#  On structures of BiHom-Superdialgebras and their derivations

Nil MANSUROĞLU 1, Bouzid Mosbahi 2 1Kırşehir Ahi Evran University, Faculty of
Arts and Science, Department of Mathematics, 40100 Kırşehir, Turkey
2Department of Mathematics, Faculty of Sciences, University of Sfax, Sfax,
Tunisia<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

BiHom-superdialgebras are clear generalization of Hom-superdialgebras. The
purpose of this note is to describe and to survey structures of BiHom-
superdialgebras. Then we derive derivations of BiHom-superdialgebras.

###### Key words and phrases:

BiHom-superdialgebras, BiHom-associative superalgebra, Hom-superdialgebra,
Derivation.

###### 2010 Mathematics Subject Classification:

16Z05, 16D70, 17A60, 17B05, 17B40

##  1\.  Introduction

The associative superdialgebras are also known as diassociative superalgebras.
In 2015, C. Wang, Q. Zhang and Z. Weiz described them as a generalization of
associative superalgebras. These authors developed many results on associative
superdialgebra in the paper [2]. As a generalization of associative
superalgebras, they have two associative products and also they comply three
other statements. On the other hand, many results of superdialgebras were
investigated in [4, 5, 6]. There are many application areas of these algebras,
for instance, physics, classical geometry and non-commutative geometry. For
more details (see [3, 7, 8, 9, 10, 11]). The principal beginning point of this
paper is based on the concept of BiHom-dialgebras introduced by Zahari and
Bakayoko in [1]. In this paper, the authors studied BiHom-associative
dialgebras and they gave the central extensions and the classifications of
BiHom-associative dialgebras with dimension $n\geq 4$. By following their
approach in [1], we present the concepts on BiHom-superdialgebras. Moreover,
we state some main properties on such algebras.

##  2\.  Structure of BiHom-superdialgebras

By using some definitions on superdialgebras and Hom-dialgebras in [6, 12], we
introduce some structure of BiHom-superdialgebras. Furthermore, we survey some
important consequences on BiHom-superdialgebras which are similar to some
statements on Hom-superdialgebras [1].

######  Definition 2.1.

Let $H$ be a superspace. If two even bilinear mapings $\dashv,\vdash:H\times
H\longrightarrow H$ satisfy the next statements

$\displaystyle\text{(i) }\quad p\vdash(q\dashv r)=(p\vdash q)\dashv r,$
$\displaystyle\text{(ii) }\quad p\dashv(q\dashv r)=(p\dashv q)\dashv
r=p\dashv(q\vdash r),$ $\displaystyle\text{(iii)}\quad p\vdash(q\vdash
r)=(p\vdash q)\vdash r=(p\dashv q)\vdash r$

for every homogeneous elements $p,q,r\in H$, then a 3-tuple
$(H,\dashv,\vdash)$ is called a superdialgebra.

######  Definition 2.2.

Given a superspace $H$. If two even bilinear mapings $\dashv,\vdash:H\times
H\longrightarrow H$ and an even superspace homomorphism
$\alpha:H\longrightarrow H$ satisfy the next properties

$\displaystyle\text{(i) }\quad\alpha(p\dashv
q)=\alpha(p)\dashv\alpha(q),\quad\alpha(p\vdash q)=\alpha(p)\vdash\alpha(q),$
$\displaystyle\text{(ii) }\quad\alpha(p)\dashv(q\dashv r)=(p\dashv
q)\dashv\alpha(r)=\alpha(p)\dashv(q\vdash r),$ $\displaystyle\text{(iii)
}\quad\alpha(p)\vdash(q\vdash r)=(p\vdash q)\vdash\alpha(r)=(p\dashv
q)\vdash\alpha(r),$ $\displaystyle\text{(iv) }\quad\alpha(p)\vdash(q\dashv
r)=(p\vdash q)\dashv\alpha(r)$

for every homogeneous elements $p,q,r\in H,$ then a tuple
$(H,\dashv,\vdash\alpha)$ is said to be a Hom-superdialgebra.

######  Definition 2.3.

Given a superspace $H$. If an even bilinear map $\cdot:H\times
H\longrightarrow H$ and two even superspace homomorphisms
$\alpha,\epsilon:H\longrightarrow H$ hold the next properties

$\displaystyle\text{(i)}\quad\alpha\cdot\epsilon=\epsilon\cdot\alpha,$
$\displaystyle\text{(ii)}\quad\alpha(p\cdot
q)=\alpha(p)\cdot\alpha(q),\>\epsilon(p\cdot q)=\epsilon(p)\cdot\epsilon(q),$
$\displaystyle\text{(iii)}\quad\alpha(p)\cdot(q\cdot r)=(p\cdot
q)\cdot\epsilon(r)$

for every homogeneous elements $p,q,r\in H,$ then a 4-tuple
$(H,\cdot,\alpha,\epsilon)$ is said to be a BiHom-associative superalgebra.

######  Definition 2.4.

Given a superspace $H$. If two even bilinear mapings $\dashv,\vdash:H\times
H\longrightarrow H$ and two even superspace homomorphisms
$\alpha,\epsilon:H\longrightarrow H$ satisfy the following axioms

$\displaystyle\text{(i)}\quad\alpha\circ\epsilon=\epsilon\circ\alpha,$
$\displaystyle\text{(ii)}\quad\alpha(p\dashv
q)=\alpha(p)\dashv\alpha(q),\quad\alpha(p\vdash q)=\alpha(p)\vdash\alpha(q),$
$\displaystyle\text{(iii)}\quad\epsilon(p\dashv
q)=\epsilon(p)\dashv\epsilon(q),\quad\epsilon(p\vdash
q)=\epsilon(p)\vdash\epsilon(q),$ $\displaystyle\text{(iv)}\quad(p\dashv
q)\dashv\epsilon(r)=\alpha(p)\dashv(q\dashv r),$
$\displaystyle\text{(v)}\quad(p\vdash
q)\dashv\epsilon(r)=\alpha(p)\vdash(q\dashv r),$
$\displaystyle\text{(vi)}\quad(p\dashv
q)\vdash\epsilon(r)=\alpha(p)\dashv(q\vdash r),$
$\displaystyle\text{(vii)}\quad(p\vdash
q)\vdash\epsilon(r)=\alpha(p)\vdash(q\vdash r)$

for every homogeneous elements $p,q,r\in H$, then a 5-tuple
$(H,\dashv,\vdash,\alpha,\epsilon)$ is called a BiHom-superdialgebra.

Here $\alpha$ and $\epsilon$ are called structure maps of $H$. Since the maps
$\alpha,\epsilon$ commute, for any integer $m,n$, we denote by

$\alpha^{m}\epsilon^{n}=\underbrace{\alpha\circ\ldots\circ\alpha}_{(m-\text{times})}\circ\underbrace{\epsilon\circ\ldots\circ\epsilon}_{(n-\text{times})}.$

Particularly, $\alpha^{0}\epsilon^{0}=Id_{H}$,
$\alpha^{1}\epsilon^{1}=\alpha\epsilon$ and $\alpha^{-m}\epsilon^{-n}$ is the
inverse of $\alpha^{m}\epsilon^{n}.$

######  Definition 2.5.

Given a BiHom-superdialgebra $(H,\dashv,\vdash,\alpha,\epsilon)$, when
$\alpha$ and $\epsilon$ are bijection, then
$(H,\dashv,\vdash,\alpha,\epsilon)$ is called a regular BiHom-superdialgebra.

######  Remark 2.1.

If $p\vdash q=p\dashv q=p\cdot q$ for any $p,q\in H$, then each BiHom-
associative superalgebra $H$ is a BiHom-superdialgebra.

######  Example 2.6.

By taking $\epsilon=\alpha$, any Hom-superdialgebra becomes a BiHom-
superdialgebra. By setting $\alpha=\epsilon=id$, any superdialgebra is a
BiHom-superdialgebra.

######  Definition 2.7.

Given two BiHom-superdialgebras $(H_{1},\dashv,\vdash,\alpha,\epsilon)$ and
$(H_{2},\dashv^{\prime},\vdash^{\prime},\alpha^{\prime},\epsilon^{\prime})$
and let $g:H_{1}\longrightarrow H_{2}$ be an even homomorphism. $g$ is said to
be a morphism of BiHom-superdialgebras if

$g\circ\alpha=\alpha^{\prime}\circ g,\quad
g\circ\epsilon=\epsilon^{\prime}\circ g$

and

$g(p)\dashv^{\prime}g(q)=g(p\dashv q),\quad g(p)\vdash^{\prime}g(q)=g(p\vdash
q)$

for every $p,q\in H_{1}$.

######  Definition 2.8.

Let $\alpha$ and $\epsilon$ be two morphisms. A BiHom-superdialgebra
$(H,\dashv,\vdash,\alpha,\epsilon)$ is said to be a multiplicative BiHom-
superdialgebra.

######  Definition 2.9.

if $\alpha$ and $\epsilon$ are bijective, we say that
$(H,\dashv,\vdash,\alpha,\epsilon)$ is a regular BiHom-superdialgebra.

######  Theorem 2.10.

Let $(H,\dashv,\vdash,\alpha,\epsilon)$ be a BiHom-superdialgebra and
$\alpha^{\prime},\epsilon^{\prime}:H\longrightarrow H$ be two commuting
endomorphisms. Consider
$\triangleleft=\dashv\circ(\alpha^{\prime}\otimes\epsilon^{\prime})$ and
$\triangleright=\vdash\circ(\alpha^{\prime}\otimes\epsilon^{\prime})$. Then,
$H_{(\alpha^{\prime},\epsilon^{\prime})}=(H,\triangleleft,\triangleright,\alpha\alpha^{\prime},\epsilon\epsilon^{\prime})$
is a BiHom-superdialgebra.

###### Proof.

Here, we will only prove one of conditions in Definition 2.4. For each
$p,q,r\in H$, we obtain

$\displaystyle(p\triangleleft
q)\triangleleft\epsilon\epsilon^{\prime}(r)-\alpha\alpha^{\prime}(p)\triangleleft(q\triangleright
r)$
$\displaystyle=\alpha^{\prime}(\alpha^{\prime}(p)\dashv\epsilon^{\prime}(q))\dashv\epsilon^{\prime}\epsilon\epsilon^{\prime}(r)-\alpha^{\prime}\alpha\alpha^{\prime}(p)\dashv\epsilon^{\prime}(\alpha^{\prime}(q)\vdash\epsilon^{\prime}(r))$
$\displaystyle=(\alpha^{\prime}\alpha^{\prime}(p)\dashv\alpha^{\prime}\epsilon^{\prime}(q))\dashv\epsilon(\epsilon^{\prime}\epsilon^{\prime})(r)-\alpha\alpha^{\prime}\alpha^{\prime}(p)\dashv(\alpha^{\prime}\epsilon^{\prime}(q)\vdash\epsilon^{\prime}\epsilon^{\prime}(r))$

Since $H$ is a BiHom-superdialgebra, by Using Definition 2.4 (v), we obtain

$\displaystyle(p\triangleleft
q)\triangleleft\epsilon\epsilon^{\prime}(r)-\alpha\alpha^{\prime}(p)\triangleleft(q\triangleright
r)$
$\displaystyle=\alpha(\alpha^{\prime}\alpha^{\prime}(p))\dashv(\alpha^{\prime}\epsilon^{\prime}(q)\vdash\epsilon^{\prime}\epsilon^{\prime}(r))-\alpha\alpha^{\prime}\alpha^{\prime}(p)\dashv(\alpha^{\prime}\epsilon^{\prime}(q)\vdash\epsilon^{\prime}\epsilon^{\prime}(r)).$

The left side is equal to zero, therefore, the proof of the part (iv) of
Definition 2.4 is completed. By using the similar argument to other axioms, we
prove the theorem. ∎

Consequently, the theorem derives the following sequence of corollaries.

######  Corollary 2.11.

Given the multiplicative BiHom-superdialgebra
$(H,\dashv,\vdash,\alpha,\epsilon)$. Then,
$(H,\dashv\circ(\alpha^{n}\otimes\epsilon^{n}),\vdash\circ(\alpha^{n}\otimes\epsilon^{n}),\alpha^{n+1},\epsilon^{n+1})$
is a multiplicative BiHom-superdialgebra as well.

###### Proof.

It is clearly sufficient to take $\alpha^{\prime}=\alpha^{n}$ and
$\epsilon^{\prime}=\epsilon^{n}$ in Theorem 2.10. Thus, this proves the
corollary. ∎

######  Corollary 2.12.

Let $(H,\dashv,\vdash,\alpha)$ be a multiplicative Hom-superdialgebra and
$\epsilon:H\longrightarrow H$ be an endomorphism of $H$. Then,
$(H,\dashv\circ(\alpha\otimes\epsilon),\vdash\circ(\alpha\otimes\epsilon),\alpha^{2},\epsilon)$
is also a BiHom-superdialgebra.

###### Proof.

By taking $\alpha^{\prime}=\alpha$, $\beta=Id_{H}$ and
$\epsilon^{\prime}=\epsilon$ in Theorem 2.10, we prove the corollary. ∎

######  Corollary 2.13.

If $(H,\dashv,\vdash,\alpha,\epsilon)$ is a regular BiHom-superdialgebra, then
$(H,\dashv\circ(\alpha^{-1}\otimes\epsilon^{-1}),\vdash\circ(\alpha^{-1}\otimes\epsilon^{-1}))$
is a superdialgebra.

