Restricted Central Extensions of Restricted Heisenberg Lie Superalgebras

TL;DR

This paper investigates the structure of restricted Heisenberg Lie superalgebras over fields of characteristic p > 2, focusing on their cohomology and central extensions, providing explicit descriptions of these extensions.

Contribution

It computes the restricted 2-cohomology spaces and describes the associated one-dimensional central extensions for restricted Heisenberg Lie superalgebras.

Findings

Computed restricted 2-cohomology spaces.

Described the structure of restricted one-dimensional central extensions.

Provided explicit classifications of these extensions.

Abstract

Restricted Heisenberg Lie superalgebras are studied over an algebraically closed field F of characteristic p > 2. We use the ordinary 1- and 2-cohomology spaces with trivial coefficients to compute the restricted 2-cohomology spaces. As an application, the corresponding restricted one-dimensional central extensions are described.

Equations

$$\{x_1,\dots ,x_{2m},x_{2m+1}\mid y_1,\dots ,y_n\}$$

$$[x_i,x_{m+i}]=[y_j,y_j]=x_{2m+1},$$

$$\{x_1,\dots ,x_n\mid y_1,\dots ,y_n,y_{n+1}\}$$

$$[x_i,y_i]=y_{n+1},$$

$$[g_1,g_2,g_3,\dots ,g_j]=[[\dots [[g_1,g_2],g_3],\dots ],g_j].$$

$$\underset{p}{\underbrace{g\dots g}}\cdot m=g^{[p]}\cdot m.$$

$$d^q:C^q(\mathfrak{g})\to C^{q+1}(\mathfrak{g})$$

$$d^0:C^0(\mathfrak{g})\to C^1(\mathfrak{g}),$$

$$d^1:C^1(\mathfrak{g})\to C^2(\mathfrak{g}),$$

$$d^2:C^2(\mathfrak{g})\to C^3(\mathfrak{g}),$$

$$+(-1{)}^{|g|(|h|+|f|)}\phi ([h,f]\wedge g).$$

$$\omega(g+h)=\omega(g)+\omega(h)+\sum_{\begin{subarray}{c}g_{i}=\mbox{\rm\scriptsize$g$ or $h$}\\ g_{1}=g,g_{2}=h\end{subarray}}\frac{1}{\#(g)}\varphi([g_{1},g_{2},g_{3},\dots,g_{p-1}]\wedge g_{p})$$

$$\eta (g,h_1+h_2)$$

$$\sum _{\begin{array}{c}{l}_1,\dots ,l_p=1\mathrm{or}2\\ l_1=1,l_2=2\end{array}}\frac{1}{\#\{l_i=1\}}\zeta (g\wedge [h_{l_1},\cdots ,h_{l_{p-1}}]\wedge h_{l_p}),$$

$$C^{2}_{res}(\mathfrak{g})=\{(\varphi,\omega)\ |\ \varphi\in C^{2}(\mathfrak{g}),\omega:\mathfrak{g}_{\bar{0}}\to\mathbb{F}\ \mbox{\rm is $\varphi$-compatible}\},$$

$$C^{3}_{res}(\mathfrak{g})=\{(\zeta,\eta)\ |\ \zeta\in C^{3}(\mathfrak{g}),\eta:\mathfrak{g}\times\mathfrak{g}_{\bar{0}}\to\mathbb{F}\ \mbox{\rm is $\zeta$-compatible}\}.$$

$${\mathrm{Hom}}_{\mathrm{Fr}}({\mathfrak{g}}_{\bar{0}},\mathbb{F})=$$

$$\mathrm{and}x,y\in {\mathfrak{g}}_{\bar{0}}\}$$

$$0⟶{\mathrm{Hom}}_{\mathrm{Fr}}({\mathfrak{g}}_{\bar{0}},\mathbb{F})\stackrel{\iota }{⟶}{C}_{res}^2(\mathfrak{g})\stackrel{\pi }{⟶}{C}^2(\mathfrak{g})⟶0,$$

$$\iota (f)=(0,f)\mathrm{and}\pi (\phi ,\omega )=\phi ,$$

$$\{(e^{i,j},\widetilde{e^{i,j}})|1\le i<j\le m\}\cup \{(e^{i,j},\widetilde{e^{i,j}})|1\le i\le m,m+1\le j\le m+n\}$$

$$\cup \{(e^{i,j},\widetilde{e^{i,j}})|m+1\le i\le j\le m+n\}\cup \{(0,{\overline{e}}^i)|1\le i\le m\}$$

$${\mathrm{ind}}^1(\psi )(g)=\psi (g^{[p]}).$$

$$d_{res}^1:C_{\ast }^1(\mathfrak{g})\to C_{\ast }^2(\mathfrak{g})$$

$$d_{res}^1(\psi )=(d^1(\psi ),{\mathrm{ind}}^1(\psi )).$$

$$H_{res}^1(\mathfrak{g})=(\mathfrak{g}/([\mathfrak{g},\mathfrak{g}]+⟨{\mathfrak{g}}_{\bar{0}}^{[p]}{⟩}_{\mathbb{F}}){)}^{\ast }.$$

