Symplectic Double Extensions for Restricted Quasi-Frobenius Lie
(Super)Algebras

Symplectic Double Extensions for Restricted
Quasi-Frobenius Lie (Super)Algebras††This paper is a contribution to the
Special Issue on Differential Geometry Inspired by Mathematical Physics in
honor of Jean-Pierre Bourguignon for his 75th birthday. The full collection is
available at https://www.emis.de/journals/SIGMA/Bourguignon.html

Sofiane BOUARROUDJ a, Quentin EHRET b and Yoshiaki MAEDA c

S. Bouarroudj, Q. Ehret and Y. Maeda

a) Division of Science and Mathematics, New York University Abu Dhabi,
a) P.O. Box 129188, Abu Dhabi, United Arab Emirates<EMAIL_ADDRESS>
b) University of Haute-Alsace, IRIMAS UR 7499, F-68100 Mulhouse, France
<EMAIL_ADDRESS>
c) Tohoku Forum for Creativity, Tohoku University, 2-1-1, Katahira, Aoba-ku,
Sendai, Japan<EMAIL_ADDRESS>
Received January 31, 2023, in final form September 11, 2023; Published online
September 28, 2023

In this paper, we present a method of symplectic double extensions for
restricted quasi-Frobenius Lie superalgebras. Certain cocycles in the
restricted cohomology represent obstructions to symplectic double extension,
which we fully describe. We found a necessary condition for which a restricted
quasi-Frobenius Lie superalgebras is a symplectic double extension of a
smaller restricted Lie superalgebra. The constructions are illustrated with a
few examples.

restricted Lie (super)algebra; quasi-Frobenius Lie (super)algebra; double
extension

17B50; 17B20

##  1 Introduction

A Lie group $G$ is called symplectic if it has a left-invariant closed 2-form
whose rank is equal to $\dim(G)$. In this case, $\mathfrak{g}=\mathrm{Lie}(G)$
would be a quasi-Frobenius Lie algebra. Specifically, there exists $\omega\in
Z^{2}(\mathfrak{g})$ such that $\omega$ is non-degenerate, see Section 2.4 for
more details. Symplectic Lie groups include, for example, abelian groups of
even dimension and the group of affine transformations of ${\mathbb{K}}^{n}$,
for $n\geq 1$, where ${\mathbb{K}}={\mathbb{R}}$ or ${\mathbb{C}}$, see [4,
34].

Medina and Revoy classified symplectic nilpotent groups by their Lie algebras
in [34]. Symplectic double extensions were introduced in their work, and they
showed that each nilpotent quasi-Frobenius Lie algebra can be obtained by a
sequence of symplectic double extensions from the trivial Lie algebra. As a
group, $(G,\omega)$ is a double extension of the group $(H,\Omega)$, meaning
$(H,\Omega)$ is a Marsden–Weinstein reduced manifold of $(G,\omega)$.
Therefore, symplectic nilmanifolds can be obtained by symplectic double
extensions.

The notion of symplectic double extension is the symplectic analog to the
notion of double extension studied earlier in [24, 33] when the bilinear form
is non-degenerate, invariant and symmetric (abbreviated NIS in [12]). In
contrast to usual double extensions, there are some cohomological obstructions
to carry out symplectic double extensions, as discussed in [34].

The construction was later generalized to arbitrary symplectic Lie groups by
Dardié and Medina [17]. They showed that such groups can be described in terms
of semi-direct products of Lie groups or symplectic reduction and principal
fiber bundles with affine fiber. Furthermore, they showed that each group
obtained by this process carries an invariant Lagrangian foliation such that
the affine structure defined by the symplectic form over each leaf is complete
with respect to the affine structure deduced from that of the symplectic
group.

In [17], several examples of symplectic double extensions were provided. In
particular, symplectic Lie groups containing a closed normal subgroup of
codimension 1 that is either the abelian group or the Heisenberg group are
obtained through the process of symplectic double extension.

The structure theory of symplectic Lie groups was further studied in depth by
Baues and Cortés in [4]. In their work, they introduced the concepts of
symplectic reduction and symplectic oxidation. Basically, these two notions
reverse each other, and the notion of symplectic oxidation is close to the
notion of generalized double extensions (compare [4, Section 2.3] and [17,
Théorème 2.7]). It was shown that every symplectic Lie group admits a sequence
of subsequent symplectic reductions leading to a unique irreducible symplectic
Lie group (referred to as the symplectic basis). Such a result was described
in [4] as a Jordan–Hölder type uniqueness theorem. In the Lie algebra context,
symplectic reduction is associated with the study of a reduction with respect
to isotropic ideals. Further, it was shown that the isomorphism class of the
simply connected symplectic basis is independent of the chosen reduction
sequence. Another remarkable result of [4] is that all symplectic Lie groups
of dimension less than or equal to 6 have a Lagrangian subgroup; moreover,
there is an eight-dimensional symplectic Lie group without a Lagrangian
subgroup.