###### Proof.

The claim follows in the case where $\alpha^{\prime}=\alpha^{-1}$ and
$\epsilon^{\prime}=\epsilon^{-1}$ of Theorem 2.10. ∎

######  Corollary 2.14.

Given a superdialgebra $(H,\dashv,\vdash)$ and two commuting homomorphisms
$\alpha,\epsilon:H\longrightarrow H$. Then,
$(H,\dashv\circ(\alpha\otimes\epsilon),\vdash\circ(\alpha\otimes\epsilon),\alpha,\epsilon)$
is a BiHom-superdialgebra.

###### Proof.

By taking $\alpha=\epsilon=Id_{H}$ and replacing $\alpha^{\prime}$ by $\alpha$
and $\epsilon^{\prime}$ by $\epsilon$ in Theorem 2.10, we obtain the result as
required. ∎

######  Definition 2.15.

Let $(H,\dashv,\vdash,\alpha,\epsilon)$ be a BiHom-superdialgebra and $S$ a
subset of $H$. If $S$ is stable under $\alpha$ and $\epsilon$ and $p\dashv
q,p\vdash q\in S$ for every $p,q\in S$, $S$ is called a BiHom-supersubalgebra
of $H$.

######  Definition 2.16.

Given a BiHom-superdialgebra $(H,\dashv,\vdash,\alpha,\epsilon)$ and a BiHom-
supersubalgebra $T$ of $H$. For every $p\in T,q\in H$, we say $T$ is a left
BiHom-ideal of $H$ if we have $p\dashv q,p\vdash q\in T$. If $q\dashv p$ and
$q\vdash p$ are in $T$, then $T$ is called a right BiHom-ideal of $H$ and $T$
is said to be a two sided BiHom-ideal of $H$ if $p\dashv q\;,\;q\dashv
p,\;p\vdash q\;,\;q\vdash p\in T$.

######  Example 2.17.

Clearly, $T=\\{0\\}$ and $T=H$ are two-sided BiHom-ideals. If
$\theta:H_{1}\longrightarrow H_{2}$ is a morphism of BiHom-superdialgebras,
then the kernel $Ker\theta$ is a two sided BiHom-ideal in $H_{1}$ and the
image $Im\theta$ is a BiHom-supersubalgebra of $H_{2}$. Moreover, if $T_{1}$
and $T_{2}$ are two sided BiHom-ideals of $H$, then $T_{1}+T_{2}$ is two sided
BiHom-ideal as well.

######  Proposition 2.18.

Given a BiHom-superdialgebra $(H,\dashv,\vdash,\alpha,\epsilon)$ and a two
sided BiHom-ideal $T$ of $(H,\dashv,\vdash,\alpha,\epsilon)$. Then,
$(H/T,\overline{\dashv},\overline{\vdash},\overline{\alpha},\overline{\epsilon})$
is a BiHom-superdialgebra where

$\overline{\alpha}(\overline{p})=\overline{\alpha(p)},\quad\overline{\epsilon}(\overline{p})=\overline{\epsilon(p)}$

and

$\overline{p}\;\overline{\dashv}\;\overline{q}=\overline{p\dashv
q},\quad\overline{p}\;\overline{\vdash}\;\overline{q}=\overline{p\vdash q}$

for all $\overline{p},\overline{q}\in H/T$.

###### Proof.

Here, We only prove right superassociativity. Similarly, the others are
proved. For every $\overline{p},\overline{q},\overline{r}\in H/T$, we obtain

$\displaystyle(\overline{p}\overline{\vdash}\overline{q})\overline{\vdash}\overline{\epsilon}(\overline{r})-\overline{\alpha}(\overline{p})\overline{\vdash}(\overline{q}\overline{\vdash}\overline{r})$
$\displaystyle=\overline{(p\vdash
q)}\overline{\vdash}\overline{\epsilon(r)}-\overline{\alpha(p)}\overline{\vdash}\overline{(q\vdash
r)}$ $\displaystyle=\overline{(p\vdash
q)\vdash\epsilon(r)-\alpha(p)\vdash(q\vdash r)}\quad\text{(by Definition
\ref{def2} (vii))}$ $\displaystyle=\overline{0}.$

Then,
$(H/T,\overline{\dashv},\overline{\vdash},\overline{\alpha},\overline{\epsilon})$
is BiHom-superdialgebra. ∎

##  3\.  Derivation of BiHom-superdialgebra

Some significant concepts corresponding to derivations of BiHom-
superdialgebras will be presented in this section.

######  Definition 3.1.

Let $(H,\mu,\alpha,\epsilon)$ be a BiHom-superalgebra. A linear maping
$d:H\longrightarrow H$ is said to be an $\alpha^{m}\epsilon^{n}$-derivation of
$(H,\mu,\alpha,\epsilon)$ if it holds

$\displaystyle\alpha\circ d$ $\displaystyle=d\circ\alpha\quad\text{and}\quad
d\circ\epsilon=\epsilon\circ d,$ $\displaystyle d\circ\mu(p,q)$
$\displaystyle=\mu(d(p),\alpha^{m}\epsilon^{n}(q))+(-1)^{|p||d|}\mu(\alpha^{m}\epsilon^{n}(p),d(q))$

for every $p,q\in H$ and for each integers $m,n$.

######  Definition 3.2.

A linear maping $d:H\longrightarrow H$ on a BiHom-superalgebra is called a
differential if

$d(p.q)=dp.q+(-1)^{|p||d|}p.dq,\quad\text{for every}\quad p,q\in H,\quad
d^{2}=0$

and

$d\circ\alpha=\alpha\circ d\quad d\circ\epsilon=\epsilon\circ d.$

######  Proposition 3.3.

Given a differential BiHom-superalgebra $(H,.,\alpha,\epsilon,d)$ and the
products $\dashv$ and $\vdash$ on $H$ by

$\displaystyle p\dashv q=\alpha(p).dq\quad and\quad p\vdash q=dp.\epsilon(q).$

Then $(H,\dashv,\vdash,\alpha,\epsilon)$ is a BiHom-superdialgebra where
$\alpha$ and $\epsilon$ are idempotant structure maps.

###### Proof.

It is easy to show the BiHom-superassociativity. We have

$\displaystyle\alpha(p)\dashv(q\dashv r)$
$\displaystyle=\alpha(p)\dashv(\alpha(q).d(r))$
$\displaystyle=\alpha(\alpha(p)).d(\alpha(q).d(r))$
$\displaystyle=\alpha(\alpha(p)).(d(\alpha(q)).d(r)+\alpha(q)d^{2}(r))$
$\displaystyle=\alpha(\alpha(p)).(d(\alpha(q)).d(r))$ $\displaystyle=(p\dashv
q)\dashv\epsilon(r)$

for any $p,q,r\in H$. By using a similar way, we obtain

$\displaystyle\alpha(p)\vdash(q\vdash r)$
$\displaystyle=\alpha(p)\vdash(d(q).\epsilon(r))$
$\displaystyle=d(\alpha(p)).\epsilon(d(q).\epsilon(r))$
$\displaystyle=\alpha(d(p)).(\epsilon(d(q)).\epsilon^{2}(r))$
$\displaystyle=(d(p).\epsilon(d(q))).\epsilon^{2}(r)$ $\displaystyle=(p\vdash
q)\vdash\epsilon(r).$

∎

######  Proposition 3.4.

Let $(H,\dashv,\vdash,\alpha,\epsilon)$ be a BiHom-superdialgebra such that
$\alpha(r)=\epsilon(r)$ for all $r\in H$. For any $p,q\in H$, we describe the
linear maping $ad^{(\alpha,\epsilon)}_{r}:H\longrightarrow H$ by

$ad^{(\alpha,\epsilon)}_{r}(p)=p\dashv\epsilon(p)-(-1)^{|p||r|}\alpha(r)\vdash
x.$

Then $ad_{r}^{(\alpha,\epsilon)}$ is a derivation of $H$.

###### Proof.

We have

$\displaystyle ad^{(\alpha,\epsilon)}_{r}(p)\dashv q+p\dashv
ad^{(\alpha,\epsilon)}_{r}(q)$
$\displaystyle=(p\dashv\epsilon(r)-(-1)^{|p||r|}\alpha(r)\vdash p)\dashv
q+p\dashv(q\dashv\epsilon(r)-(-1)^{|q||r|}\alpha(r)\vdash q)$
$\displaystyle=(p\dashv\epsilon(r))\dashv q-(-1)^{|p||r|}(\alpha(r)\vdash
p)\dashv q$
$\displaystyle+p\dashv(q\dashv\epsilon(r))-(-1)^{|q||p|}p\dashv(\alpha(r)\vdash
q)$ $\displaystyle=p\dashv(\epsilon(r)\dashv
q)-(-1)^{|p||r|}\alpha(r)\vdash(p\dashv q)$ $\displaystyle+(p\dashv
q)\dashv\epsilon(r)-(-1)^{|q||r|}p\dashv(\alpha(r)\vdash q)$
$\displaystyle=(p\dashv
q)\dashv\epsilon(r)-(-1)^{|p||r|}\alpha(r)\vdash(p\dashv q)$
$\displaystyle=ad^{(\alpha,\epsilon)}_{r}(p\dashv q).$

As a result, $ad_{r}^{(\alpha,\epsilon)}$ is a derivation. ∎

######  Definition 3.5.

Given a BiHom-superdialgebra $(H,\dashv,\vdash,\alpha,\epsilon)$. A linear
maping $d:H\longrightarrow H$ is called a $\alpha^{m}\epsilon^{n}$-derivation
of $H$, if it holds

$\displaystyle\text{(i)}\quad d\circ\alpha=\alpha\circ d,\epsilon\circ
d=\quad\quad d\circ\epsilon,$ $\displaystyle\text{(ii)}\quad d(p\dashv
q)=\alpha^{m}\epsilon^{n}(p)\dashv
d(q)+(-1)^{|p||d|}d(p)\dashv\alpha^{m}\epsilon^{n}(q),$
$\displaystyle\text{(iii)}\quad d(p\vdash q)=\alpha^{m}\epsilon^{n}(p)\vdash
d(q)+(-1)^{|p||d|}d(p)\vdash\alpha^{m}\epsilon^{n}(q)$

for each $p,q\in H$ and for each integers $m,n$.

The set of all $\alpha^{m}\epsilon^{n}$-derivation of $H$ is represented by
$Der_{\alpha^{m}\epsilon^{n}}(H)$ and the set of all derivation on BiHom-
superdialgebra is represented by

$Der(H)=\bigoplus_{m,n\geq 0}Der_{\alpha^{m}\epsilon^{n}}(H).$

We define, for any $d,d^{\prime}\in Der(H)$, the even bilinear maping
$[,]:Der(H)\times Der(H)\longrightarrow Der(H)$ by

$[d,d^{\prime}]=d\circ d^{\prime}-(-1)^{|d||d^{\prime}|}d^{\prime}\circ d.$

######  Proposition 3.6.

For each $d\in(Der_{\alpha^{m}\epsilon^{n}}(H))_{i}$ and
$d^{\prime}\in(Der_{\alpha^{m}\epsilon^{n}}(H))_{j}$, then

$[d,d^{\prime}]\in Der_{\alpha^{m+s}\epsilon^{n+t}}(H)_{|d|+|d^{\prime}|},$

where $m+s,n+t\geq-1\>\text{and}\>(i,j)\in\mathbb{Z}^{2}_{2}$.

###### Proof.

For each $p,q\in H$, we have

$\displaystyle[d,d^{\prime}](\mu(p,q))$ $\displaystyle=(d\circ
d^{\prime}-(-1)^{|d||d^{\prime}|}d^{\prime}\circ d)(\mu(p,q))$
$\displaystyle=d\circ d^{\prime}\mu(p,q)-(-1)^{|d||d^{\prime}|}d^{\prime}\circ
d\mu(p,q)$
$\displaystyle=d(\mu(d^{\prime}(p),\alpha^{s}\epsilon^{t}(q))+(-1)^{|p||d^{\prime}|}\mu(\alpha^{s}\epsilon^{t}(p),d^{\prime}(q))-(-1)^{|d||d^{\prime}|}d^{\prime}(\mu(d(p),\alpha^{k}\epsilon^{l}(q))$
$\displaystyle+(-1)^{|d||p|}\mu(\alpha^{s}\epsilon^{t}(p),d(q)))$
$\displaystyle=\mu(d\circ
d^{\prime}(p),\alpha^{m}\epsilon^{n}\alpha^{s}\epsilon^{t}(q))+(-1)^{|d|(|d^{\prime}|+|p|)}\mu(\alpha^{m}\epsilon^{n}d^{\prime}(p),d(\alpha^{s}\epsilon^{t}(q)))$
$\displaystyle+(-1)^{|p||d^{\prime}|}\mu(d(\alpha^{s}\epsilon^{t}(p),\alpha^{m}\epsilon^{n}d^{\prime}(q)))+(-1)^{|p||d^{\prime}|+|p||d|}\mu(\alpha^{m}\epsilon^{n}\alpha^{s}\epsilon^{t}(p),d\circ
d^{\prime}(q)))$ $\displaystyle-(-1)^{|d||d^{\prime}|}\mu(d^{\prime}\circ
d(p),\alpha^{s}\epsilon^{t}\alpha^{m}\epsilon^{n}(q)))-(-1)^{|p||d^{\prime}|}\mu(\alpha^{s}\epsilon^{t}d(p),d^{\prime}\alpha^{m}\epsilon^{n}(q))$
$\displaystyle-(-1)^{|d||d^{\prime}|+|p||d|}\mu(d^{\prime}(\alpha^{m}\epsilon^{n}(p),\alpha^{s}\epsilon^{t}d(q))-(-1)^{|d||d^{\prime}|+|p||d|+|d||d^{\prime}|}\mu(\alpha^{s}\epsilon^{t}(p)\alpha^{m}\epsilon^{n}(p),d^{\prime}\circ
d(q)).$