$${\mathrm{ind}}^2(\phi ,\omega )(g,h)=\phi (g\wedge h^{[p]})-\phi ([g,\underset{p-1}{\underbrace{h,\dots ,h}}],h).$$

$$d_{res}^2(\phi ,\omega )=(d^2(\phi ),{\mathrm{ind}}^2(\phi ,\omega )).$$

$$H_{res}^q(\mathfrak{g})=H_{res}^q(\mathfrak{g},\mathbb{F})=\ker (d_{res}^q)/\mathrm{im}(d_{res}^{q-1}),$$

$$0$$

$$\stackrel{H}{⟶}{\mathrm{Hom}}_{\mathrm{Fr}}({\mathfrak{g}}_{\bar{0}},H^1(\mathfrak{g})).$$

$$D_{\psi }(g)=\psi (g^{[p]})$$

$$H_{\phi }(g)\cdot h=\phi (g\wedge (\mathrm{ad}g{)}^{p-1}(h))-\phi (g^{[p]}\wedge h)$$

Full text

Keywords: restricted Lie superalgebra; Heisenberg superalgebra; restricted cohomology; restricted central extension

Mathematics Subject Classification: 17B50; 17B56

1 Introduction

Given a symplectic form on a symplectic space, we can define its Heisenberg Lie algebra. It is a two-step nilpotent Lie algebra with a one-dimensional center. Due to its application in the commutative relations of quantum mechanics, Heisenberg Lie algebras have become an important research object in modern mathematics. The concept of Heisenberg Lie algebras was generalized to Lie superalgebras by considering its supersymplectic form, which is called Heisenberg Lie superalgebra. That is, a two-step nilpotent Lie superalgebra with a one-dimensional center. In [19], the authors showed the following classification theorem.

Theorem 1.1**.**

(see [19, Section 3]) Finite-dimensional complex Lie superalgebras of Heisenberg type are classified into two types described as follows (we write the even basis elements first separated from the odd ones by a vertical bar):

$(1)$* $\mathfrak{h}_{m,n}$ is the $(2m+1,n)$-dimensional vector superspace spanned by the elements*

[TABLE]

with the non-vanishing Lie brackets

[TABLE]

for $1\leq i\leq m$ and $1\leq j\leq n$.

$(2)$* $\mathfrak{ba}_{n}$ is the $(n,n+1)$-dimensional vector superspace spanned by the elements*

[TABLE]

with the non-vanishing Lie brackets

[TABLE]

for $1\leq i\leq n$.

The cohomology theory of Heisenberg Lie algebras and Heisenberg Lie superalgebras was developed by many authors, see [1, 4, 11, 20, 21] for examples. In particular, Bai and Liu computed the trivial cohomology of Heisenberg Lie superalgebras over fields of characteristic zero in [1].

Based on the fact that the $p$-powers of inner derivations of associative algebras are still inner derivations, the notion of restricted Lie algebras over fields of characteristic $p>0$ was introduced and Jacobson proved that a Lie algebra is restrictable if and only if the $p$-powers of the inner derivations are still inner derivations, see [15, 16]. The notion was generalized to Lie superalgebras called restricted Lie superalgebras or Lie $p$-superalgebras in [17]. Different from restricted Lie algebras, a Lie superalgebra is restrictable if and only if the $p$-powers of the even inner derivations are still inner derivations [18]. As a result, two-step nilpotent Lie superalgebras, particularly Heisenberg Lie superalgebras, are restrictable. The restricted cohomology theory of restricted Lie algebras was first considered by Hochschild in [14]. For a restricted Lie algebra, he associated the usual Chevalley-Eilenberg cohomology with the restricted cohomology by the canonical projection from its universal enveloping algebra to the restricted universal enveloping algebra. Moreover, Evans and Fuchs constructed a cochain complex for restricted cohomology up to order 3 in [12]. It has been proved to be an effective tool to compute restricted cohomology. By means of the cochain complex, Evans and Fialowski, etc. computed the restricted cohomology and restricted central extensions for restricted Witt algebras, restricted affine nilpotent Lie algebras, restricted simple Lie algebras, restricted filiform Lie algebras and restricted Heisenberg Lie algebras, see [6, 7, 8, 9, 10, 11]. Moreover, the central extensions of symmetric modular Lie superalgebras have been studied in [3]. Recently, the restricted cohomology theory of restricted Lie superalgebras was introduced by Bouarroudj and Ehret [2].

In this paper, restricted Heisenberg Lie superalgebras are studied. We compute the restricted 1-, 2-cohomology spaces with trivial coefficients and the corresponding restricted one-dimensional central extensions of restricted Heisenberg Lie superalgebras. The structure of the paper is as follows. In section 2, we recall the definitions of restricted Lie superalgebras, the Chevalley-Eilenberg cochain complex and restricted cochain complex in the case of trivial coefficients. We also recall the six-term exact sequence associated the ordinary cohomology with restricted cohomology and the correspondence between restricted one dimensional central extensions and restricted 2-cohomology. In section 3, we compute the restricted cohomology and restricted one-dimensional central extensions of $\mathfrak{h}^{\lambda}_{m,n}$, the restricted Heisenberg Lie superalgebra with the even center. In section 4, we compute the restricted cohomology and restricted one-dimensional central extensions of $\mathfrak{ba}^{0}_{n}$, the restricted Heisenberg Lie superalgebra with the odd center. We only consider central extensions obtained by even cocycles, because the central extensions obtained by the odd ones are trivial.

