{j}|~{}|\alpha|+|u|=r+1,~{}j\in\mathrm{I}\}$ and $\xi:=|\pi|+n$. For $i\in\mathrm{I}_{0}$, we abbreviate $x^{(\varepsilon_{i})}$ to $x_{i}$, where $\varepsilon_{i}$ is denoted the $m$-tuple with 1 as the i-th entry and 0 elsewhere.

Hereafter, suppose $m=2r+1$ is odd and $n=2t$ is even. Let $\mathrm{J}=\mathrm{I}\setminus\{m\}$ and $\mathrm{J}_{0}=\mathrm{I}_{0}\setminus\{m\}$. For $i\in\mathrm{J}$, put

[TABLE]

Define a linear mapping $D_{K}:\mathcal{O}(m,n)\rightarrow W(m,n)$ by means of

[TABLE]

The restricted linear mapping of $D_{K}$ on $\mathcal{O}(m,n;\underline{t})$ still is denoted by $D_{K}$, that is

[TABLE]

Let $\widetilde{K}(m,n;\underline{t})$ denote the image of $\mathcal{O}(m,n;\underline{t})$ under $D_{K}$. Consider the derived algebra of $\widetilde{K}(m,n;\underline{t})$:

[TABLE]

The derived algebra $K(m,n;\underline{t})$ is a finite dimensional simple Lie superalgebra, which is called the contact Lie superalgebra. We define a Lie bracket $\langle\cdot,\cdot\rangle$ on the tensor superalgebra $\mathcal{O}(m,n;\underline{t})$ by

[TABLE]

for all $f,g\in\mathcal{O}(m,n;\underline{t})$. Since $D_{K}$ is injective and $D_{K}(\langle f,g\rangle)=[D_{K}(f),D_{K}(g)]$, there exists an isomorphism, that is,

[TABLE]

For convenience, we use $W$ and $W_{r}$ denote $W(m,n;\underline{t})$ and its $\mathbb{Z}$-graded subspace $W(m,n;\underline{t})_{r}$, respectively, ${K}(m,n;\underline{t})$ is denoted by $K$.

3 Skew-symmetry Super-biderivation of $K(m,n;\underline{t})$

Lemma 3.1**.**

[10]** Let $L$ be a Lie superalgebra. Suppose that $\phi$ is a skew-symmetric super-biderivation on $L$, then

[TABLE]

for any homogenous element $x,y,u,v\in L$.

Lemma 3.2**.**

[10]** Let $L$ be a Lie superalgebra. Suppose that $\phi$ is a skew-symmetric super-biderivation on $L$. If $\mathrm{p}(x)+\mathrm{p}(y)=\overline{0}$, then

[TABLE]

for any homogenous element $x,y\in L$.

Lemma 3.3**.**

[10]** Let $L$ be a Lie superalgebra. Suppose that $\phi$ is a skew-symmetric super-biderivation on $L$. If $[x,y]=0$, then $\phi(x,y)\in C_{L}([L,L])$, where $C_{L}([L,L])$ is the centralizer of $[L,L]$ .

Lemma 3.4**.**

Let $K$ denote the contact Lie superalgebra. Suppose $\phi$ is a skew-symmetric super-biderivation on $K$. If $\langle x,y\rangle=0$ for $x,y\in K$, then $\phi(x,y)=0.$

Proof. .

Since $K$ is a simple Lie superalgebra, it is obvious that $K=\langle K,K\rangle$ and $C(K)=0$. if $\langle x,y\rangle=0$ for $x,y\in K$, by Lemma 3.3, we obtain $\phi(x,y)\in C_{K}(\langle K,K\rangle)=C(K)=0.$ ∎

Set $T_{K}=\mathrm{span}_{\mathbb{F}}\{x_{i}x_{i^{\prime}}~{}|~{}i\in\mathrm{J}\}$. Obviously, $T_{K}\subseteq K(m,n;\underline{t})_{0}\bigcap K(m,n;\underline{t})_{\bar{0}}$. $T_{K}$ is an abelian subalgebra of $K$. For any $x^{(\alpha)}x^{u}\in K$, we have

[TABLE]

where $\delta_{(\mathrm{P})}=1$ if the proposition $\mathrm{P}$ is true, $=0$ if the proposition $\mathrm{P}$ is false. Fixed an $m$-tuple $\alpha$, where $\alpha\in\mathbb{N}^{m}$, $0\leq\alpha\leq\pi$ and $u\in\mathbb{B}$, we define a linear function $(\alpha+\langle u\rangle):T\rightarrow\mathbb{F}$ such that

[TABLE]

Further, $K$ has a weight space decomposition with respect to $T_{K}$:

[TABLE]

Lemma 3.5**.**

Suppose that $\phi$ is a $\mathbb{Z}_{2}$-homogeneous skew-symmetric super-biderivation on $K$. Let $x^{(\alpha)}x^{u}\in K$ such that

[TABLE]

for any $x_{i}x_{i^{\prime}}\in T_{K}$.