In [5], Bazzoni et al. introduced the notion of complex symplectic oxidation,
and obtained certain complex symplectic Lie algebras of dimension $4n+4$ from
those of dimension $4n$. The method has been used to describe all nilpotent
complex symplectic Lie algebras in eight dimensions.

Recently, Bajo and Benayadi in [3], generalized the idea of symplectic double
extensions studied in [16] even when the center is trivial in the case of an
abelian para-Kähler Lie algebra $\mathfrak{g}$ which is automatically
solvable. They also showed that such a Lie algebra can be obtained by a
sequence of generalized double extensions from the affine Lie algebra. In
fact, the case where $\mathfrak{g}$ is solvable but not necessarily abelian
para-Kähler has not yet been solved.

It is shown in [14] that these results can be superized for Lie superalgebras.
In this case, there are four cases to consider: either the bilinear form or
the derivation can be even or odd. It was also shown in [14] that Filiform Lie
superalgebras are quasi-Frobenius, and the 4-dimensional quasi-Frobenius Lie
superalgebras are classified based on Backhouse’s classification [2]. As far
as we know, nothing has been done about supergroups.

Lie groups and Lie algebras play an important role in geometry, they provide
relevant examples of manifolds and an algebraic characterization of some
structures.

###  1.1 Restricted Lie superalgebras and double extensions

As far as we know, Jacobson introduced the concept of a restricted Lie algebra
[30]. Roughly speaking, one requires the existence of an endomorphism on the
modular Lie algebra that resembles the $p$-th power mapping $x\mapsto x^{p}$
in associative algebras. Lie algebras associated with algebraic groups over
fields with positive characteristic are restricted, and this class resembles
the characteristic 0 case, see [38, 40]. Superization of the notion of
restrictedness was studied by several authors, see, for example, [12, 23, 36,
37, 41, 42, 43, 44] and especially [13], where new phenomena were observed in
characteristic 2. The restrictedness of simple (and close to simple) Lie
superalgebras with Cartan matrices was fully studied in [15].

Cohomology theory for restricted Lie algebras was introduced by Hochschild
[29]. While higher order cohomology is difficult to calculate, it can be
calculated at lower order, see [22]. An attempt of a superization of the
restricted cohomology was given in [43]. These constructions will be used in
this paper to describe some restricted cocycles and their triviality.

The double extensions of Lie superalgebras equipped with a NIS (also called
quadratic Lie superalgebra) was initiated in [6], then followed by a series of
papers refining the results, see [7] and references therein. Among the best
known examples are the affine Kac–Moody algebras, which are double extensions
of loop algebras. In the supersetting, the challenge is to prove that every
Lie superalgebra with a NIS is a double extension of a smaller Lie
superalgebra, since the Levi decomposition does not hold for Lie
superalgebras. Moreover, the double extension of modular Lie superalgebras in
characteristic 2 requires new features as demonstrated in [8, 9]. The double
extension of restricted Lie superalgebras equipped with NIS was studied
recently in [10], and several examples, mainly Lie superalgebras with Cartan
matrices, were studied based on [12, 15]. This paper is the ‘symplectic’
analog of the paper [10]. Here, we construct double extensions for restricted
quasi-Frobenius Lie superalgebras, and find necessary conditions for the
double extension to be restricted too. The main novelty here is that the
obstructions to carrying out symplectic double extensions are captured by the
restricted cohomology, in contrast to the ‘non-restricted’ case, where the
obstructions are captured by the Chevalley–Eilenberg cohomology, see Theorems
3.1, 3.4, 4.1, and 4.3. With only 1-cocycles or 2-cocycles to deal with, the
restricted cohomology will be quite manageable.

This is how the paper is structured. Basic concepts of restricted Lie
superalgebras, cohomology of restricted Lie superalgebras, and quasi-Frobenius
structures are discussed in Section 2. To make the paper self-contained, we
have recalled some concepts, even though they are well-known. Sections 3 and 4
is devoted to the construction of symplectic double extensions. The main
results are given in Theorems 3.1, 3.4, 4.1, and 4.3. The converse of these
theorems are given by Theorems 3.2, 3.5, 4.2 and 4.4. In Section 5, we provide
a few examples of symplectic double extensions, borrowed from [14, 27].

##  2 Restricted Lie superalgebras and restricted cohomology

Hereafter, $\mathbb{K}$ is an arbitrary field of characteristic
$\mathrm{char}(\mathbb{K})=p>2$. We refer to [40] for a thorough study of
restricted Lie algebras.