Since any two of maps $d,d^{\prime},\alpha,\epsilon$ commute, we have

$\displaystyle d^{\prime}\circ\epsilon^{n}=\epsilon^{n}\circ d^{\prime},\quad
d^{\prime}\circ\alpha^{m}=\alpha^{m}\circ d^{\prime},$ $\displaystyle
d\circ\epsilon^{t}=\epsilon^{t}\circ d,\quad d\circ\alpha^{s}=\alpha^{s}\circ
d.$

Therefore, we have

$[d,d^{\prime}](\mu(p,q))=\mu([d,d^{\prime}](p),\alpha^{m+s}\epsilon^{n+t}(q))+(-1)^{|p|(|d^{\prime}|+|d|)}\mu(\alpha^{m+s}\beta^{n+t}(p),[d,d^{\prime}](q)).$

In addition, it is easy to see that

$\displaystyle[d,d^{\prime}]\circ\alpha$ $\displaystyle=(d\circ
d^{\prime}-(-1)^{|d||d^{\prime}|}d^{\prime}\circ d)\circ\alpha$
$\displaystyle=d\circ
d^{\prime}\circ\alpha-(-1)^{|d||d^{\prime}|}d^{\prime}\circ d\circ\alpha$
$\displaystyle=\alpha\circ d\circ d^{\prime}-(-1)^{|d||d^{\prime}|}\alpha\circ
d^{\prime}\circ d$ $\displaystyle=\alpha\circ[d,d^{\prime}]$

and

$\displaystyle[d,d^{\prime}]\circ\epsilon$ $\displaystyle=(d\circ
d^{\prime}-(-1)^{|d||d^{\prime}|}d^{\prime}\circ d)\circ\epsilon$
$\displaystyle=d\circ
d^{\prime}\circ\epsilon-(-1)^{|d||d^{\prime}|}d^{\prime}\circ d\circ\epsilon$
$\displaystyle=\epsilon\circ d\circ
d^{\prime}-(-1)^{|d||d^{\prime}|}\epsilon\circ d^{\prime}\circ d$
$\displaystyle=\epsilon\circ[d,d^{\prime}]$

which occurs that

$[d,d^{\prime}]\in(Der_{\alpha^{m+s},\epsilon^{n+t}}(D))_{(|d||d^{\prime}|)}$

by taking $\mu=\dashv$ and $\mu=\vdash$ respectively.
Consequently, for each integers $k$ and $l$, represent by

$Der(H)=(\bigoplus_{m,n}Der_{\alpha^{m},\epsilon^{n}}(H))_{0}+(\bigoplus_{m,n}Der_{\alpha^{m},\epsilon^{n}}(H))_{1}.$

∎

######  Definition 3.7.

Let $(H,\dashv,\vdash,\alpha,\epsilon)$ be a BiHom-superdialgebra and
$\gamma,\delta,\lambda$ elements in $\mathbb{C}$. A linear maping
$d:H\longrightarrow H$ is a generalized $\alpha^{m}\epsilon^{n}$-derivation or
a $(\gamma,\delta,\lambda)-(\alpha^{m}\epsilon^{n})$-derivation of $H$ if for
every $p,q\in H$, we have

$\displaystyle\text{(i)}\quad d\circ\alpha=\alpha\circ
d,d\circ\epsilon=\epsilon\circ d,$
$\displaystyle\text{(ii)}\quad\gamma(d(p\dashv
q))=\delta(d(p)\dashv\alpha^{m}\epsilon^{n}(q))+(-1)^{|p||d|}\lambda(\alpha^{m}\epsilon^{n}(p)\dashv
d(q)),$ $\displaystyle\text{(iii)}\quad\gamma(d(p\vdash
q))=\delta(d(p)\vdash\alpha^{m}\epsilon^{n}(q))+(-1)^{|p||d|}\lambda(\alpha^{m}\epsilon^{n}(p)\vdash
d(q)).$

Denote by $Der^{\gamma,\delta,\lambda}(H)$ the set of
$(\gamma,\delta,\lambda)-(\alpha^{m}\epsilon^{n})$-derivations of the BiHom-
superdialgebra $(H,\dashv,\vdash,\alpha,\epsilon)$.

######  Proposition 3.8.

Let $d\in Der_{(\alpha^{m},\epsilon^{n})}^{(\gamma,\delta,\lambda)}(H)\quad$
and $\quad d^{\prime}\in
Der_{(\alpha^{m^{\prime}},\epsilon^{n^{\prime}})}^{(\gamma^{\prime},\delta^{\prime},\lambda^{\prime})}(H)$.
Then

$[d,d^{\prime}]\in
Der_{(\alpha^{m+m^{\prime}},\epsilon^{n+n^{\prime}})}^{(\gamma+\gamma^{\prime},\delta+\delta^{\prime},\lambda+\lambda^{\prime})}(H).$

###### Proof.

Applying the similar way to Proposition 3.6 completes the proof. ∎

######  Definition 3.9.

Let $(H,\dashv,\vdash,\alpha,\epsilon)$ be a BiHom-superdialgebra. A linear
maping $d:H\longrightarrow H$ is said to be an $\alpha^{k}\epsilon^{l}$-quasi
derivation of the BiHom-superdialgebra $(H,\dashv,\vdash,\alpha,\epsilon)$ if
there is a derivation $d^{\prime}:H\longrightarrow H$

$\displaystyle\text{(i)}\quad d\circ\alpha=\alpha\circ d,\epsilon\circ d=\quad
d\circ\epsilon,$ $\displaystyle\text{(ii)}\quad
d^{\prime}\circ\alpha=\alpha\circ d^{\prime},\epsilon\circ d^{\prime}=\quad
d^{\prime}\circ\epsilon$ $\displaystyle\text{(iii)}\quad d^{\prime}(p\dashv
q)=d(p)\dashv\alpha^{m}\epsilon^{n}(q)+(-1)^{|p||d|}(\alpha^{m}\epsilon^{n}(p)\dashv
d(q)),$ $\displaystyle\text{(iv)}\quad d^{\prime}(p\vdash
q)=d(p)\vdash\alpha^{m}\epsilon^{n}(q)+(-1)^{|p||d|}(\alpha^{m}\epsilon^{n}(p)\vdash
d(q))$

for all homogeneous elements $p,q\in H$.

Denote by $QDer_{\alpha^{m}\epsilon^{n}}(H)$ the set of
$\alpha^{m}\epsilon^{n}$-quasi derivations of $H$. The set of all Quasi-
derivations of $H$ is given by

$QDer(H)=(\bigoplus_{m,n\geq
0}QDer_{\alpha^{m},\epsilon^{n}}(H))_{\overline{0}}+(\bigoplus_{m,n\geq
0}QDer_{\alpha^{m},\epsilon^{n}}(H))_{\overline{1}}.$

## References

  * 1. Zahari, A., Bakayoko, I. (2023). On BiHom-associative dialgebras. Open Journal of Mathematical Sciences.
  * 2. Chunyue, W. A. N. G., Zhang, Q., Zhu, W. E. İ. (2015). Hom-Leibniz superalgebras and hom-Leibniz poisson superalgebras. Hacettepe Journal of Mathematics and Statistics, 44(5), 1163-1179.
  * 3. Graziani, G., Makhlouf, A., Menini, C., Panaite, F. (2015). BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications. 11, 086.
  * 4. Liu, D. and Hu, N. (2004). Leibniz central extensions on some infinite dimensional Lie algebras, Comm. Alg., 32 (6), 2385-2405.
  * 5. Liu, D. and Hu, N. (2005) Steinberg Leibniz algebras and superalgebras, J. Algebra., 283 (1), 199-221.
  * 6. Liu, D. and Hu, N. (2006) Leibniz superalgebras and central extensions, J. Algebra Appl., 5 (6), 765-780.
  * 7. Loday, J. L. Une version non commutative des alge‘bres de Lie: Les alge‘bres de Leibniz, Enseign. Math., 39 (1) 269-294, 1993.
  * 8. Loday, J.I. (2001). Dialgebras, In Dialgebras and Related Operads. (pp 7-66) Springer, Berlin, Heidelberg.
  * 9. Makhlouf, A., Zahari, A. (2020). Structure and classification of Hom-associative algebras. Acta et Commentationes Universitatis Tartuensis de Mathematica. 24 (1), 79-102.
  * 10. Pozhidaev, A.P. (2008). Dialgebras and related triple systems. Siberian Mathematical Journal. 49 (4), 696-708.
  * 11. Rikhsiboev, I.M., Rakhimov, I.S., Basri, W. (2014). Diassociative algebras and their derivations. Journal of Physics: Conference Series ( Vol 553, No. 1, p 012006) IOP publishing.
  * 12. Yau, D. Envelopping algebras of Hom-Lie algebras, J. Gen. Lie Theory Appl., 2(2), 95-108, 2008.


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#  Restricted cohomology of restricted Lie superalgebras

Jixia Yuan,1,2,111Supported by NNSF of China (11601135), Natural Science
Foundation of Heilongjiang Province of China (QC2017002) and Project funded by
China Postdoctoral Science Foundation(2018M630311) Liangyun
Chen1,222Correspondence<EMAIL_ADDRESS>(L. Chen); Supported by NNSF
of China (11771069 and 12071405) and Yan Cao3,333Supported by NNSF of China
(11801121), Natural Science Foundation of Heilongjiang Province of China
(QC2018006)

1\. School of Mathematical and Statistics, Northeast Normal University
Changchun 130024, China

2\. School of Mathematical Sciences, Heilongjiang University
Harbin 150080, China

3\. Department of Mathematics, Harbin University of Science and Technology
Harbin 150080, China

> Abstract. Suppose the ground field $\mathbb{F}$ is an algebraically closed
> field characteristic of $p>2$. In this paper, we investigate the restricted
> cohomology theory of restricted Lie superalgebras. Algebraic interpretations
> of low dimensional restricted cohomology of restricted Lie superalgebra are
> given. We show that there is a family of restricted model filiform Lie
> superalgebra $L_{p,p}^{\lambda}$ structures parameterized by elements
> $\lambda\in\mathbb{F}^{p}.$ We explicitly describe both the $1$-dimensional
> ordinary and restricted cohomology superspaces of $L_{p,p}^{\lambda}$ with
> coefficients in the $1$-dimensional trivial module and show that these
> superspaces are equal. We also describe the $2$-dimensional ordinary and
> restricted cohomology superspaces of $L_{p,p}^{\lambda}$ with coefficients
> in the $1$-dimensional trivial module and show that these superspaces are
> unequal.
>
>
> Mathematics Subject Classification. 17B30, 17B56.
>
> Keywords. restricted Lie superalgebras, restricted cohomology, restricted
> model filiform Lie superalgebras.

##  1\. Introduction

As a natural generalization of Lie algebras, Lie superalgebras play an
important role in the theoretical physics and mathematics. Let
$L=L_{\bar{0}}\oplus L_{\bar{1}}$ be a restricted Lie superalgebra. Then
$L_{\bar{0}}$ is a restricted Lie algebra and $L_{\bar{1}}$ is a restricted
$L_{\bar{0}}$-module. Therefore, restricted Lie superalgebras play a central
role in the theory of modular Lie superalgebras, just as in the modular Lie
algebra situation.

Cohomology is a very important tool in the research of topology, smooth vector
fields, holomorphic functions, etc. Cohomology theory is from the study of the
topology of Lie groups and vector fields on Lie groups by Cartan. The standard
complex of Lie algebra was originally constructed by Chevalley and Eilenberg
in [6]. Many conclusions on cohomology of Lie algebras were given in [15],
etc. Hochschild first considered the cohomology theory of restricted Lie
algebras in [16]. In dissertation [8], Evans improved on the results in [16]
and obtained a cochain complex that is capable of making computations. In
recent years, restricted cohomology theory of restricted Lie algebras have
aroused the interest of a great many researchers, see [9, 10, 11, 12]. Shortly
after the birth of supersymmetry most basic constructions and results of the
classical theory of Lie algebras are generalized to the case of Lie
superalgebras. Fuks and Leites in [13] calculated the cohomology of the
classical Lie superalgebras with trivial coefficients. In [24], Scheunert and
Zhang introduced in detail the concepts of cohomology of Lie superalgebra. In
[25], Su and Zhang explicitly computed the $1,2$-dimensional cohomology of the
classical Lie superalgebras $\mathfrak{sl}_{m|n}$ and $\mathfrak{osp}_{2|2n}$
with coefficients in the finite dimensional irreducible modules and the Kac
modules. In addition, the works of Boe, Kujawa, Nakano, Bagci, Lehrer,
Poletaeva etc also promoted the development of cohomology theory of Lie
superalgebras, see [1, 5, 20]. The papers of Iwai and Shimada [18], May [22]
and Priddy [23] from the 1960s address the problem of constructing a
resolution for computing the cohomology of restricted graded Lie algebras.
Strictly speaking, these earlier works were written in the context of
$\mathbb{Z}$-graded Lie algebras, also the same methods may apply to Lie
superalgebras by reducing all of the gradings modulo $2$. For a discussion of
this earlier work that is specifically in the context of Lie superalgebras,
see [7]. The resolutions constructed in these earlier works are not entirely
explicit (owing to the inductive nature of the constructions), but the
resolutions are relatively explicit in low degrees. In recent years,
cohomology theory of restricted Lie superalgebras have aroused the interest of
some researchers, see [2].