2 Restricted Lie superalgebras and cohomology

In this section, we recall the definitions of a restricted Lie superalgebra, the corresponding Chevalley-Eilenberg cochain complex and restricted cochain complex [2]. Everywhere in this section, $\mathbb{F}$ denotes an algebraically closed field of characteristic $p>2$ and $\mathfrak{g}$ denotes a finite-dimensional Lie superalgebra over $\mathbb{F}$ with an ordered basis $\{e_{1},\ldots,e_{m}\mid e_{m+1},\ldots,e_{m+n}\}$ (we write the even basis elements first separated from the odd ones by a vertical bar). Write $e^{k}$, $e^{i,j}$ and $e^{u,v,w}$ for the dual vectors of the basis vectors $e_{k}\in\mathfrak{g}$, $e_{i,j}=e_{i}\wedge e_{j}\in\wedge^{2}\mathfrak{g}$ and $e_{u,v,w}=e_{u}\wedge e_{v}\wedge e_{w}\in\wedge^{3}\mathfrak{g}$, respectively. In the case of $p=3$, we add $[g,[g,g]]=0$ for all $g\in\mathfrak{g}_{\bar{1}}$ as a part of the definition of Lie superalgebras. For $j\geq 2$ and $g_{1},\ldots,g_{j}\in\mathfrak{g}$, we denote the $j$-fold bracket

[TABLE]

2.1 Restricted Lie superalgebras

A restricted Lie algebra $\mathfrak{g}$ over a field $\mathbb{F}$ of positive characteristic $p$ is a Lie algebra $\mathfrak{g}$ over $\mathbb{F}$ together with a map $[p]:\mathfrak{g}\to\mathfrak{g}$, written $g\mapsto g^{[p]}$, such that for all $a\in\mathbb{F}$, and all $g,h\in\mathfrak{g}$

• (1)

$(ag)^{[p]}=a^{p}g^{[p]}$;

• (2)

$(g+h)^{[p]}=g^{[p]}+h^{[p]}+\sum\limits_{i=1}^{p-1}s_{i}(g,h)$, where $is_{i}(g,h)$ is the coefficient of $t^{i-1}$ in the formal expression $(\mathrm{ad}(tg+h))^{p-1}(g)$; and

• (3)

$(\mathrm{ad}\ g)^{p}=\mathrm{ad}\ g^{[p]}$.

The map $[p]$ is called a $[p]$-operator on $\mathfrak{g}$. Let $(\mathfrak{g},[p])$ be a restricted Lie algebra. A $\mathfrak{g}$-module $M$ is called restricted if, for $g\in\mathfrak{g}$ and $m\in M$,

[TABLE]

Definition 2.1**.**

A restricted Lie superalgebra is a Lie superalgebra $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ with a map $[p]:\mathfrak{g}_{\bar{0}}\rightarrow\mathfrak{g}_{\bar{0}}$ such that $(\mathfrak{g}_{\bar{0}},[p])$ is a restricted Lie algebra and $\mathfrak{g}_{\bar{1}}$ is a restricted $\mathfrak{g}_{\bar{0}}$-module with respect to the bracket.

A Lie superalgebra $\mathfrak{g}$ is called restrictable if there exists a map $[p]:\mathfrak{g}_{\bar{0}}\rightarrow\mathfrak{g}_{\bar{0}}$ such that $(\mathfrak{g},[p])$ is a restricted Lie superalgebra. The superization of Jacobson’s theorem [2, 15, 16] shows that a Lie superalgebra $\mathfrak{g}$ with a basis $\{e_{1},\ldots,e_{m}\mid e_{m+1},\ldots,e_{m+n}\}$ is restrictable if and only if $(\mathop{\rm ad}e_{i})^{p}$ is an inner derivation for all $1\leq i\leq m$. As a result, all two-step nilpotent Lie superalgebras, particularly Heisenberg Lie superalgebras, are restrictable.