Proof. .

The equation by Lemma 3.4, it follows that $\phi(x_{i}x_{i^{\prime}},x_{j}x_{j^{\prime}})=0$ for any $i,~{}j\in\mathrm{J}$ from $[x_{i}x_{i^{\prime}},x_{j}x_{j^{\prime}}]=0$. Note that $\mathrm{p}(x_{l}x_{l^{\prime}})=\overline{0}$ for all $l\in\mathrm{J}$, then all $x^{(\alpha)}x^{u}\in K$, it is clear that

[TABLE]

The proof is completed. ∎

Remark 3.6**.**

Due to Lemma 3.5, we can find that any $\mathbb{Z}_{2}$-homogeneous skew-symmetric super-biderivation on $K$ is an even bilinear map. Since $\phi(x_{i}x_{i^{\prime}},x^{(\alpha)}x^{u})$ and $x^{(\alpha)}x^{u}$ have the same $\mathbb{Z}_{2}$-degree. Then the $\mathbb{Z}_{2}$-degree of $\phi$ is even.

Lemma 3.7**.**

[17]** Let $M=\{x^{(\kappa_{i}\varepsilon_{i})}|0\leq\kappa_{i}\leq\pi_{i},i\in\mathrm{I}_{0}\}$ and $N=\{x_{i}|i\in\mathrm{I}_{1}\}$. Then $K$ is generated by $M\cup N$.

Lemma 3.8**.**

Let $i\in\mathrm{J}_{0}$, $j\in\mathrm{I}_{1}$ and $q_{i}\in\mathbb{N}$, $1\leq q_{i}\leq\pi_{i}$. Then the following statements hold:

• (1)

$K(m,n;\underline{t})_{(0)}=\sum\limits_{0\leq\alpha\leq\pi,~{}\bar{u}\in\mathbb{B}}\mathbb{F}(\mathop{\prod\limits_{l\in\mathrm{J}_{0}}}\limits_{\alpha_{l^{\prime}}-\alpha_{l}\equiv 0~{}(\mathrm{mod}~{}p)}x^{(\alpha_{l}\varepsilon_{l})})x^{(\alpha_{m}\varepsilon_{m})}x^{\bar{u}};$**

• (2)

$K(m,n;\underline{t})_{(q_{i}\varepsilon_{i})}=\sum\limits_{0\leq\alpha\leq\pi,~{}\bar{u}\in\mathbb{B}}\mathbb{F}(\mathop{\prod\limits_{l\in\mathrm{J}_{0}\backslash\{i,{i^{\prime}}\}}}\limits_{\alpha_{{l^{\prime}}}-\alpha_{l}\equiv 0~{}(\mathrm{mod}~{}p)}x^{(\alpha_{l}\varepsilon_{l})})x^{(\alpha_{i}\varepsilon_{i})}x^{(\overline{(\alpha_{i}-q_{i})}\varepsilon_{i^{\prime}})}x^{(\alpha_{m}\varepsilon_{m})}x^{\bar{u}};$**

• (3)

$K(m,n;\underline{t})_{(\langle j\rangle)}=\sum\limits_{0\leq\alpha\leq\pi,~{}\bar{u}\in\mathbb{B}}\mathbb{F}(\mathop{\prod\limits_{l\in\mathrm{J}_{0}}}\limits_{\alpha_{{l^{\prime}}}-\alpha_{l}\equiv 0~{}(\mathrm{mod}~{}p)}x^{(\alpha_{l}\varepsilon_{l})})x^{(\alpha_{m}\varepsilon_{m})}x_{j}x^{\bar{u}};$**

where $i$ and ${i^{\prime}}$ are both in $\bar{u}$ for $i\in\mathrm{J}$, and $\alpha_{l}^{\overline{q}}$ is denoted some integer and $\alpha_{l}^{\overline{q}}\equiv q~{}(\mathrm{mod}~{}p)$.

Proof. .