In the study of the reducibility of the varieties of nilpotent Lie algebras,
Vergne introduced the concept of filiform Lie algebras, see [26]. Since then,
the study of the filiform Lie algebras, especially the model filiform Lie
algebra, has become an important subject. As what happens in the Lie case,
(model) filiform Lie superalgebra is an important subject of Lie superalgebra.
Many conclusions on cohomology of the model filiform Lie superalgebra were
given in [14, 4, 21].

In this paper, we are interested in the cohomology theory of restricted Lie
superalgebras. We give algebraic interpretations of restricted cohomology of
restricted Lie superalgebra $L$: The superspace of all restricted outer
superderivations is equal to $1$-dimensional restricted cohomology of $L$ with
coefficients in the adjoint module; Let superspaces $M,N$ be two restricted
$L$-modules, then equivalence classes of extensions of $N$ by $M$ are in one
to one correspondence with $1$-dimensional restricted cohomology of $L$ with
coefficients in the module $\mathrm{Hom}_{\mathbb{F}}(N,M)$; Equivalence
classes of restricted central extensions of $L$ are in one to one
correspondence with $2$-dimensional restricted cohomology of $L$ with
coefficients in the trivial module. We show that there is a family
$L_{p,p}^{\lambda}$ of restricted model filiform Lie superalgebra structures
parameterized by elements $\lambda\in\mathbb{F}^{\lambda}.$ We explicitly
describe both the $1,2$-dimensional ordinary and restricted cohomology of
$L_{p,p}^{\lambda}$ with coefficients in the $1$-dimensional trivial module.

This paper is organized as follows. In Section 2, we give some necessary
concepts and notations. In Section 3, we give algebraic interpretations of
restricted cohomology of restricted Lie superalgebra. In Section 4, we
determine both the $1,2$-dimensional ordinary and restricted cohomology of
restricted model filiform Lie superalgebra with coefficients in the
$1$-dimensional trivial module.

##  2\. Preliminaries

Let $\mathbb{F}$ be an algebraically closed field of characteristics $p>2$.
All unspecified vector superspaces are taken over $\mathbb{F}$. Suppose
$\mathbb{Z}_{2}:=\\{\bar{0},\bar{1}\\}$ is the additive group of order 2.
Suppose $\mathfrak{A}:=\mathfrak{A}_{\bar{0}}\oplus\mathfrak{A}_{\bar{1}}$ is
a superalgebra. We write $|x|:=\theta$ for the parity of a
$\mathbb{Z}_{2}$-homogeneous element $x\in\mathfrak{A}_{\theta},$
$\theta\in\mathbb{Z}_{2}.$

###  2.1. Restricted Lie superalgebras

Let us recall the definitions of Lie superalgebras and restricted Lie
superalgebras [19, 27].

######  Definition 2.1.

A Lie superalgebra is a vector superspace $L=L_{\bar{0}}\oplus L_{\bar{1}}$
with an even bilinear mapping $[\cdot,\cdot]:L\times L\longrightarrow L$
satisfying the following axioms:

$\displaystyle[x,y]=-(-1)^{|x||y|}[y,x],$
$\displaystyle[x,[y,z]]=[[x,y],z]+(-1)^{|x||y|}[y,[x,z]]$

for all $x,y,z\in L.$

######  Definition 2.2.

A restricted Lie superalgebras is a Lie superalgebra $L=L_{\bar{0}}\oplus
L_{\bar{1}}$ together with a map $[p]:L_{\bar{0}}\longrightarrow L_{\bar{0}}$,
denoted by $x\longmapsto x^{[p]}$, that satisfies:

  * (1)

$(L_{\bar{0}},[p])$ is a restricted Lie algebra,

  * (2)

$L_{\bar{1}}$ is a restricted $L_{\bar{0}}$-module.

###  2.2. Ordinary and restricted cohomology of restricted Lie superalgebras

Let us give some definitions relative to cohomology of Lie superalgebras[24]
and restricted cohomology of Lie superalgebras. For details of ordinary
cohomologies we refer to [3, 13, 24], which are the foundational papers on
cohonology of Lie superalgebras. By [7, 8, 18, 22, 23], we derive the types of
$1,2$-dimensional restricted cohomology of restricted Lie superalgebras.

Let $L$ be a Lie superalgebra and $M=M_{\bar{0}}\oplus M_{\bar{1}}$ be a
$L$-module. For $q\geq 0$, we let $\bigwedge^{q}L$ be the $q$-th super-
exterior product of $L$, that is $\bigwedge^{q}L$ is the $q$-fold tensor
product of $L$ modulo the $L$-submodule generated by the elements of the form:

$x_{1}\otimes\cdots\otimes x_{k}\otimes x_{k+1}\otimes\cdots\otimes
x_{q}+(-1)^{|x_{k}||x_{k+1}|}x_{1}\otimes\cdots\otimes x_{k+1}\otimes
x_{k}\otimes\cdots\otimes x_{q}$

for $x_{1},\ldots,x_{q}\in L.$ Set

$C^{q}(L;M)=\mathrm{Hom}_{\mathbb{F}}(\bigwedge^{q}L,M).$

There spaces $C^{q}(L;M)$ are naturally $\mathbb{Z}_{2}$-graded. Note that

$C^{0}(L;M)=\mathrm{Hom}_{\mathbb{F}}(\mathbb{F},M)\cong M.$

Let $d^{q}:C^{q}(L;M)\longrightarrow C^{q+1}(L;M)$ be given by the formula:

$\displaystyle d^{q}(\varphi)(x_{1},\ldots,x_{q+1})$ $\displaystyle=$
$\displaystyle\sum_{1\leq i<j\leq
q+1}(-1)^{\sigma_{ij}(x_{1},\ldots,x_{q+1})}\varphi([x_{i},x_{j}],x_{1},\ldots,\hat{x}_{i},\ldots\hat{x}_{j},\ldots,x_{q+1})$
(2.1)
$\displaystyle+\sum_{i=1}^{q+1}(-1)^{\gamma_{i}(x_{1},\ldots,x_{q+1},\varphi)}x_{i}\varphi(x_{1},\ldots,\hat{x}_{i},\ldots,x_{q+1}),$

where $x_{1},\ldots,x_{q+1}\in L,$ $\varphi\in C^{q}(L;M)$ and

$\sigma_{ij}(x_{1},\ldots,x_{q+1})=i+j+|x_{i}|(|x_{1}|+\cdots+|x_{i-1}|)+|x_{j}|(|x_{1}|+\cdots+|x_{j-1}|+|x_{i}|),$
$\gamma_{i}(x_{1},\ldots,x_{q+1},\varphi)=i+1+|x_{i}|(|x_{1}|+\cdots+|x_{i-1}|+|\varphi|).$

A direct verification shows that $d^{q}d^{q-1}=0$ and $|d^{q}|=\bar{0}.$ The
elements of kernel of $d^{q}$ are called $q$-dimensional cocycles and the
elements of image of $d^{q-1}$ are called $q$-dimensional coboundaries. We
will denote the $q$-dimensional cocycles and coboundaries by $Z^{q}(L;M)$ and
$B^{q}(L;M)$, respectively.

######  Definition 2.3.

We called $H^{q}(L;M)=Z^{q}(L;M)/B^{q}(L;M)$ is an ordinary cohomology of $L$
with coefficients in the module $M.$

Let $q\leq 1,$ define $C_{*}^{q}(L;M)=C^{q}(L;M)$. If $\varphi\in C^{2}(L;M)$
and $\omega:L_{\bar{0}}\longrightarrow M$, we say that $\omega$ has the
$*$-property with respect to $\varphi$ if for all $\lambda\in\mathbb{F}$ and
all $x,y\in L_{\bar{0}}$

  * (i)

$\omega(\lambda x)=\lambda^{p}\omega(x)$,

  * (ii)

$\omega(x+y)=\omega(x)+\omega(y)+\sum\limits_{x_{i}=x\;or\;y\atop
x_{1}=x,x_{2}=y}\frac{1}{\sharp(x)}\sum_{k=0}^{p-2}(-1)^{k}x_{p}\cdots
x_{p-k+1}\varphi([x_{1},\ldots,x_{p-k-1}],x_{p-k}),$ where $\sharp(x)$ is the
number of factors $x_{i}$ equal to $x$ and

$[x_{1},\ldots,x_{p-k-1}]=[[[\cdots[[x_{1},x_{2}],x_{3}],\cdots],x_{p-k-2}],x_{p-k-1}].$

Set

$C_{*}^{2}(L;M)=\\{(\varphi,\omega)\mid\varphi\in
C^{2}(L;M),\omega:L_{\bar{0}}\longrightarrow M\;\mbox{has
the}\;*\mbox{-property w.r.t.}\;\varphi\\}.$

If $\alpha\in C^{3}(L;M)$ and $\beta:L_{\bar{0}}\times
L_{\bar{0}}\longrightarrow M$, we say that $\beta$ has the $**$-property
w.r.t. $\alpha$ if for all $\lambda\in\mathbb{F}$ and all $x,y,y_{1},y_{2}\in
L_{\bar{0}}$

  * (i)

$\beta(x,y)$ is linear with respect to $x$,

  * (ii)

$\beta(x,\lambda y)=\lambda^{p}\beta(x,y)$,

  * (iii)

$\displaystyle\beta(x,y_{1}+y_{2})$ $\displaystyle=$
$\displaystyle\beta(x,y_{2})+\beta(x,y_{2})-\sum\limits_{h_{i}=y_{1}\;or\;y_{2}\atop
h_{1}=y_{1},h_{2}=y_{2}}\frac{1}{\sharp(y_{1})}\sum_{j=0}^{p-2}(-1)^{j}\sum_{k=1}^{j}C_{j}^{k}h_{p}\cdots
h_{p-k+1}\cdot$
$\displaystyle\alpha([x,h_{p-k},\ldots,h_{p-j+1}],[h_{1},\ldots,h_{p-j-1}],h_{p-j}).$

Set

$C_{*}^{3}(L;M)=\\{(\alpha,\beta)\mid\alpha\in
C^{3}(L;M),\beta:L_{\bar{0}}\times L_{\bar{0}}\longrightarrow M\;\mbox{has
the}\;**\mbox{-property w.r.t.}\;\alpha\\}.$

Let $q\leq 3$, the $\mathbb{Z}_{2}$-grading of $C^{q}(L;M)$ is inherited by

$C_{*}^{q}(L;M)=C_{*}^{q}(L;M)_{\bar{0}}\oplus C_{*}^{q}(L;M)_{\bar{1}},$

where

$C_{*}^{q}(L;M)_{\theta}=\\{(\alpha,\beta)\in
C_{*}^{q}(L;M)\mid|\alpha|=\theta\\}\;\mbox{for
all}\;\theta\in\mathbb{Z}_{2}.$

Now, an element $\varphi\in C_{*}^{1}(L;M)$ induces a map
$\mathrm{ind}^{1}(\varphi):L_{\bar{0}}\longrightarrow M$ by the formula

$\displaystyle\mathrm{ind}^{1}(\varphi)(x)=\varphi(x^{[p]})-x^{p-1}\varphi(x)\;\mbox{for
all}\;x\in L_{\bar{0}}.$ (2.2)

Since $d^{1}\varphi\mid_{L_{\bar{0}}\times L_{\bar{0}}}\in
C^{2}(L_{\bar{0}};M)$, then the [8, Lemma 3.7] shows that
$\mathrm{ind}^{1}(\varphi)$ satisfies the $*$-property w.r.t. $d^{1}\varphi.$
An element $(\alpha,\beta)\in C_{*}^{2}(L;M)$ induces a map
$\mathrm{ind}^{2}(\alpha,\beta):L_{\bar{0}}\times L_{\bar{0}}\longrightarrow
M$ by the formula

$\displaystyle\mathrm{ind}^{2}(\alpha,\beta)(x,y)=\alpha(x,y^{[p]})-\sum_{i+j=p-1}(-1)^{i}y^{i}\alpha([x,\underbrace{y,\ldots,y}_{j}],y)+x\beta(y)\;\mbox{for
all}\;x,y\in L_{\bar{0}}.$ (2.3)

Since $d^{2}\alpha\mid_{L_{\bar{0}}\times L_{\bar{0}}\times L_{\bar{0}}}\in
C^{3}(L_{\bar{0}};M)$, then the [8, Lemma 3.10] shows that
$\mathrm{ind}^{2}(\alpha,\beta)$ satisfies the $**$-property w.r.t.
$d^{2}\alpha.$ The restricted differentials are defined by

$\displaystyle d_{*}^{0}:C_{*}^{0}(L;M)\longrightarrow
C_{*}^{1}(L;M)\;\;\;\;d_{*}^{0}=d^{0},$ (2.4) $\displaystyle
d_{*}^{1}:C_{*}^{1}(L;M)\longrightarrow
C_{*}^{2}(L;M)\;\;\;d_{*}^{1}(\varphi)=(d^{1}\varphi,\mathrm{ind}^{1}(\varphi)),$
(2.5) $\displaystyle d_{*}^{2}:C_{*}^{2}(L;M)\longrightarrow
C_{*}^{3}(L;M)\;\;\;d_{*}^{2}(\alpha,\beta)=(d^{2}\alpha,\mathrm{ind}^{2}(\alpha,\beta)).$
(2.6)

Let $q\leq 2.$ We can show that $d_{*}^{q}d_{*}^{q-1}=0$ and
$|d_{*}^{q}|=\bar{0}$. The elements of kernel of $d_{*}^{q}$ are called
$q$-dimensional restricted cocycles and the elements of image of $d_{*}^{q-1}$
are called $q$-dimensional restricted coboundaries. We will denote the
$q$-dimensional restricted cocycles and restricted coboundaries by
$Z_{*}^{q}(L;M)$ and $B_{*}^{q}(L;M)$, respectively.