2.2 Chevalley-Eilenberg Lie superalgebra cohomology

We only describe the Chevalley-Eilenberg cochain spaces $C^{q}(\mathfrak{g})=C^{q}(\mathfrak{g},\mathbb{F})$ for $q=0,1,2,3$ and differentials $d^{q}:C^{q}(\mathfrak{g})\to C^{q+1}(\mathfrak{g})$ for $q=0,1,2$. The full description of the complex can be found in [5, 13]. Set $C^{0}(\mathfrak{g})=\mathbb{F}$ and $C^{q}(\mathfrak{g})=C^{q}(\mathfrak{g})_{\bar{0}}\oplus C^{q}(\mathfrak{g})_{\bar{1}}=(\wedge^{q}\mathfrak{g}^{*})_{\bar{0}}\oplus(\wedge^{q}\mathfrak{g}^{*})_{\bar{1}}$ for $q=1,2,3$. The differentials

[TABLE]

are defined for $\psi\in C^{1}(\mathfrak{g})$, $\varphi\in C^{2}(\mathfrak{g})$ and $g,h,f\in\mathfrak{g}$ by

[TABLE]

The maps $d^{q}$ satisfy $d^{q}d^{q-1}=0$ and $H^{q}(\mathfrak{g})=H^{q}(\mathfrak{g},\mathbb{F})=Z^{q}(\mathfrak{g},\mathbb{F})/B^{q}(\mathfrak{g},\mathbb{F})$, where $Z^{q}(\mathfrak{g},\mathbb{F})=\ker(d^{q})$ and $B^{q}(\mathfrak{g},\mathbb{F})=\mathop{\rm im}(d^{q-1})$. A cochain is called a cocycle (resp. coboundary) if it is in $Z^{q}(\mathfrak{g},\mathbb{F})$ (resp. $B^{q}(\mathfrak{g},\mathbb{F})$).

2.3 Restricted Lie superalgebra cohomology

Since we are interested in central extensions, we only consider trivial coefficients and the full description can be found in [2].

Given $\varphi\in C^{2}(\mathfrak{g})$, a map $\omega:\mathfrak{g}_{\bar{0}}\to\mathbb{F}$ is * $\varphi$-compatible* if for all $g,h\in\mathfrak{g}_{\bar{0}}$ and all $a\in\mathbb{F}$

$\omega(ag)=a^{p}\omega(g)$ and

[TABLE]

where $\#(g)$ is the number of factors $g_{i}$ equal to $g$.

Similar to the Lie algebras case (see [7]), for $\varphi\in C^{2}(\mathfrak{g})$, we can assign the values of $\omega$ arbitrarily on a basis for $\mathfrak{g}_{\bar{0}}$ and use (1) to define $\omega:\mathfrak{g}_{\bar{0}}\to\mathbb{F}$ that is $\varphi$-compatible and the map $\omega$ is unique because its values are completely determined by $\omega(e_{i})$ and $\varphi$. In particular, given $\varphi$, we can define $\tilde{\varphi}(e_{i})=0$ for all $i$ and use (1) to determine a unique $\varphi$-compatible map $\widetilde{\varphi}:\mathfrak{g}_{\bar{0}}\to\mathbb{F}$. Note that, in general, $\widetilde{\varphi}\neq 0$ but $\widetilde{\varphi}(0)=0$. Moreover, If $\varphi_{1},\varphi_{2}\in C^{2}(\mathfrak{g})$ and $a\in\mathbb{F}$, then $\widetilde{(a\varphi_{1}+\varphi_{2})}=a\widetilde{\varphi}_{1}+\widetilde{\varphi}_{2}$.

If $\zeta\in C^{3}(\mathfrak{g})$, then a map $\eta:\mathfrak{g}\times\mathfrak{g}_{\bar{0}}\to\mathbb{F}$ is * $\zeta$-compatible* if for all $a\in\mathbb{F}$ and all $g\in\mathfrak{g},h,h_{1},h_{2}\in\mathfrak{g}_{\bar{0}}$, $\eta(\cdot,h)$ is linear in the first coordinate, $\eta(g,ah)=a^{p}\eta(g,h)$ and

[TABLE]

where $\#\{l_{i}=1\}$ is the number of factors $l_{i}$ equal to 1. The restricted cochain spaces are defined as $C^{0}_{res}(\mathfrak{g})=C^{0}(\mathfrak{g})$, $C^{1}_{res}(\mathfrak{g})=C^{1}(\mathfrak{g})$,

[TABLE]

[TABLE]

We define

[TABLE]

to be the space of * Frobenius homomorphisms* from $\mathfrak{g}_{\bar{0}}$ to $\mathbb{F}$. For $1\leq i\leq m$, define $\overline{e}^{i}:\mathfrak{g}_{\bar{0}}\to\mathbb{F}$ by $\overline{e}^{i}\left(\sum_{j=1}^{m}a_{j}e_{j}\right)=a_{i}^{p}.$ The set $\{\overline{e}^{i}\ |\ 1\leq i\leq m\}$ is a basis for the space of Frobenius homomorphisms ${\rm Hom}_{\rm Fr}(\mathfrak{g}_{\bar{0}},\mathbb{F})$.