(1) We first discuss the vector of the same weight with 1 in $K$ with respect to $T_{K}$. Since we have the equation

[TABLE]

For any $l\in\mathrm{J}$, in contrast with equation (3.1), we get that

[TABLE]

Then if $l\in\mathrm{J_{0}},$ it is obvious that is $\alpha_{l^{\prime}}-\alpha_{l}\equiv 0~{}(\mathrm{mod}~{}p)$. If $l\in\mathrm{I_{1}}$, it is obvious that $l$ and $l^{{}^{\prime}}$ are both in $\bar{u}$. It proves that

[TABLE]

(2) Without loss of generality, we choose a fixed element $i\in\mathrm{J}$. For any $l\in\mathrm{J}$, we have the equation

[TABLE]

For any $l\in\mathrm{J}$, by equation(3.1) we have that

[TABLE]

Then we try to discuss the choice of $l\in\mathrm{J}.$ If $l\in\mathrm{J_{0}}\setminus\{i,i^{{}^{\prime}}\}$, it is obvious that $\alpha_{l^{\prime}}-\alpha_{l}\equiv 0~{}(\mathrm{mod}~{}p)$. If $l=i$, it is obvious that $\alpha_{l^{\prime}}-\alpha_{l}\equiv-q_{i}~{}(\mathrm{mod}~{}p)$. If $l\in\mathrm{I_{1}}$, we have that $l$ and $l^{\prime}$ are both in $\bar{u}$. So we proves that

[TABLE]

(3) Without loss of generality, we choose a fixed element $i\in\mathrm{J}$. For any $l\in\mathrm{J}$, we have the equation

[TABLE]

By equation (3.1), for any $l\in\mathrm{J}$, we have that

[TABLE]

Then we try to discuss the choice of $l\in\mathrm{J}.$ If $l\in\mathrm{J_{0}}$, it is obvious that $\alpha_{l^{\prime}}-\alpha_{l}\equiv 0~{}(\mathrm{mod}~{}p)$. If $l\in\mathrm{I_{1}}$, we have that $l$ and $l^{\prime}$ are both in $\bar{u}$. It proves that

[TABLE]

∎

Lemma 3.9**.**

Suppose that $\phi$ is a $\mathbb{Z}_{2}$-homogeneous skew-symmetric super-biderivation on $K$. For any $x_{l}x_{l^{\prime}}\in T_{K}$ and $x^{(q_{m}\varepsilon_{m})}\in M$, where $0\leq q_{m}\leq\pi_{m}$, we have

[TABLE]

Proof. .

When $q_{m}=0$, by Lemma 3.4, it is obvious that $\phi(x_{l}x_{l^{\prime}},1)=0$ for $l\in\mathrm{I}$ from the equation $\langle x_{l}x_{l^{\prime}},1\rangle=0$. When $q_{m}\neq 0$, we have that

[TABLE]

Hence, we have that

[TABLE]

The proof is completed. ∎

Lemma 3.10**.**

Suppose that $\phi$ is a $\mathbb{Z}_{2}$-homogeneous skew-symmetric super-biderivation on $K$. For any $i\in\mathrm{J}$ and $x_{l}x_{l^{\prime}}\in T_{K}$, there is an element $\lambda_{i}\in\mathbb{F}$ such that

[TABLE]

where $\lambda_{i}$ is dependent on the second component.

Proof. .

Without loss of generality, we choose a fixed element $i\in\mathrm{J}$. By Lemma 3.4, it is obvious that $\phi(x_{l}x_{l^{\prime}},x_{i})=0$ for $l\in\mathrm{J}\setminus\{i,{i^{\prime}}\}$ from $\langle x_{l}x_{l^{\prime}},x_{i}\rangle=0$. So we only need to discuss the case with the condition $l=i$.

When $q_{i}=1$, by Lemma 3.4 (2), we can suppose that

[TABLE]

It is obvious that

[TABLE]

By computing the equation, we find that $a(\alpha,\bar{u})=0$ if $\alpha_{m}>0$. Putting $l\in\mathrm{J}\backslash\{i,i^{\prime}\}$, we have that

[TABLE]

By computing the equation, we find that $a(\alpha,\bar{u})=0$ if $\alpha_{l}>0$ for $l\in\mathrm{J}_{0}\backslash\{i,i^{\prime}\}$ or $|\bar{u}|>0$. Then we can suppose that

[TABLE]

Since $\mathrm{p}(x_{i}x_{i^{\prime}})+\mathrm{p}(x_{i})=\overline{0}$ for any $i\in\mathrm{I}_{0}$, by Lemma 3.2, we have

[TABLE]

By computing the equation, we find that $a(\alpha)=0$ if $\overline{\alpha_{i}-1}>0$. Hence we get that $\alpha_{i}=1$. Let $\lambda_{i}=a(\varepsilon_{i})$. From what has been discussed above, for any $i\in\mathrm{J}_{0}$ we have that

[TABLE]

where $\lambda_{i}$ is dependent on the second component.