######  Definition 2.4.

We called $H_{*}^{q}(L;M)=Z_{*}^{q}(L;M)/B_{*}^{q}(L;M)$ is $q$-dimensional
restricted cohomology of $L$ with coefficients in the module $M.$

Note that $H^{0}(L;M)=H_{*}^{0}(L;M).$

##  3\. Algebraic interpretations

In this section, we give the algebraic interpretations of low dimensional
restricted cohomology of restricted Lie superalgebras and show that
equivalence classes of these objects are naturally corresponding to the
restricted cohomology of restricted Lie superalgebras defined above.

######  Definition 3.1.

Let $L$ be a restricted Lie superalgebra. An homogeneous linear map
$D:L\longrightarrow L$ is called a restricted superderivation of $L$ with
parity $|D|$ if for all $x,y\in L$, $z\in L_{\bar{0}}$

$\displaystyle D([x,y])=(-1)^{|D||x|}[x,D(y)]+[D(x),y],$ (3.1) $\displaystyle
D(z^{[p]})=(\mathrm{ad}z)^{p-1}D(z).$ (3.2)

Write $(\mathrm{Der}_{\mathrm{res.}})_{\bar{0}}(L)$ (resp.
$(\mathrm{Der}_{\mathrm{res.}})_{\bar{1}}(L)$) for the set of all restricted
superderivations with parity $\bar{0}$ (resp. $\bar{1}$) of $L$. Denote

$\mathrm{Der}_{\mathrm{res.}}(L)=(\mathrm{Der}_{\mathrm{res.}})_{\bar{0}}(L)\oplus(\mathrm{Der}_{\mathrm{res.}})_{\bar{1}}(L).$

For $x\in L$, we have $\mathrm{ad}x\in\mathrm{Der}_{\mathrm{res.}}(L)$ and
then $\mathrm{ad}L\vartriangleleft\mathrm{Der}_{\mathrm{res.}}(L).$ We will
call restricted superderivations of the form $\mathrm{ad}x$ inner and elements
of the $\mathrm{Der}_{\mathrm{res.}}(L)/\mathrm{ad}L$ outer.

######  Theorem 3.2.

Let $L$ is a restricted Lie superalgebra, then

$\mathrm{Der}_{\mathrm{res.}}(L)/\mathrm{ad}L=H^{1}_{*}(L;L).$

###### Proof.

(1) Let $\varphi\in C_{*}^{1}(L;L)$. By the equation (2.5) we know that
$\varphi$ is a $1$-dimensional restricted cocycle if and only if
$d^{1}\varphi=0$ and $\mathrm{ind}^{1}(\varphi)=0.$

  * (1.1)

If $d^{1}\varphi=0$, then by the equation (2.1) we have

$\displaystyle 0=d^{1}\varphi(x,y)$ $\displaystyle=$
$\displaystyle-\varphi([x,y])+(-1)^{|x||\varphi|}[x,\varphi(y)]-(-1)^{|y|(|\varphi|+|x|)}[y,\varphi(x)]$
$\displaystyle=$
$\displaystyle-\varphi([x,y])+(-1)^{|x||\varphi|}[x,\varphi(y)]+[\varphi(x),y]$

for all $x,y\in L.$ So $\varphi$ satisfies the equation (3.1).

  * (1.2)

If $\mathrm{ind}^{1}(\varphi)=0$, then by the equation (2.2) we have

$\displaystyle
0=\mathrm{ind}^{1}(\varphi)(z)=\varphi(z^{[p]})-z^{p-1}\varphi(z)$

for all $z\in L_{\bar{0}}.$ So $\varphi$ satisfies the equation (3.2).

In summary, $\varphi$ is a $1$-dimensional restricted cocycle if and only if
$\varphi\in\mathrm{Der}_{\mathrm{res.}}(L),$ that is

$Z^{1}_{*}(L;L)=\mathrm{Der}_{\mathrm{res.}}(L).$

(2) Let $\psi\in B^{1}_{*}(L;L)$, then there is $x\in C^{0}_{*}(L;L)\cong L$
such that $\psi=d^{0}_{*}(x).$ So by the equations (2.1) and (2.4) we have

$\psi(y)=d^{0}_{*}(x)(y)=(-1)^{|x||y|}[y,x]=-[x,y]$

for all $y\in L.$ Therefore, $\psi=-\mathrm{ad}x.$ It follows that
$B^{1}_{*}(L;L)=\mathrm{ad}L.$

The proof is complete. ∎

######  Definition 3.3.

Let $L$ be a restricted Lie superalgebra and superspaces $M,N$ be two
restricted $L$-modules. A restricted extension of $N$ by $M$ is an exact
sequence

$\displaystyle 0\longrightarrow
M\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}E\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}N\longrightarrow
0$ (3.3)

of restricted $L$-modules and homomorphisms. Two extensions of $N$ by $M$ are
equivalent if there is an isomorphism of restricted $L$-modules
$\sigma:E_{1}\longrightarrow E_{2}$ so that

$\displaystyle\begin{array}[c]{ccccccccc}0&\longrightarrow&M&\stackrel{{\scriptstyle\iota_{1}}}{{\longrightarrow}}&E_{1}&\stackrel{{\scriptstyle\pi_{1}}}{{\longrightarrow}}&N&\longrightarrow&0\\\
&&\parallel&\circlearrowright&\downarrow\scriptstyle{\sigma}&\circlearrowright&\parallel&&\\\
0&\longrightarrow&M&\stackrel{{\scriptstyle\iota_{2}}}{{\longrightarrow}}&E_{2}&\stackrel{{\scriptstyle\pi_{2}}}{{\longrightarrow}}&N&\longrightarrow&0\end{array}$
(3.7)

commutes, and we ask for a description of the set $\mathrm{Ext_{res.}}(N,M)$
of equivalence classes of extensions of $N$ by $M$.

Let $L$ be a restricted Lie superalgebras and $M,N$ be two restricted
$L$-modules. For $x\in L$, $\phi\in\mathrm{Hom}_{\mathbb{F}}(N,M)$ and $n\in
N$ we define

$\displaystyle(x\phi)(n)=x\phi(n)-(-1)^{|\phi||x|}\phi(xn).$ (3.8)

Then $\mathrm{Hom}_{\mathbb{F}}(N,M)$ is a restricted $L$-module and we have
the following Theorem.

######  Theorem 3.4.

Let $L$ be a restricted Lie superalgebras and $M,N$ be two restricted
$L$-modules. Then $\mathrm{Ext_{res.}}(N,M)$ is in one to one correspondence
$H_{*}^{1}(L;\mathrm{Hom}_{\mathbb{F}}(N,M))$. In particular,
$\mathrm{Ext_{res.}}(\mathbb{F},M)$ is in one to one correspondence
$H_{*}^{1}(L;M)$.

###### Proof.

(1)Given an extension (3.3) of $N$ by $M$ and choose a $\mathbb{F}$-linear map
$\rho$ such that $\pi\rho=\mathrm{Id}_{N}.$ We define an element
$\varphi_{\rho}\in C_{*}^{1}(L;\mathrm{Hom}_{\mathbb{F}}(N,M))$ by the formula

$\displaystyle\varphi_{\rho}(x)(n)=(-1)^{|x||\rho|}x\rho(n)-\rho(xn)\;\mbox{for
all}\;x\in L,n\in N,$ (3.9)

that is $\varphi_{\rho}(x)=(-1)^{|x||\rho|}x\rho.$ We claim that
$d^{1}_{*}\varphi_{\rho}=(d^{1}\varphi_{\rho},\mathrm{ind}^{1}(\varphi_{\rho}))=0$,
that is $\varphi_{\rho}\in Z_{*}^{1}(L;\mathrm{Hom}_{\mathbb{F}}(N,M))$.

  * (1.1)

For all $x,y\in L$ and $n\in N$, by the equations (2.1), (3.8) and (3.9) we
have

$\displaystyle d^{1}\varphi_{\rho}(x,y)(n)$ $\displaystyle=$
$\displaystyle-\varphi_{\rho}([x,y])(n)+(-1)^{|x||\varphi_{\rho}|}(x\varphi_{\rho}(y))(n)$
$\displaystyle-(-1)^{|y|(|x|+|\varphi_{\rho}|)}(y\varphi_{\rho}(x))(n)$
$\displaystyle=$
$\displaystyle-(-1)^{(|x|+|y|)|\rho|}[x,y]\rho(n)+\rho([x,y]n)$
$\displaystyle+(-1)^{|x||\varphi_{\rho}|}x\varphi_{\rho}(y)(n)-(-1)^{|y||x|}\varphi_{\rho}(y)(xn)$
$\displaystyle-(-1)^{|y|(|x|+|\varphi_{\rho}|)}y\varphi_{\rho}(x)(n)+\varphi_{\rho}(x)(yn)$
$\displaystyle=$
$\displaystyle-(-1)^{(|x|+|y|)|\rho|}[x,y]\rho(n)+\rho([x,y]n)$
$\displaystyle+(-1)^{|x||\varphi_{\rho}|}(-1)^{|y||\rho|}xy\rho(n)-(-1)^{|x||\varphi_{\rho}|}x\rho(yn)$
$\displaystyle-(-1)^{|y||x|}(-1)^{|y||\rho|}y\rho(xn)+(-1)^{|y||x|}\rho(yxn)$
$\displaystyle-(-1)^{|y|(|x|+|\varphi_{\rho}|)}(-1)^{|x||\rho|}yx\rho(n)+(-1)^{|y|(|x|+|\varphi_{\rho}|)}y\rho(xn)$
$\displaystyle+(-1)^{|x||\rho|}x\rho(yn)-\rho(xyn)$ $\displaystyle=$
$\displaystyle[-(-1)^{(|x|+|y|)|\rho|}[x,y]\rho(n)+(-1)^{|x||\varphi_{\rho}|}(-1)^{|y||\rho|}xy\rho(n)$
$\displaystyle-(-1)^{|y|(|x|+|\varphi_{\rho}|)}(-1)^{|x||\rho|}yx\rho(n)]$
$\displaystyle+[\rho([x,y]n)+(-1)^{|y||x|}\rho(yxn)-\rho(xyn)]$
$\displaystyle[-(-1)^{|x||\varphi_{\rho}|}x\rho(yn)+(-1)^{|x||\rho|}x\rho(yn)]$
$\displaystyle[-(-1)^{|y||x|}(-1)^{|y||\rho|}y\rho(xn)+(-1)^{|y|(|x|+|\varphi_{\rho}|)}y\rho(xn)].$

Note that $|\varphi_{\rho}|=|\rho|$. Then $d^{1}\varphi_{\rho}(x,y)(n)=0,$
that is $d^{1}\varphi_{\rho}=0.$

  * (1.2)

For all $x\in L_{\bar{0}}$, by the equations (2.2) and (3.9) we have

$\mathrm{ind}^{1}(\varphi_{\rho})(x)=\varphi_{\rho}(x^{[p]})-x^{p-1}\varphi_{\rho}(x)=x^{[p]}\rho-x^{p}\rho=0.$

Then $\mathrm{ind}^{1}(\varphi_{\rho})=0.$

(2) Conversely, if $\varphi\in Z_{*}^{1}(L;\mathrm{Hom}_{\mathbb{F}}(N,M))$,
we construct an extension of $N$ by $M$ as follows. We set $E=N\oplus M$ and

$\displaystyle x(n,m)=(xn,xm+(-1)^{|\varphi|(|x|+|n|)}\varphi(x)(n)),$ (3.10)

where $x\in L$, $n\in N$ and $m\in M.$ We claim that $E$ is a restricted
$L$-module and

$\displaystyle 0\longrightarrow M\hookrightarrow E\twoheadrightarrow
N\longrightarrow 0$

is a restricted extension of $N$ by $M$.

  * (2.1)

Since $\varphi\in Z_{*}^{1}(L;\mathrm{Hom}_{\mathbb{F}}(N,M))$, then by the
equation (2.1) we have

$\displaystyle\varphi([x,y])=(-1)^{|x||\varphi|}x\varphi(y)-(-1)^{|y|(|x|+|\varphi|)}y\varphi(x)$
(3.11)

for all $x,y\in L$. By the equations (3.10) and (3.11) we have

$\displaystyle(xy-(-1)^{|x||y|}yx)(n,m)$ $\displaystyle=$ $\displaystyle
x(yn,ym+(-1)^{|\varphi|(|y|+|n|)}\varphi(y)(n))$
$\displaystyle-(-1)^{|x||y|}y(xn,xm+(-1)^{|\varphi|(|x|+|n|)}\varphi(x)(n))$
$\displaystyle=$
$\displaystyle(xyn,xym+(-1)^{|\varphi|(|y|+|n|)}x\varphi(y)(n)+(-1)^{|\varphi|(|x|+|y|+|n|)}\varphi(x)(yn))$
$\displaystyle-(-1)^{|x||y|}(yxn,yxm+(-1)^{|\varphi|(|x|+|n|)}y\varphi(x)(n)+(-1)^{|\varphi|(|y|+|x|+|n|)}\varphi(y)(xn))$
$\displaystyle=$
$\displaystyle([x,y]n,[x,y]m+(-1)^{|\varphi|(|y|+|n|)}(x\varphi(y))(n)-(-1)^{|x||y|+|\varphi|(|x|+|n|)}(y\varphi(x))(n))$
$\displaystyle=$
$\displaystyle([x,y]n,[x,y]m+(-1)^{|\varphi|(|[x,y]|+|n|)}\varphi([x,y])(n))=[x,y](n,m)$

for all $n\in N$ and $m\in M.$

  * (2.2)

Since $\varphi\in Z_{*}^{1}(L;\mathrm{Hom}_{\mathbb{F}}(N,M))$, then

$\displaystyle\varphi(x^{[p]})=x^{p-1}\varphi(x)$ (3.12)

for all $x\in L_{\bar{0}}.$ By the equations (3.10) and (3.12) we have

$\displaystyle x^{[p]}(n,m)$ $\displaystyle=$
$\displaystyle(x^{[p]}n,x^{[p]}m+(-1)^{|\varphi||n|}\varphi(x^{[p]})(n))$
$\displaystyle=$
$\displaystyle(x^{p}n,x^{p}m+(-1)^{|\varphi||n|}x^{p-1}\varphi(x)(n))=x^{p}(n,m)$

for all $n\in N$ and $m\in M.$

  * (2.3)

Evidently the canonical inclusion and projection are restricted Lie
superalgebra homomorphisms with this module structure so that we have an
extension of $N$ by $M$.