Note that a map $\omega:\mathfrak{g}_{\bar{0}}\to\mathbb{F}$ is [math]-compatible if and only if $\omega\in{\rm Hom}_{\rm Fr}(\mathfrak{g}_{\bar{0}},\mathbb{F})$, then we have the exact sequence

[TABLE]

where the maps $\iota:{\rm Hom}_{\rm Fr}(\mathfrak{g}_{\bar{0}},\mathbb{F})\to C^{2}_{res}(\mathfrak{g})$ and $\pi:C^{2}_{res}(\mathfrak{g})\to C^{2}(\mathfrak{g})$ are given by

[TABLE]

for $f\in{\rm Hom}_{\rm Fr}(\mathfrak{g}_{\bar{0}},\mathbb{F})$ and $(\varphi,\omega)\in C^{2}_{res}(\mathfrak{g})$. Recall $\mathfrak{g}$ is a finite-dimensional Lie superalgebra with an ordered basis $\{e_{1},\ldots,e_{m}\mid e_{m+1},\ldots,e_{m+n}\}$. Then $\dim C^{2}_{res}(\mathfrak{g})=\dim C^{2}(\mathfrak{g})+\dim\mathfrak{g}_{\bar{0}}$ and

[TABLE]

[TABLE]

is a basis for $C^{2}_{res}(\mathfrak{g})$. We will use this basis in all computations that follow.

Define $d_{res}^{0}=d^{0}$. For $\psi\in C^{1}_{res}(\mathfrak{g})$, define the map ${\rm ind}^{1}(\psi):\mathfrak{g}_{\bar{0}}\to\mathbb{F}$ by

[TABLE]

The map ${\rm ind}^{1}(\psi)$ is $d^{1}(\psi)$-compatible for all $\psi\in C^{1}_{res}(\mathfrak{g})$, and the differential

[TABLE]

is defined by

[TABLE]

By definition,

[TABLE]

For $(\varphi,\omega)\in C^{2}_{res}(\mathfrak{g})$, define the map ${\rm ind}^{2}(\varphi,\omega):\mathfrak{g}\times\mathfrak{g}_{\bar{0}}\to\mathbb{F}$ by the formula

[TABLE]

The map ${\rm ind}^{2}(\varphi,\omega)$ is $d^{2}(\varphi)$-compatible for all $\varphi\in C^{2}(\mathfrak{g})$, and the differential $d^{2}_{res}:C^{2}_{res}(\mathfrak{g})\to C^{3}_{res}(\mathfrak{g})$ is defined by

[TABLE]

Note that if $\omega_{1}$ and $\omega_{2}$ are both $\varphi$-compatible, then ${\rm ind}^{2}(\varphi,\omega_{1})={\rm ind}^{2}(\varphi,\omega_{2})$. The following result is a straightforward superization of [11, Lemma 2.1].

Lemma 2.2**.**

If $(\varphi,\omega)\in C^{2}_{res}(\mathfrak{g})$ and $\varphi=d^{1}(\psi)$ with $\psi\in C^{1}(\mathfrak{g})$, then $(\varphi,{\rm ind}^{1}(\psi))\in C^{2}_{res}(\mathfrak{g})$ and ${\rm ind}^{2}(\varphi,\omega)={\rm ind}^{2}(\varphi,{\rm ind}^{1}(\psi))$.

These maps $d_{res}^{q}$ satisfy $d_{res}^{q}d_{res}^{q-1}=0$ and we define

[TABLE]

where $q=1,2$.

2.4 The six-term exact sequence

Suppose that $(\mathfrak{g},[p])$ is a restricted Lie superalgebra. There is a six-term exact sequence, which is superized version of [14, p. 575] and [22]:

[TABLE]

The maps $D:H^{1}(\mathfrak{g})\to{\rm Hom}_{\rm Fr}(\mathfrak{g}_{\bar{0}},\mathbb{F})$ and $H:H^{2}(\mathfrak{g})\to{\rm Hom}_{\rm Fr}(\mathfrak{g}_{\bar{0}},H^{1}(\mathfrak{g}))$ in (5) are given by

[TABLE]

and

[TABLE]

where $g\in\mathfrak{g}_{\bar{0}}$ and $h\in\mathfrak{g}$.

Proof.

The proof is similar to Theorems 2.1, 3.1 and 3.2 in [14]. ∎

2.5 Restricted one-dimensional central extension

The definition of restricted central extensions was introduced in [23]. Let $(\mathfrak{g},[p])$ be a restricted Lie superalgebra and $M$ be a strongly abelian restricted Lie superalgebra (i.e., $[M,M]=0$ and $M_{\bar{0}}^{[p]}=0$). A restricted extension of $\mathfrak{g}$ by $M$ is a short exact sequence of restricted Lie superalgebras

[TABLE]

In the above diagram, the maps $\iota$ and $\pi$ are even restricted Lie superalgebra homomorphisms. If $\iota(M)$ is contained in the center of $\mathfrak{G}$, then the extension is called central. Two restricted central extensions of $\mathfrak{g}$ by $M$ are called equivalent if they can be included in the usual commutative diagram. That is, there exists a restricted Lie superalgebra homomorphism $\sigma:\mathfrak{G}_{1}\to\mathfrak{G}_{2}$ that fixes $M$ elementwise and $\pi_{2}\sigma=\pi_{1}$.