Similarly, we choose a fixed element $j\in\mathrm{I}_{1}$. By Lemma 3.8 (3), we can suppose that

[TABLE]

where $a(\alpha,\bar{u},j)\in\mathbb{F}$. By the definition of the shew-symmetric super-biderivation, we have

[TABLE]

By computing the equation, we find that $a(\alpha,\bar{u},j)=0$ if $\alpha_{m}>0$. For $k\in\mathrm{J}\backslash\{j,j^{\prime}\}$, it is obvious that

[TABLE]

Putting $k\in\mathrm{I}_{1}$, we can deduce $a(\alpha,\bar{u},j)=0$ if $|\bar{u}|>0$. Putting $k\in\mathrm{J}_{0}$, we have that $a(\alpha,\bar{u},j)=0$ if $\alpha_{k}>0$. Let $a(0,0,j)=\lambda_{j}$. Hence, for any $j\in\mathrm{I}_{1}$, we have that

[TABLE]

The proof is completed. ∎

Lemma 3.11**.**

Suppose that $\phi$ is a $\mathbb{Z}_{2}$-homogenous skew-symmetric super-biderivation on $K$. For any $x^{(q_{i}\varepsilon_{i})}\in M$, where $1\leq q_{i}\leq\pi_{i}$, $i\in\mathrm{J}_{0}$, there is an element $\lambda_{i}\in\mathbb{F}$ such that

[TABLE]

Proof. .

Without loss of generality, we choose a fixed element $i\in\mathrm{I}_{0}$. By Lemma 3.4, it is obvious that $\phi(x_{l}x_{l^{\prime}},x^{(q_{i}\varepsilon_{i})})=0$ for $l\in\mathrm{J}\setminus\{i,i^{\prime}\}$ from $\langle x_{l}x_{l^{\prime}},x^{(q_{i}\varepsilon_{i})}\rangle=0$. So we only need to consider the condition that $l=i$, it is clear that $\langle x_{i}x_{i^{\prime}},x^{(q_{i}\varepsilon_{i})}\rangle=-q_{i}x^{(q_{i}\varepsilon_{i})}$. If $q_{i}>1$, by Lemma 3.1 and 3.10, we have

[TABLE]

Because of $\mathcal{C}_{K_{-1}}(K)=\{f\in K|\langle f,x_{i}\rangle=0,\forall~{}i\in\mathrm{I}\}=K_{-2}=\mathbb{F}1$, the equation (3.2) implies that

[TABLE]

where $\lambda_{i}$ is denoted in Lemma 3.10 and $b\in\mathbb{F}$. Since $\phi(x_{i}x_{i^{\prime}},x^{(q_{i}\varepsilon_{i})})\in K(m,n;\underline{t})_{(q_{i}\varepsilon_{i})}$ by Lemma 3.5. It is easily seen from Lemma 3.8 (2) that $K_{-2}\cap K_{(q_{i}\varepsilon_{i})}=\emptyset$ for $q_{i}>1$. So $b=0$ and

[TABLE]

where $\lambda_{i}$ is dependent on the second component. ∎

Lemma 3.12**.**

Suppose that $\phi$ is a $\mathbb{Z}_{2}$-homogenous skew-symmetric super-biderivation on $K$. For any $x^{(q_{m}\varepsilon_{m})}\in M$, where $0\leq q_{m}\leq\pi_{m}$, there is an element $\lambda\in\mathbb{F}$ such that

[TABLE]

Proof. .