(3) We claim that $\mathrm{Ext_{res.}}(N,M)$ is in one to one correspondence
$H_{*}^{1}(L;\mathrm{Hom}_{\mathbb{F}}(N,M))$.

  * (3.1)

Suppose that $\varphi_{1},\varphi_{2}\in
Z^{1}_{*}(L;\mathrm{Hom}_{\mathbb{F}}(N,M))_{\theta}$, where
$\theta\in\mathbb{Z}_{2},$ $E_{1},E_{2}$ denote the corresponding extensions
of $N$ by $M$. If $f\in\mathrm{Hom}_{\mathbb{F}}(N,M)_{\theta}$ satisfies
$d^{0}_{*}f=\varphi_{2}-\varphi_{1}$. We claim that $E_{1}$ and $E_{2}$ are
equivalent. We define a map

$\displaystyle\sigma:E_{1}\longrightarrow
E_{2}\;\;\;\sigma(n,m)=(n,m-(-1)^{\theta|n|}f(n))$ (3.13)

for all $n\in N$ and $m\in M.$ Clearly this map is an isomorphism of vector
spaces making the diagram (3.7) commutes.

    * (3.1.1)

Let $x\in L$, $n\in N$ and $m\in M$. Then by the equations (2.4), (3.8),
(3.10) and (3.13) we have

$\displaystyle\sigma(x(n,m))$ $\displaystyle=$
$\displaystyle\sigma(xn,xm+(-1)^{\theta(|x|+|n|)}\varphi_{1}(x)(n))$
$\displaystyle=$
$\displaystyle(xn,xm+(-1)^{\theta(|x|+|n|)}\varphi_{1}(x)(n)-(-1)^{\theta(|n|+|x|)}f(xn))$
$\displaystyle=$
$\displaystyle(xn,xm+(-1)^{\theta(|x|+|n|)}(\varphi_{2}(x)(n)-d^{0}_{*}f(x)(n))-(-1)^{\theta(|n|+|x|)}f(xn))$
$\displaystyle=$
$\displaystyle(xn,xm+(-1)^{\theta(|x|+|n|)}(\varphi_{2}(x)(n)-(-1)^{|x|\theta}(xf)(n))-(-1)^{\theta(|n|+|x|)}f(xn))$
$\displaystyle=$
$\displaystyle(xn,xm+(-1)^{\theta(|x|+|n|)}\varphi_{2}(x)(n)-(-1)^{\theta|n|}xf(n))$
$\displaystyle=$ $\displaystyle x(n,m-(-1)^{\theta|n|}f(n))$ $\displaystyle=$
$\displaystyle x\sigma(n,m).$

So $\sigma$ is a $L$-module homomorphism.

    * (3.1.2)

Let $x\in L_{\bar{0}}$, $n\in N$ and $m\in M$. Then by the equations (2.4),
(3.8), (3.10) and (3.13) we have

$\displaystyle\sigma(x^{[p]}(n,m))$ $\displaystyle=$
$\displaystyle\sigma(x^{[p]}n,x^{[p]}m+(-1)^{\theta|n|}\varphi_{1}(x^{[p]})(n))$
$\displaystyle=$
$\displaystyle\sigma(x^{p}n,x^{p}m+(-1)^{\theta|n|}x^{p-1}\varphi_{1}(x)(n))$
$\displaystyle=$
$\displaystyle(x^{p}n,x^{p}m+(-1)^{\theta|n|}x^{p-1}\varphi_{1}(x)(n)-(-1)^{\theta|n|}f(x^{p}n))$
$\displaystyle=$
$\displaystyle(x^{p}n,x^{p}m+(-1)^{\theta|n|}x^{p-1}(\varphi_{2}(x)(n)-d^{0}_{*}f(x)(n))-(-1)^{\theta|n|}f(x^{p}n))$
$\displaystyle=$
$\displaystyle(x^{p}n,x^{p}m+(-1)^{\theta|n|}x^{p-1}\varphi_{2}(x)(n)-(-1)^{|n|\theta}(x^{p}f)(n))-(-1)^{\theta|n|}f(x^{p}n))$
$\displaystyle=$
$\displaystyle(x^{p}n,x^{p}m+(-1)^{\theta|n|}x^{p-1}\varphi_{2}(x)(n)-(-1)^{\theta|n|}x^{p}f(n))$
$\displaystyle=$ $\displaystyle x^{p}(n,m-(-1)^{\theta|n|}f(n))$
$\displaystyle=$ $\displaystyle x^{p}\sigma(n,m).$

So $\sigma$ is a homomorphism of restricted $L$-modules.

In summary, $\sigma$ is an isomorphism of restricted $L$-modules. Therefore,
$E_{1}$ and $E_{2}$ are equivalent.

  * (3.2)

Since $d^{0}_{*}=d^{0}$ and
$C^{1}_{*}(L;\mathrm{Hom}_{\mathbb{F}}(N,M))=C^{1}(L;\mathrm{Hom}_{\mathbb{F}}(N,M)),$
it follows from the classical cohomology theory that restricted cocycle whose
cohomology class depends only on the equivalence class of the extension.

The proof is complete. ∎

######  Definition 3.5.

We say that a restricted Lie superalgebra $K$ is strongly abelian if in
addition to $[K,K]=0$, we also have $K_{\bar{0}}^{[p]}=0.$

######  Definition 3.6.

Let $L$ be a restricted Lie superalgebras and $K$ be a strongly abelian
restricted Lie superalgebras. A restricted extension of $L$ by $K$ is an exact
sequence

$\displaystyle 0\longrightarrow
K\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}E\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}L\longrightarrow
0$ (3.14)

of restricted Lie superalgebras and homomorphisms. Two extensions of $L$ by
$K$ being equivalent if there is an isomorphism of restricted Lie
superalgebras $\sigma:E_{1}\longrightarrow E_{2}$ so that

$\displaystyle\begin{array}[c]{ccccccccc}0&\longrightarrow&K&\stackrel{{\scriptstyle\iota_{1}}}{{\longrightarrow}}&E_{1}&\stackrel{{\scriptstyle\pi_{1}}}{{\longrightarrow}}&L&\longrightarrow&0\\\
&&\parallel&\circlearrowright&\downarrow\scriptstyle{\sigma}&\circlearrowright&\parallel&&\\\
0&\longrightarrow&K&\stackrel{{\scriptstyle\iota_{2}}}{{\longrightarrow}}&E_{2}&\stackrel{{\scriptstyle\pi_{2}}}{{\longrightarrow}}&L&\longrightarrow&0\end{array}$

commutes.

Since $K$ is a strongly abelian restricted Lie superalgebra and
$\iota(K)\vartriangleleft E$, then a restricted extension of $L$ by $K$ gives
the structure of a $L$-module by the action $x\cdot k=[\tilde{x},\iota(k)]$,
where $x\in L,$ $k\in K$ and $\tilde{x}\in E$ is any element satisfying
$\pi(\tilde{x})=x$. Since $\pi(\tilde{x}^{[p]})=\pi(\tilde{x})^{[p]}=x^{[p]},$
so

$\displaystyle x^{[p]}\cdot
k=[\widetilde{x^{[p]}},k]=[\tilde{x}^{[p]},k]=(\mathrm{ad}\tilde{x})^{p}(k)=\underbrace{x\cdots
x}_{p}\cdot k$

for all $x\in L_{\bar{0}}$ and $k\in K.$ Then $K$ is a restricted $L$-module.
We remark here that if $\iota(K)$ is contained in the center of $E$, then $K$
is a trivial $L$-module. Such an extension is called central.

######  Theorem 3.7.

Let $L$ be a restricted Lie superalgebras and $K$ be strongly abelian
restricted Lie superalgebras. Then the set of equivalence classes of
restricted center extensions of $L$ by $K$ is in one to one correspondence
with $H^{2}_{*}(L;K)$. In particular, $H^{2}_{*}(L;\mathbb{F})$ is in one to
one correspondence with equivalence classes of $1$-dimensional restricted
center extensions of $L$.

###### Proof.

  * (1)

Given an extension (3.14) of $L$ by $K$ and choose a $\mathbb{F}$-linear map
$\rho$ such that $\pi\rho=\mathrm{Id}_{L}.$ Since $\pi$ is a homomorphism of
restricted Lie superalgebras, so

$[\rho(x),\rho(y)]-\rho([x,y]),\rho(x)^{[p]}-\rho(x^{[p]})\in\mathrm{Ker}\pi=\mathrm{Im}\iota.$

Then we can define $\alpha_{\rho}:L\times L\longrightarrow K$ and
$\beta_{\rho}:L_{\bar{0}}\longrightarrow K$ with the formula

$\displaystyle\alpha_{\rho}(x,y)=\iota^{-1}([\rho(x),\rho(y)]-\rho([x,y])),$
(3.16) $\displaystyle\beta_{\rho}(z)=\iota^{-1}(\rho(z)^{[p]}-\rho(z^{[p]})),$

where $x,y\in L$ and $z\in L_{\bar{0}}.$ Similar to [8, Theorem 3.7] we can
prove that $\beta_{\rho}$ has the $*$-property with respect to
$\alpha_{\rho}.$ Then $(\alpha_{\rho},\beta_{\rho})\in C^{2}_{*}(L;K).$ We
claim that $(\alpha_{\rho},\beta_{\rho})\in Z^{2}_{*}(L;K)$.

    * (1.1)

Let $x_{1},x_{2},x_{3}\in L.$ By the equations (2.1) and (3.16) we have

$\displaystyle d^{2}\alpha_{\rho}(x_{1},x_{2},x_{3})$ $\displaystyle=$
$\displaystyle-\alpha_{\rho}([x_{1},x_{2}],x_{3})+(-1)^{|x_{2}||x_{3}|}\alpha_{\rho}([x_{1},x_{3}],x_{2})$
$\displaystyle-(-1)^{|x_{1}|(|x_{2}|+|x_{3}|)}\alpha_{\rho}([x_{2},x_{3}],x_{1})$
$\displaystyle=$
$\displaystyle\iota^{-1}(-[\rho([x_{1},x_{2}]),\rho(x_{3})]+(-1)^{|x_{2}||x_{3}|}[\rho([x_{1},x_{3}]),\rho(x_{2})]$
$\displaystyle-(-1)^{|x_{1}|(|x_{2}|+|x_{3}|)}[\rho([x_{2},x_{3}]),\rho(x_{1})]$
$\displaystyle+\rho([x_{1},x_{2}],x_{3})-(-1)^{|x_{2}||x_{3}|}\rho([x_{1},x_{3}],x_{2})$
$\displaystyle+(-1)^{|x_{1}|(|x_{2}|+|x_{3}|)}\rho([x_{2},x_{3}],x_{1})).$

Since $\iota^{-1}([\rho(x_{1}),\rho(x_{2})]-\rho([x_{1},x_{2}]))\in K$ and
$[\iota(K),E]=0,$ so

$\iota^{-1}([[\rho(x_{1}),\rho(x_{2})],\rho(x_{3})])=\iota^{-1}([\rho([x_{1},x_{2}]),\rho(x_{3})]).$

Then

$\displaystyle d^{2}\alpha_{\rho}(x_{1},x_{2},x_{3})$ $\displaystyle=$
$\displaystyle\iota^{-1}(-[[\rho(x_{1}),\rho(x_{2})],\rho(x_{3})]+(-1)^{|x_{2}||x_{3}|}[[\rho(x_{1}),\rho(x_{3})],\rho(x_{2})]$
$\displaystyle-(-1)^{|x_{1}|(|x_{2}|+|x_{3}|)}[[\rho(x_{2}),\rho(x_{3})],\rho(x_{1})]$
$\displaystyle+\rho([[x_{1},x_{2}],x_{3}]-(-1)^{|x_{2}||x_{3}|}[[x_{1},x_{3}],x_{2}])+(-1)^{|x_{1}|(|x_{2}|+|x_{3}|)}[[x_{2},x_{3}],x_{1}]))$
$\displaystyle=$ $\displaystyle 0,$

that is $d^{2}\alpha_{\rho}=0.$

    * (1.2)