Let $(\mathfrak{g},[p])$ be a restricted Lie superalgebra and $M$ a restricted $\mathfrak{g}$-module. We define a subspace of $C^{2}_{res}(\mathfrak{g},M)$ by

[TABLE]

Since $B_{res}^{2}(\mathfrak{g},M)_{\bar{0}}=d_{res}^{1}(C^{1}_{res}(\mathfrak{g},M)_{\bar{0}})\subseteq C^{2}_{res}(\mathfrak{g},M)^{+}$, we define

[TABLE]

and

[TABLE]

In the case of $M=\mathbb{F}$, we have $C^{2}_{res}(\mathfrak{g})^{+}=C^{2}_{res}(\mathfrak{g})$ and $H_{res}^{2}(\mathfrak{g})^{+}_{\bar{0}}=H_{res}^{2}(\mathfrak{g})_{\bar{0}}.$ Moreover, we have the following theorem.

Theorem 2.3**.**

[2, Theorem 3.5.3]** Let $(\mathfrak{g},[p])$ be a restricted Lie superalgebra. Then $H_{res}^{2}(\mathfrak{g})_{\bar{0}}$ is in one to one correspondence with the equivalence classes of restricted one-dimensional central extensions of $\mathfrak{g}$.

Now we consider restricted central extensions of $\mathfrak{g}$ by a one-dimensional space $\mathbb{F}c$. By Theorem 2.3, these restricted one-dimensional central extensions can be determined by restricted 2-cohomology. In particular, if $(\varphi,\omega)\in C^{2}_{res}(\mathfrak{g})_{\bar{0}}$ is a restricted even 2-cocycle, then the corresponding restricted one-dimensional central extension $\mathfrak{G}=\mathfrak{g}\oplus\mathbb{F}c$ has the bracket and $[p]$-operator defined for all $g,h\in\mathfrak{g}$ and $g_{0}\in\mathfrak{g}_{\bar{0}}$ by

[TABLE]

where $[\cdot,\cdot]_{\mathfrak{g}}$ and $[p]_{\mathfrak{g}}$ denote the bracket of $\mathfrak{g}$ and $[p]$-operator of $\mathfrak{g}$, respectively.

3 Restricted Lie superalgebra $\mathfrak{h}^{\lambda}_{m,n}$

Recall the Heisenberg Lie superalgebra with the even center $\mathfrak{h}_{m,n}$ having a basis

[TABLE]

with non-zero brackets of basis elements (in particular, $x_{2m+1}$ spans the even center),

[TABLE]

for $m,n\geq 1$, $1\leq i\leq m$ and $1\leq j\leq n$. The even part of $\mathfrak{h}_{m,n}$ is exactly the Heisenberg Lie algebra $\mathfrak{h}_{m}$ [19, Section 3] and the restricted structures of $\mathfrak{h}_{m}$, parameterized by elements $\lambda\in\mathbb{F}^{2m+1}$, were studied in [11, Section 3.1]. If

[TABLE]

and $g=\displaystyle\sum^{2m+1}_{i=1}a_{i}x_{i}\in(\mathfrak{h}_{m,n})_{\bar{0}}=\mathfrak{h}_{m}$, then

[TABLE]

Since $[(\mathfrak{h}_{m,n})_{\bar{0}},(\mathfrak{h}_{m,n})_{\bar{1}}]=0$, $(\mathfrak{h}_{m,n},[p])$ is a restricted Lie superalgebra by Definition 2.1. We denote the corresponding restricted Lie superalgebra by $\mathfrak{h}^{\lambda}_{m,n}$.

3.1 Restricted cohomology $H^{q}_{res}(\mathfrak{h}_{m,n}^{\lambda})$ for $q=1,2$

The cohomology of the Heisenberg Lie superalgebras $\mathfrak{h}_{m,n}$ with trivial coefficients in characteristic [math] was studied in [1, Theorem 4.1] and it is easy to check the results on $H^{1}(\mathfrak{h}_{m,n})$ and $H^{2}(\mathfrak{h}_{m,n})$ still hold for characteristic $p>2$:

Lemma 3.1**.**

For $p>2$,

(1) $H^{1}(\mathfrak{h}_{m,n})=(\mathfrak{h}_{m,n}/\mathbb{F}x_{2m+1})^{\ast};$

(2) $H^{2}(\mathfrak{h}_{m,n})=\bigwedge^{2}(\mathfrak{h}_{m,n}/\mathbb{F}x_{2m+1})^{\ast}/(\mathbb{F}dx^{2m+1})$ where $dx^{2m+1}=\sum^{m}_{i=1}x^{i,m+i}-2^{-1}\sum_{i=1}^{n}y^{i,i}$.