When $q_{m}=1$, we suppose that

[TABLE]

where $c_{(\alpha,u)}\in\mathbb{F}$. For $k\in\mathrm{J_{0}}$, by the definition of the skew-symmetric super-biderivation, we have the equation

[TABLE]

By computing the equation, we find that $c_{(\alpha,u)}=0$ if $\alpha_{m}-\varepsilon_{m}\geq 0$. We suppose that

[TABLE]

where $\widehat{m}$ represents an m-tuple with 0 as the m-th entry. For $k\in\mathrm{J}$, by the definition of the skew-symmetric biderivation, we have the equation

[TABLE]

By computing the equation, we find that $c_{\alpha_{(\widehat{m},u)}}=0$ if $\alpha_{\widehat{m}}-\varepsilon_{k^{\prime}}>0$ or $|u|>0$. Then we can suppose that

[TABLE]

Set $\lambda=\frac{c_{0}}{2}$, then we can get that

[TABLE]

When $q_{m}\geq 2$, we suppose that

[TABLE]

By Lemma 3.1 and the conclusion of the case $q_{m}=1$, we have the equation

[TABLE]

By computing the equation, we find that $c_{(\alpha,u)}=0$ if $\alpha-(q_{m}-1)\varepsilon_{m}\neq 0$ or $|u|>0$. And $c_{(q_{m}-1)}=2\lambda$. We suppose that

[TABLE]

For any $i\in\mathrm{J_{0}}$, by Lemma 3.1, we have the equation

[TABLE]

Since $\mathcal{C}_{K_{-1}}(K)=K_{-2}$, we have that

[TABLE]

Then we have $c_{\alpha_{\widehat{m}}}=0$ for $\alpha_{\widehat{m}}>0$ and $\lambda=\lambda_{i}$ for $i\in\mathrm{J_{0}}$. Then we can get that

[TABLE]

Utilizing the definition of the skew-symmetry biderivation, by Lemma 3.4, we have that

[TABLE]

It is obvious that $c_{0}=0$ for $p>2$. So we can get that

[TABLE]

The proof is complete. ∎

Remark 3.13**.**

We claim that $\lambda_{1}=\cdots=\lambda_{m}=\cdots=\lambda_{m+n}$. Choose two mutually different elements $i,j\in\mathrm{J}$. Since the characteristic $p>3$, there are two positive integers $q_{i}$ and $q_{m}$, which are greater than 1 and are neither congruent to 0 modulo $p$, such that we have

[TABLE]

By direct calculation, it is easily seen that $\lambda_{i}=\lambda$ for any $i\in\mathrm{I}$. Set $\lambda:=\lambda_{1}=\cdots=\lambda_{m}=\lambda_{m+1}=\cdots=\lambda_{m+n}$. Then we can conclude that for any $x^{(q_{i}\varepsilon_{i})}\in M$, $1\leq q_{i}\leq\pi_{i}$ and $x_{l}x_{l^{\prime}}\in T$, there is an element $\lambda\in\mathbb{F}$ such that

[TABLE]

where $\lambda$ depends on neither $x^{(q_{i}\varepsilon_{i})}$ nor $x_{l}x_{l^{\prime}}$.

Theorem 3.14**.**

Let $K$ be the contact Lie superalgebra $K(m,n;\underline{t})$ over the prime field $\mathbb{F}$ of the characteristic $p>3$, where $m,~{}n\in\mathbb{N}+1$ and $\underline{t}=(t_{1},t_{2},\ldots,t_{m})$ is an $m$-tuple of positive integers. Then

[TABLE]

Proof. .

Suppose that $\phi$ is a skew-symmetric super-biderivation on $K$. By Lemmas 3.9 and 3.10, there is an element $\lambda\in\mathbb{F}$ such that $\phi(x_{i}x_{i^{\prime}},x_{i})=\lambda\langle x_{i}x_{i^{\prime}},x_{i}\rangle$ for all $i\in\mathrm{J}$. For any $x^{(\alpha)}x^{u},x^{(\beta)}x^{v}\in K$ and $x_{l}x_{l^{\prime}}\in T_{k}$, by Lemma 3.1 and Remark 3.6, we have the equation

[TABLE]

Since $\mathcal{C}_{K_{-1}}(K)=K_{-2}$, we have that

[TABLE]

where $\lambda$ is denoted in Remark 3.13 and $b\in\mathbb{F}$. By Lemma 3.1 and Remark 3.6, we have

[TABLE]

Then $b=0$. Hence, $\phi(x^{(\alpha)}x^{u},x^{(\beta)}x^{v})=\lambda\langle x^{(\alpha)}x^{u},x^{(\beta)}x^{v}\rangle$  for any $x^{(\alpha)}x^{u},x^{(\beta)}x^{v}\in K$ and $\phi$ is an inner super-biderivation. ∎

Acknowledgements The authors would like to thank the referee for valuable comments and suggestions on this article.