Since $K$ is a trivial $L$-module, then by equations (2.3) and (3.16) we have

$\displaystyle\mathrm{ind}^{2}(\alpha_{\rho},\beta_{\rho})(x,y)$
$\displaystyle=$
$\displaystyle\alpha_{\rho}(x,y^{[p]})-\sum_{i+j=p-1}(-1)^{i}y^{i}\alpha_{\rho}([x,\underbrace{y,\ldots,y}_{j}],y)+x\beta_{\rho}(y)$
$\displaystyle=$
$\displaystyle\iota^{-1}([\rho(x),\rho(y^{[p]})]-\rho([x,y^{[p]}]))-\alpha_{\rho}([x,\underbrace{y,\ldots,y}_{p-1}],y)$
$\displaystyle=$
$\displaystyle\iota^{-1}([\rho(x),\rho(y)^{[p]}]-\rho([x,y^{[p]}]))-\alpha_{\rho}([x,\underbrace{y,\ldots,y}_{p-1}],y)$
$\displaystyle=$
$\displaystyle\iota^{-1}([\rho(x),\underbrace{\rho(y),\ldots,\rho(y)}_{p}]-\rho([x,\underbrace{y,\ldots,y}_{p}]))$
$\displaystyle-\iota^{-1}([\rho([x,\underbrace{y,\ldots,y}_{p-1}]),\rho(y)]+\rho([x,\underbrace{y,\ldots,y}_{p}]))=0$

for all $x,y\in L_{\bar{0}}.$

By the equation (2.6) we have $d^{2}_{*}(\alpha_{\rho},\beta_{\rho})=0.$

  * (2)

For each $(\alpha,\beta)\in Z^{2}_{*}(L;K),$ one can construct a restricted
central extension $L$ by $K$ as follows. We define $L_{\alpha,\beta}=K\oplus
L$ as a vector space and we define the Lie bracket and p-operator in
$L_{\alpha,\beta}$ by the formula

$\displaystyle[(k_{1},l_{1}),(k_{2},l_{2})]=(\alpha(l_{1},l_{2}),[l_{1},l_{2}]),$
(3.17) $\displaystyle(k,l)^{[p]}=(\beta(l),l^{[p]}),$ (3.18)

where $k\in K_{\bar{0}},k_{1},k_{2}\in K$ and $l\in L_{\bar{0}},l_{1},l_{2}\in
L.$

    * (2.1)

The equation (3.17) is well known that the Jacobi identity is equivalent to
$d^{2}\alpha=0.$

    * (2.2)

The equation (3.18) is a $p$-operator precisely because $\beta$ has the
$*$-property with respect to $\alpha$ and $\mathrm{ind}^{2}(\alpha,\beta)=0.$

  * (3)

Next we show that for $(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2})\in
Z^{2}_{*}(L;K),$ such that the central extensions $L_{\alpha_{1},\beta_{1}}$
and $L_{\alpha_{2},\beta_{2}}$ are equivalent, we have
$(\alpha_{1}-\alpha_{2},\beta_{1}-\beta_{2})\in B^{2}_{*}(L;K)$. Let $\sigma$
be an isomorphism of restricted Lie superalgebras and $\rho_{1},\rho_{2}$ be
two $\mathbb{F}$-linear maps such that

$\displaystyle\begin{array}[c]{ccccccccc}0&\longrightarrow&K&\stackrel{{\scriptstyle\iota_{1}}}{{\longrightarrow}}&L_{\alpha_{1},\beta_{1}}&\stackrel{{\scriptstyle\pi_{1}}}{{\longrightarrow}}&L&\longrightarrow&0\\\
&&\parallel&\circlearrowright&\downarrow\scriptstyle{\sigma}&\circlearrowright&\parallel&&\\\
0&\longrightarrow&K&\stackrel{{\scriptstyle\iota_{2}}}{{\longrightarrow}}&L_{\alpha_{2},\beta_{2}}&\stackrel{{\scriptstyle\pi_{2}}}{{\longrightarrow}}&L&\longrightarrow&0\end{array}$
(3.22)

commutes and $\pi_{1}\rho_{1}=\pi_{2}\rho_{2}=\mathrm{Id}_{L}$. Since

$\pi_{2}(\sigma\rho_{1}-\rho_{2})=\pi_{2}\sigma\rho_{1}-\pi_{2}\rho_{2}=\pi_{1}\rho_{1}-\pi_{2}\rho_{2}=0,$

so
$(\sigma\rho_{1}-\rho_{2})(L)\subseteq\mathrm{Ker}(\pi_{2})=\mathrm{Im}(\iota_{2}).$
Then $\varphi=\iota_{2}^{-1}(\sigma\rho_{1}-\rho_{2})\in C^{1}_{*}(L;K).$ We
claim that $d^{1}_{*}(\varphi)=(\alpha_{2}-\alpha_{1},\beta_{2}-\beta_{1}).$

    * (3.1)

By proof of [17, Lemma 4.1], we have $d^{1}\varphi=\alpha_{2}-\alpha_{1}.$

    * (3.2)

For all $x\in L_{\bar{0}},$ the same methods were given in the proof of [8,
Lemma 3.25] show that

$\displaystyle\mathrm{ind}^{1}(\varphi)(x)=(\beta_{2}-\beta_{1})(x).$

  * (4)

Let $(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2}),\in Z^{2}_{*}(L;K),$ such
that $(\alpha_{2},\beta_{2})-(\alpha_{1},\beta_{1})\in B^{2}_{*}(L;K)$, that
is there exists $\varphi\in C^{1}_{*}(L;K)$ satisfying
$(\alpha_{2},\beta_{2})-(\alpha_{1},\beta_{1})=d^{1}_{*}\varphi.$ Now we prove
that the extensions $L_{\alpha_{1},\beta_{1}}$ and $L_{\alpha_{2},\beta_{2}}$
are equivalent. Let us define $\sigma:L_{\alpha_{1},\beta_{1}}\longrightarrow
L_{\alpha_{2},\beta_{2}}$ by

$\displaystyle\sigma(k,l)=(k-\varphi(l),l)$ (3.23)

for all $k\in K$ and $l\in L.$ It is clear that $\sigma$ is bijective and such
that diagram (3.22) commutes. We claim $\sigma$ is a homomorphism of
restricted Lie superalgebras.

    * (4.1)

Let us check that $\sigma$ is a homomorphism of Lie superalgebras. By the
equations (3.17) and (3.23) we have

$\displaystyle[\sigma(k_{1},l_{1}),\sigma(k_{2},l_{2})]=[(k_{1}-\varphi(l_{1}),l_{1}),(k_{2}-\varphi(l_{2}),l_{2})]$
$\displaystyle=$
$\displaystyle(\alpha_{2}(l_{1},l_{2}),[l_{1},l_{2}])=(\alpha_{1}(l_{1},l_{2})+d^{1}\varphi(l_{1},l_{2}),[l_{1},l_{2}])$
$\displaystyle=$
$\displaystyle(\alpha_{1}(l_{1},l_{2})-\varphi[l_{1},l_{2}],[l_{1},l_{2}])=\sigma(\alpha_{1}(l_{1},l_{2}),[l_{1},l_{2}])$
$\displaystyle=$ $\displaystyle\sigma[(k_{1},l_{1}),(k_{2},l_{2})]$

for all $k_{1},k_{2}\in K$ and $l_{1},l_{2}\in L.$

    * (4.2)

Next we show that $\sigma$ is restricted. By the equations (3.18) and (3.23)
we have

$\displaystyle\sigma((k,l)^{[p]})$ $\displaystyle=$
$\displaystyle\sigma(\beta_{1}(l),l^{[p]})=(\beta_{1}(l)-\varphi(l^{[p]}),l^{[p]})$
$\displaystyle=$
$\displaystyle(\beta_{2}(l),l^{[p]})=(k-\varphi(l),l)^{[p]}=(\sigma(k,l))^{[p]}$

for all $k\in K_{\bar{0}}$ and $l\in L_{\bar{0}}.$

The proof is complete. ∎

##  4\. Ordinary and restricted cohomology of the restricted model filiform
Lie superalgebras

In this Section, we determine both the $1,2$-dimensional ordinary and
restricted cohomology of restricted model filiform Lie superalgebra with
coefficients in the $1$-dimensional trivial module.

###  4.1. Restricted model filiform Lie superalgebras

Fix a pair of positive integers $n,m$, let

$L_{n,m}=\mathrm{span}_{\mathbb{F}}\\{X_{1},\ldots,X_{n}\mid
Y_{1},\ldots,Y_{m}\\}$

and Lie super-brackets are given by

$\displaystyle[X_{1},X_{i}]=X_{i+1},\;\;2\leq i\leq
n-1,\;\;[X_{1},Y_{j}]=Y_{j+1},\;\;1\leq j\leq m-1$

with all other brackets are zero. We call $L_{n,m}$ the model filiform Lie
superalgebra and $\\{X_{1},\ldots,X_{n}\mid Y_{1},\ldots,Y_{m}\\}$ the
standard basis of $L_{n,m}.$ Next we show that $L_{p,p}$ admits the structure
of a restricted Lie superalgebra. If we let
$\lambda=(\lambda_{1},\ldots,\lambda_{p})\in\mathbb{F}^{p}$, and set

$(\sum_{k=1}^{p}a_{k}X_{k})^{[p]}=(\sum_{k=1}^{p}a_{k}^{p}\lambda_{k})X_{p},$

where $a_{k}\in\mathbb{F},$ then by [10], we know that $(L_{p,p})_{\bar{0}}$
is a restricted Lie algebra. Since

$\displaystyle[(\sum_{k=1}^{p}a_{k}X_{k})^{[p]},Y]=[(\sum_{k=1}^{p}a_{k}^{p}\lambda_{k})X_{p},Y]=0=\mathrm{ad}(\sum_{k=1}^{p}a_{k}X_{k})^{p}(Y),$

where $Y\in(L_{p,p})_{\bar{1}}$, then $L_{p,p}$ is a restricted Lie
superalgebra, we denote by $L_{p,p}^{\lambda}.$

###  4.2. Cochain Complexes with Trivial Coefficients

For ordinary cohomology of $L_{p,p}^{\lambda}$ with coefficients in the
$1$-dimensional trivial module, the relevant cochain superspaces (with dual
bases) are:

$\displaystyle C^{0}(L_{p,p}^{\lambda};\mathbb{F})\;\;\;\;\Omega^{0}$
$\displaystyle=$ $\displaystyle\\{1\\},$ $\displaystyle
C^{1}(L_{p,p}^{\lambda};\mathbb{F})\;\;\;\Omega^{1}$ $\displaystyle=$
$\displaystyle\\{X^{i},Y^{j}\mid 1\leq i,j\leq p\\},$ $\displaystyle
C^{2}(L_{p,p}^{\lambda};\mathbb{F})\;\;\;\Omega^{2}$ $\displaystyle=$
$\displaystyle\\{X^{i,j},X^{k}Y^{l},Y^{s,t}\mid 1\leq i<j\leq p,1\leq k,l\leq
p,1\leq s\leq t\leq p\\},$ $\displaystyle
C^{3}(L_{p,p}^{\lambda};\mathbb{F})\;\;\;\Omega^{3}$ $\displaystyle=$
$\displaystyle\\{X^{i,j,k},X^{r,s}Y^{t},X^{u}Y^{v,w},Y^{x,y,z}\mid 1\leq
i<j<k\leq p,1\leq r<s\leq p,$ $\displaystyle 1\leq t,u\leq p,1\leq v\leq w\leq
p,1\leq x\leq y\leq z\leq p\\}.$

By the equation (2.1) we have

$\displaystyle
d^{1}(X^{i})=-X^{1,i-1},d^{1}(Y^{j})=-X^{1}Y^{j-1},d^{1}(X^{1})=d^{1}(X^{2})=d^{1}(Y^{1})=0,i\geq
3,j\geq 2,$ (4.1) $\displaystyle d^{2}(X^{1,i})$ $\displaystyle=$
$\displaystyle 0,d^{2}(X^{2,3})=0,d^{2}(X^{2,j})=-X^{1,2,j-1},i\geq 2,j\geq
4,$ (4.2) $\displaystyle d^{2}(X^{i,i+1})$ $\displaystyle=$
$\displaystyle-X^{1,i-1,i+1},d^{2}(X^{j,k})=-X^{1,j-1,k}-X^{1,j,k-1},i,j\geq
3,k-1>j,$ (4.3) $\displaystyle d^{2}(X^{1}Y^{i})$ $\displaystyle=$
$\displaystyle 0,d^{2}(X^{2}Y^{1})=0,d^{2}(X^{2}Y^{j})=-X^{1,2}Y^{j-1},i\geq
1,j\geq 2,$ (4.4) $\displaystyle d^{2}(X^{i}Y^{1})$ $\displaystyle=$
$\displaystyle-X^{1,i-1}Y^{1},d^{2}(X^{i}Y^{j})=-X^{1,i-1}Y^{j}-X^{1,i}Y^{j-1},i\geq
3,j\geq 2,$ (4.5) $\displaystyle d^{2}(Y^{1,1})$ $\displaystyle=$
$\displaystyle
0,d^{2}(Y^{1,i})=-X^{1}Y^{1,i-1},d^{2}(Y^{i,i})=-2X^{1}Y^{i-1,i},i\geq 2,$
(4.6) $\displaystyle d^{2}(Y^{i,j})$ $\displaystyle=$
$\displaystyle-X^{1}Y^{i-1,j}-X^{1}Y^{i,j-1},j>i\geq 2.$ (4.7)

For restricted cohomology of $L_{p,p}^{\lambda}$ with coefficients in the
$1$-dimensional trivial module, the relevant cochain superspaces (with bases)
are:

$\displaystyle C^{0}_{*}(L_{p,p}^{\lambda};\mathbb{F})\;\;\;\;\Omega^{0},$
$\displaystyle C^{1}_{*}(L_{p,p}^{\lambda};\mathbb{F})\;\;\;\;\Omega^{1},$
$\displaystyle
C^{2}_{*}(L_{p,p}^{\lambda};\mathbb{F})\;\;\;\;\\{(0,X^{i}_{*}),(\varphi,\varphi_{*})\mid
1\leq i\leq p,\varphi\in\Omega^{2}\\},$

where $X^{k}_{*}(\sum_{i=1}^{p}a_{i}X_{i})=a_{k}^{p},$ $a_{k}\in\mathbb{F},$
$\varphi_{*}:(L_{p,p}^{\lambda})_{\bar{0}}\longrightarrow\mathbb{F}$ that
vanishes on the basis and has the $*$-property w.r.t. $\varphi.$

###  4.3. The cohomology $H^{1}(L_{p,p}^{\lambda},\mathbb{F})$ and
$H^{1}_{*}(L_{p,p}^{\lambda},\mathbb{F})$

In this subsection we study $1$-dimensional restricted cohomology of
$L_{p,p}^{\lambda}$ with coefficients in the $1$-dimensional trivial module.