Let us now consider the restricted Lie superalgebra $\mathfrak{h}_{m,n}^{\lambda}$. The $[p]$-operator formula (9) implies $\langle(\mathfrak{h}^{\lambda}_{m,n})_{\bar{0}}^{[p]}\rangle=\mathbb{F}x_{2m+1}$. Then, it follows from Equation (3) that

[TABLE]

the classes of $\{x^{1},\ldots,x^{2m}\mid y^{1},\ldots,y^{n}\}$ form a basis and the map $H^{1}_{res}(\mathfrak{h}_{m,n}^{\lambda})\to H^{1}(\mathfrak{h}_{m,n}^{\lambda})$ is an isomorphism. Besides that, Equations (7) and (9) give $H=0$ in the sequence (5) and the six-term exact sequence (5) reduces to the exact sequence

[TABLE]

It follows $H^{2}_{res}(\mathfrak{h}_{m,n}^{\lambda})\cong{\rm Hom}_{\rm Fr}((\mathfrak{h}_{m,n}^{\lambda})_{\bar{0}},\mathbb{F})\oplus H^{2}(\mathfrak{h}_{m,n}^{\lambda}).$ Part (2) of Lemma 3.1 implies the following theorem:

Theorem 3.2**.**

For $m,n\geq 1$ and any form $\lambda:(\mathfrak{h}_{m,n})_{\bar{0}}\to\mathbb{F}$, $\mathrm{sdim}\ H^{2}_{res}(\mathfrak{h}_{m,n}^{\lambda})=(2m^{2}+m+\frac{n^{2}+n}{2},2mn)$ and a basis consists of the classes of the cocycles

[TABLE]

[TABLE]

3.2 Restricted one-dimensional central extensions of

$\mathfrak{h}_{m,n}^{\lambda}$

With the Equations (8) together with Theorem 3.2 we can explicitly describe the restricted one-dimensional central extensions of $\mathfrak{h}_{m,n}^{\lambda}$. Let $g=\sum a_{i}x_{i}+\sum b_{i}y_{i}$, $h=\sum a_{i}^{\prime}x_{i}+\sum b^{\prime}_{i}y_{i}$ and $g_{0}=\sum c_{i}x_{i}$ denote arbitrary elements of $\mathfrak{h}_{m,n}^{\lambda}$ and $(\mathfrak{h}_{m,n}^{\lambda})_{\bar{0}}$, respectively.

If $1\leq i\leq 2m+1$ and $\mathfrak{H}_{i}=\mathfrak{h}_{m,n}^{\lambda}\oplus\mathbb{F}c$ denotes the one-dimensional restricted central extension of $\mathfrak{h}_{m,n}^{\lambda}$ by a one-dimensional space $\mathbb{F}c$ determined by the cohomology class of the restricted cocycle $(0,\overline{x}^{i})$, then Equations (8) gives the (non-zero) bracket and $[p]$-operator

[TABLE]

The central extensions $\mathfrak{H}_{i}$ form a basis for the $(2m+1)$-dimensional space of restricted one-dimensional central extensions that split as ordinary Lie superalgebra extensions (c.f. [7]).

Since $p>2$, the $(p-1)$-fold bracket in Equation (1) is a multiple of $x_{2m+1}$. If $1\leq s<t\leq 2m$ and $1\leq k\leq l\leq n$, the maps $x^{s,t}$ and $y^{k,l}$ vanish on $x_{2m+1}\wedge\mathfrak{h}_{m,n}^{\lambda}$ so that Equation (1) implies that $\widetilde{x^{s,t}}$ and $\widetilde{y^{k,l}}$ are $p$-semilinear, and hence $\widetilde{x^{s,t}}=\widetilde{y^{k,l}}=0$. It follows that for any basis element in Theorem 3.2 of the form $(x^{s,t},\widetilde{x^{s,t}})=(x^{s,t},0)$ and $(y^{k,l},\widetilde{y^{k,l}})=(y^{k,l},0)$. The corresponding restricted one-dimensional central extension $\mathfrak{X}_{s,t}$ determined by the cohomology class of the restricted cocycle $(x^{s,t},0)$ has (non-zero) bracket and $[p]$-operator

[TABLE]

The corresponding restricted one-dimensional central extension $\mathfrak{Y}_{k,l}$ determined by the cohomology class of the restricted cocycle $(y^{k,l},0)$ has (non-zero) bracket and $[p]$-operator

[TABLE]

The central extensions $\mathfrak{X}_{s,t}$ and $\mathfrak{Y}_{k,l}$ form a basis for the $(2m^{2}-m+\frac{n^{2}+n}{2}-1)$-dimensional space of restricted one-dimensional central extensions that do not split as ordinary Lie superalgebra extensions and the Lie algebras $(\mathfrak{Y}_{k,l})_{\bar{0}}$ cannot be obtained from the central extensions of Heisenberg Lie algebras [11].

4 Restricted Lie superalgebra $\mathfrak{ba}^{0}_{n}$

Recall the Heisenberg Lie superalgebra with the odd center $\mathfrak{ba}_{n}$ having a basis

[TABLE]

with non-zero brackets of basis elements (in particular, $y_{n+1}$ spans the odd center),

[TABLE]

for $n\geq 1$ and $1\leq i\leq n$. Suppose that $[p]$ is a $[p]$-operator on $(\mathfrak{ba}_{n})_{\bar{0}}$ such that $(\mathfrak{ba}_{n},[p])$ is a restricted Lie superalgebra. Then $[(\mathfrak{ba}_{n})_{\bar{0}}^{[p]},(\mathfrak{ba}_{n})_{\bar{1}}]=0$ by Definition 2.1. Then $(\mathfrak{ba}_{n})_{\bar{0}}^{[p]}=0$. That is, $[p]=0$. It means that the only $[p]$-operator is 0. We denote the corresponding restricted Lie superalgebra by $\mathfrak{ba}^{0}_{n}$.