######  Theorem 4.1.

Let $\lambda\in\mathbb{F}^{p}.$ Then we have

$H^{1}(L_{p,p}^{\lambda},\mathbb{F})=H^{1}_{*}(L_{p,p}^{\lambda},\mathbb{F})$

and the classes of $\\{X^{1},X^{2},Y^{1}\\}$ form a basis.

###### Proof.

It follows easily from (4.1) that $\mathrm{dim}(\mathrm{Ker}d^{1})=3$ and
$\\{X^{1},X^{2},Y^{1}\\}$ is a basis for this kernel. So that

$H^{1}(L_{p,p}^{\lambda},\mathbb{F})=\mathrm{Ker}d^{1}=\mathrm{span}_{\mathbb{F}}\\{X^{1},X^{2},Y^{1}\\}.$

Now, let $\varphi\in H^{1}(L_{p,p}^{\lambda},\mathbb{F})$ and
$\varphi=a_{1}X^{1}+a_{2}X^{2}+a_{3}Y^{1}$, where
$a_{1},a_{2},a_{3}\in\mathbb{F}$. Then by the equation (2.2) we have

$\displaystyle\mathrm{ind}^{1}(\varphi)(\sum_{i=1}^{p}b_{i}X_{i})=\varphi(\sum_{i=1}^{p}(b_{i}X_{i})^{[p]})=(a_{1}X^{1}+a_{2}X^{2}+a_{3}Y^{1})(\sum_{k=1}^{p}b_{k}^{p}\lambda_{k}X_{p})=0,$

where $b_{i}\in\mathbb{F}.$ So $H^{1}(L_{p,p}^{\lambda},\mathbb{F})\subseteq
H^{1}_{*}(L_{p,p}^{\lambda},\mathbb{F}).$ The proof is complete. ∎

###  4.4. The cohomology $H^{2}(L_{p,p}^{\lambda},\mathbb{F})$ and
$H^{2}_{*}(L_{p,p}^{\lambda},\mathbb{F})$

In this subsection we study $1$-dimensional restricted cohomology of
$L_{p,p}^{\lambda}$ with coefficients in the $1$-dimensional trivial module.
For $i\in\\{5,7,\ldots,p+2\\},$ $2\leq j\leq p$ and $k\in\\{2,4,\ldots,p+1\\}$
we put

$\varphi_{i}=\sum_{r=2}^{\lfloor\frac{i}{2}\rfloor}(-1)^{r}X^{r,i-r},\psi_{j}=\sum_{s=2}^{j}(-1)^{s}X^{s}Y^{j-s+1},\phi_{k}=\sum_{t=1}^{\frac{k}{2}-1}(-1)^{t}Y^{t,k-t}+(-1)^{\frac{k}{2}}\frac{1}{2}Y^{\frac{k}{2},\frac{k}{2}}.$

Then we have the following Theorem.

######  Theorem 4.2.

Let $\lambda\in\mathbb{F}^{p}.$ Then we have
$\mathrm{dim}H^{2}(L_{p,p}^{\lambda},\mathbb{F})=2p+1$ and the cohomology
classes of cocycles

$\Omega^{4}=\\{X^{1,p},X^{1}Y^{p},\varphi_{5},\varphi_{7},\ldots,\varphi_{p+2},\psi_{2},\ldots,\psi_{p},\phi_{2},\phi_{4},\ldots,\phi_{p+1}\\}$

form a basis.

###### Proof.

It follows easily from the equation(4.1) that
$\mathrm{dim}(\mathrm{Im}d^{1})=2p-3$ and

$\Omega^{5}=\\{X^{1,i},X^{1}Y^{j}\mid 2\leq i\leq p-1,1\leq j\leq p-1\\}$

is a basis for this image. By the equations (4.2)–(4.7) we have

$\displaystyle
d^{2}\varphi_{i}=-X^{1,2,i-3}+\sum_{r=3}^{\lfloor\frac{i}{2}\rfloor-1}(-1)^{r}(-X^{1,r-1,i-r}-X^{1,r,i-r-1})-(-1)^{\lfloor\frac{i}{2}\rfloor}X^{1,\lfloor\frac{i}{2}\rfloor-1,i-\lfloor\frac{i}{2}\rfloor}=0,$
$\displaystyle
d^{2}\psi_{j}=-X^{1,2}Y^{j-2}+\sum_{s=3}^{j-1}(-1)^{s}(-X^{1,s-1}Y^{j-s+1}-X^{1,s}Y^{j-s})-(-1)^{j}X^{1,j-1}Y^{1}=0,$
$\displaystyle
d^{2}\phi_{k}=X^{1}Y^{1,k-2}+\sum_{t=2}^{\frac{k}{2}-1}(-1)^{t}(-X^{1}Y^{t-1,k-t}-X^{1}Y^{t,k-t-1})-(-1)^{\frac{k}{2}}X^{1}Y^{\frac{k}{2}-1,\frac{k}{2}}=0,$

for $i\in\\{5,7,\ldots,p+2\\},$ $2\leq j\leq p$ and
$k\in\\{2,4,\ldots,p+1\\}.$ Then

$\Omega^{6}=\\{X^{1,i},X^{1}Y^{j},\varphi_{5},\varphi_{7},\ldots,\varphi_{p+2},\psi_{i},\phi_{2},\phi_{4},\ldots,\phi_{p+1}\mid
2\leq i\leq p,1\leq j\leq p\\}\subseteq\mathrm{Ker}d^{2}.$

Let $\varphi\in\mathrm{Ker}d^{2}$. By the equations (4.2)–(4.7), we can
suppose that

$\varphi\in\mathrm{span}_{\mathbb{F}}\\{X^{i,j}\mid 1\leq i<j\leq
p\\}\cup\mathrm{span}_{\mathbb{F}}\\{X^{k}Y^{l}\mid 1\leq k,l\leq
p\\}\cup\mathrm{span}_{\mathbb{F}}\\{Y^{s,t}\mid 1\leq s\leq t\leq p\\}.$

  * (1)

Suppose $\varphi\in\mathrm{span}_{\mathbb{F}}\\{X^{i,j}\mid 1\leq i<j\leq
p\\}.$ Then it is clear that any cocycle element has to include either the
basis element $X^{1,i}\;\;(2\leq i\leq p)$, and in this case this is a cocycle
element, or it has to have one element of type $X^{2,j}\;(3\leq j\leq p)$ in
the combination, and all those are combination of
$\varphi_{5},\varphi_{7},\ldots,\varphi_{p+2}$.

  * (2)

Suppose $\varphi\in\mathrm{span}_{\mathbb{F}}\\{X^{k}Y^{l}\mid 1\leq k,l\leq
p\\}.$ Then it is clear that any cocycle element has to include either the
basis element $X^{1}Y^{i}\;(1\leq i\leq p)$, and in this case this is a
cocycle element, or it has to have one element of type $X^{2}Y^{j}\;(1\leq
j\leq p)$ in the combination, and all those are combination of
$\psi_{2},\ldots,\psi_{p}$.

  * (3)

Suppose $\varphi\in\mathrm{span}_{\mathbb{F}}\\{Y^{s,t}\mid 1\leq s\leq t\leq
p\\}.$ Then it is clear that any cocycle element has to have one element of
type $Y^{1,i}\;\;(1\leq i\leq p)$ in the combination, and all those are
combination of $\phi_{2},\phi_{4},\ldots,\phi_{p+1}$.

The linear independence of the cocycle elements in $\Omega^{6}$ are clear.
Then $\mathrm{Ker}d^{2}=\mathrm{span}_{\mathbb{F}}\Omega^{6}$ and
$\mathrm{dim}\mathrm{Ker}d^{2}=4p-2.$ The proof is complete. ∎

######  Theorem 4.3.

Let $\lambda=0.$ Then we have
$\mathrm{dim}H_{*}^{2}(L_{p,p}^{\lambda},\mathbb{F})=3p+1$ and the cohomology
classes of cocycles

$\\{(\varphi,\varphi_{*}),(0,X^{i}_{*})\mid\varphi\in\Omega^{4},1\leq i\leq
p\\}$

form a basis.

###### Proof.

  * (1)

If $\lambda=0,$ then $\mathrm{ind}^{2}=0.$ So for that every $2$-cocycle
$\varphi\in C^{2}(L_{p,p}^{\lambda},\mathbb{F})$, we have
$d^{2}_{*}(\varphi,\varphi_{*})=(d^{2}\varphi,\mathrm{ind}^{2}\varphi_{*})=0,$
that is $(\varphi,\varphi_{*})\in\mathrm{Ker}d^{2}_{*},$ where
$\varphi_{*}:(L_{p,p}^{\lambda})_{\bar{0}}\longrightarrow\mathbb{F}$ that
vanishes on the basis and has the $*$-property with respect to $\varphi.$ The
proof of Theorem 4.2 shows that linearly independent subset

$\\{(\varphi,\varphi_{*}),(0,X^{i}_{*})\mid\varphi\in\Omega^{6},\mid 1\leq
i\leq p\\}$

is a basis of $\mathrm{Ker}d^{2}_{*}.$

  * (2)

Let $\psi\in C^{1}_{*}(L_{p,p}^{\lambda},\mathbb{F}).$ Then
$d^{1}_{*}(\psi)=(d^{1}\psi,\mathrm{ind}^{1}(\psi)),$ where
$\mathrm{ind}^{1}(\psi)(x)=\psi(x^{[p]})$ for all
$x\in(L_{p,p}^{\lambda})_{\bar{0}}.$ Since $\lambda=0,$ so
$\mathrm{ind}^{1}(\psi)=\psi_{*}.$ It follows easily from the proof of Theorem
4.2

$\mathrm{Im}d^{1}_{*}=\mathrm{span}_{\mathbb{F}}\\{(\varphi,\varphi_{*})\mid\varphi\in\Omega^{5}\\}.$

The proof is complete. ∎

######  Theorem 4.4.

Let $\lambda\neq 0.$ Then we have
$\mathrm{dim}H_{*}^{2}(L_{p,p}^{\lambda},\mathbb{F})=3p-1$ and the cohomology
classes of cocycles

$\\{(\varphi,\varphi_{*}),(0,X^{i}_{*})\mid\varphi\in\Omega^{4}-\\{X^{1,p},\varphi_{p+2}\\},1\leq
i\leq p\\}$

form a basis.

###### Proof.

  * (1)

Let $(\varphi,\omega)\in C^{2}_{*}(L_{p,p}^{\lambda},\mathbb{F})$ and

$\varphi=\sum_{1\leq i<j\leq p}a_{i,j}X^{i,j}+\sum_{1\leq k,l\leq
p}b_{k,l}X^{k}Y^{l}+\sum_{1\leq s\leq t\leq p}c_{s,t}Y^{s,t}\in
C^{2}(L_{p,p}^{\lambda},\mathbb{F}),$

where $a_{ij},b_{k,l},c_{s,t}\in\mathbb{F}.$ By the equation (2.3) we have

$\mathrm{ind}^{2}(\varphi,\omega)(X_{i},X_{j})=\varphi(X_{i},X_{j}^{[p]})=\lambda_{j}a_{i,p}$

for all $1\leq i,j\leq p.$ Since $\lambda\neq 0,$ so
$d^{2}_{*}(\varphi,\omega)=(d^{2}\varphi,\mathrm{ind}^{2}(\varphi,\omega))=0$
if and only if $\varphi\in\mathrm{Ker}d^{2}$ and $a_{1,p}=\cdots=a_{p-1,p}=0$.
This observation, together with the proof of Theorem 4.2, proves the following

$\\{(\varphi,\varphi_{*}),(0,X^{i}_{*})\mid\varphi\in\Omega^{6}-\\{X^{1,p},\varphi_{p+2}\\}\mid
1\leq i\leq p\\}$

is a basis of $\mathrm{Ker}d^{2}_{*}.$

  * (2)

By the equation (2.5) we have

$\displaystyle
d^{1}_{*}(X^{1})=d^{1}_{*}(X^{2})=(0,0),d^{1}_{*}(X^{i})=(-X^{1,i-1},(-X^{1,i-1})_{*}),3\leq
i\leq p-1,$ $\displaystyle
d^{1}_{*}(X^{p})=(-X^{1,p-1},(-X^{1,p-1})_{*}+\sum_{k=1}^{p}\lambda_{k}X^{k}_{*}),$
$\displaystyle
d^{1}_{*}(Y^{1})=(0,0),d^{1}_{*}(Y^{j})=(-X^{1}Y^{j-1},(-X^{1}Y^{j-1})_{*}),2\leq
j\leq p.$

Then

$\mathrm{Im}d^{1}_{*}=\mathrm{span}_{\mathbb{F}}\\{(\varphi,\varphi_{*}),(X^{1,p-1},(X^{1,p-1})_{*}-\sum_{k=1}^{p}\lambda_{k}X^{k}_{*})\mid\varphi\in\Omega^{5}-\\{X^{1,p-1}\\}\\}.$

The proof is complete. ∎

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