4.1 Restricted cohomology $H_{res}^{q}(\mathfrak{ba}^{0}_{n})$ for $q=1,2$

The cohomology of the Heisenberg Lie superalgebras $\mathfrak{ba}_{n}$ with trivial coefficients in characteristic [math] was studied in [1, Theorem 4.2] and it is easy to check the results on $H^{1}(\mathfrak{ba}_{n})$ and $H^{2}(\mathfrak{ba}_{n})$ still hold for characteristic $p>2$:

Lemma 4.1**.**

For $p>2$,

(1) $H^{1}(\mathfrak{ba}_{n})=(\mathfrak{ba}_{n}/\mathbb{F}y_{n+1})^{\ast};$

*(2) $H^{2}(\mathfrak{ba}_{n})=\bigwedge^{2}(\mathfrak{ba}_{n}/\mathbb{F}y_{n+1})^{\ast}/(\mathbb{F}dy^{n+1})\oplus\delta_{1,n}(x^{1}\wedge y^{2}),$

where $\delta_{1,n}=1$ when $n=1$ and $\delta_{1,n}=0$ otherwise, $dy^{n+1}=\sum^{n}_{i=1}x^{i}\wedge y^{i}$.*

Let us now consider the restricted Lie superalgebra $\mathfrak{ba}_{n}^{0}$. Since $[p]=0$, it follows that $H^{1}(\mathfrak{ba}_{n}^{0})=H^{1}_{res}(\mathfrak{ba}_{n}^{0})$ and $H=0$ in the sequence (5). With a similar argument to $\mathfrak{h}_{m,n}^{\lambda}$, we have the following theorem:

Theorem 4.2**.**

For $n\geq 1$, $\mathrm{sdim}\ H^{2}_{res}(\mathfrak{ba}_{n}^{0})=(n^{2}+n,n^{2}-1+\delta_{1,n})$ and a basis consists of the classes of the cocycles

[TABLE]

[TABLE]

[TABLE]

4.2 Restricted one-dimensional central extensions of

$\mathfrak{ba}^{0}_{n}$

Let $g=\sum a_{i}x_{i}+\sum b_{i}y_{i}$, $h=\sum a_{i}^{\prime}x_{i}+\sum b^{\prime}_{i}y_{i}$ and $g_{0}=\sum c_{i}x_{i}$ denote arbitrary elements of $\mathfrak{ba}^{0}_{n}$ and $(\mathfrak{ba}^{0}_{n})_{\bar{0}}$, respectively.

If $1\leq i\leq n$ and $\mathfrak{H}_{i}=\mathfrak{ba}^{0}_{n}\oplus\mathbb{F}c$ denotes the one-dimensional restricted central extension of $\mathfrak{ba}^{0}_{n}$ by a one-dimensional space $\mathbb{F}c$ determined by the cohomology class of the restricted cocycle $(0,\overline{x}^{i})$, then Equations (8) gives the (non-zero) bracket and $[p]$-operator

[TABLE]

The central extensions $\mathfrak{H}_{i}$ form a basis for the $n$-dimensional space of restricted one-dimensional central extensions that split as ordinary Lie superalgebra extensions (c.f. [7]).

Since $p>2$, the $(p-1)$-fold bracket in Equation (1) is a multiple of $y_{n+1}$. If $1\leq s\leq t\leq n$, the maps $x^{s,t}$ and $y^{s,t}$ vanish on $y_{n+1}\wedge\mathfrak{ba}^{0}_{n}$ so that Equation (1) implies that $\widetilde{x^{s,t}}$ and $\widetilde{y^{s,t}}$ are $p$-semilinear, and hence $\widetilde{x^{s,t}}=\widetilde{y^{s,t}}=0$. It follows that for any basis element in Theorem 4.2 of the form $(x^{s,t},\widetilde{x^{s,t}})=(x^{s,t},0)$ and $(y^{s,t},\widetilde{y^{s,t}})=(y^{s,t},0)$. The corresponding restricted one-dimensional central extension $\mathfrak{X}_{s,t}$ determined by the cohomology class of the restricted cocycle $(x^{s,t},0)$ has (non-zero) bracket and $[p]$-operator

[TABLE]

The corresponding restricted one-dimensional central extension $\mathfrak{Y}_{s,t}$ determined by the cohomology class of the restricted cocycle $(y^{s,t},0)$ has (non-zero) bracket and $[p]$-operator

[TABLE]

The central extensions $\mathfrak{X}_{s,t}$ and $\mathfrak{Y}_{s,t}$ form a basis for the $n^{2}$-dimensional space of restricted one-dimensional central extensions that do not split as ordinary Lie superalgebra extensions.

**Acknowledgment

**The author thanks the referee for the suggestions which improved the presentation.