###  2.1 Restricted Lie algebras

Let $\mathfrak{a}$ be a finite-dimensional modular Lie algebra over
$\mathbb{K}$. Following [30, 40], a map
$[p]\colon\mathfrak{a}\rightarrow\mathfrak{a}$, $a\mapsto a^{[p]}$ is called a
$p$-structure of $\mathfrak{a}$ and $\mathfrak{a}$ is said to be restricted if

$\displaystyle\mathrm{ad}_{a^{[p]}}=(\mathrm{ad}_{a})^{p}\qquad\text{for all}\
a\in\mathfrak{a},$ $\displaystyle(\alpha
a)^{[p]}=\alpha^{p}a^{[p]}\qquad\text{for all}\ a\in\mathfrak{a}\ \text{and}\
\alpha\in\mathbb{K},$ $\displaystyle(a+b)^{[p]}=a^{[p]}+b^{[p]}+\sum_{1\leq
i\leq p-1}s_{i}(a,b),$

where the $s_{i}(a,b)$ can be obtained from

$\displaystyle(\mathrm{ad}_{\lambda a+b})^{p-1}(a)=\sum_{1\leq i\leq
p-1}is_{i}(a,b)\lambda^{i-1}.$

To investigate $p$-structures on a Lie algebra, it is useful to use the
following theorem, due to Jacobson.

######  Theorem 2.1 ([30]).

Let $(e_{j})_{j\in J}$ be a basis of $\mathfrak{a}$ such that there are
$f_{j}\\!\\!\in\\!\mathfrak{a}$ satisfying
${(\mathrm{ad}_{e_{j}})^{p}\\!=\\!\mathrm{ad}_{f_{j}}}$. Then, there exists
exactly one $[p]$-map $[p]\colon\mathfrak{a}\rightarrow\mathfrak{a}$ such that

$e_{j}^{[p]}=f_{j}\qquad\text{for all}\ j\in J.$

In the sequel, we will need the following lemma.

######  Lemma 2.2.

The following equality holds

$\displaystyle\sum_{i=1}^{p-1}s_{i}(a,b)$
$\displaystyle=\sum_{\underset{x_{p-1}=b,~{}x_{p}=a}{x_{k}=a\text{ or
}b}}\frac{1}{\pi(a)}[x_{1},[x_{2},[\dots,[x_{p-1},x_{p}]\dots]]],$

where $\pi\\{a\\}$ stands for the number of terms $x_{k}$ equal to $a$.

###### Proof.

One can check, by expanding the expression $(\mathrm{ad}_{\lambda
a+b})^{p-1}(a)$ and noticing that $[\lambda a+b,a]=[b,a]$, that the
coefficient of $\lambda^{i-1}$ is given by

$\sum_{\underset{\sharp\\{k,~{}x_{k}=a\\}=i-1}{x_{k}=a\text{ or
}b}}[x_{1},[x_{2},[\dots,[x_{p-2},[b,a]]\dots]]].$ (2.1)

The expression (2.1) contains $i$ variables “$a$”, so
$i=\mathrm{card}\\{k,\,x_{k}=a\\}+1.$ It follows that

$\displaystyle s_{i}(a,b)$
$\displaystyle=\sum_{\underset{\text{card}\\{k,~{}x_{k}=a\\}=i-1}{x_{k}=a\text{
or }b}}\frac{1}{i}[x_{1},[x_{2},[\dots,[x_{p-2},[b,a]]\dots]]]$
$\displaystyle=\sum_{\underset{\text{card}\\{k,~{}x_{k}=a\\}=i-1}{x_{k}=a\text{
or }b}}\frac{1}{\pi\\{a\\}}[x_{1},[x_{2},[\dots,[x_{p-2},[b,a]]\dots]]].$

Therefore,

$\displaystyle\sum_{i=1}^{p-1}s_{i}(a,b)$
$\displaystyle=\sum_{i=1}^{p-1}\sum_{\underset{\text{card}\\{k,~{}x_{k}=a\\}=i}{x_{k}=a\text{
or }b}}\frac{1}{\pi\\{a\\}}[x_{1},[x_{2},[\dots,[x_{p-2},[b,a]]\dots]]]$
$\displaystyle=\sum_{\underset{x_{p-1}=b,~{}x_{p}=a}{x_{k}=a\text{ or
}b}}\frac{1}{\pi\\{a\\}}[x_{1},[x_{2},[\dots,[x_{p-1},x_{p}]\dots]]].$ ∎

Let $\mathfrak{a}$ be a restricted Lie algebra. An $\mathfrak{a}$-module $M$
is called restricted if

$\underbrace{a\cdots a}_{p\text{~{}~{}times}}\cdot m=a^{[p]}\cdot
m\qquad\text{for all}\ a\in\mathfrak{a}\ \text{and any}\ m\in M.$

###  2.2 Restricted Lie superalgebras

Let $\mathfrak{a}=\mathfrak{a}_{\bar{0}}\oplus\mathfrak{a}_{\bar{1}}$ be a
finite-dimensional Lie superalgebra defined over a field of characteristic
$p>2$. Since $[a,[a,a]]=0$, for all $a\in\mathfrak{a}_{\bar{1}}$, does not
follow from the Jacobi identity for $p=3$, we require it as part of our
definition. In [10], we presented two examples of superalgebras for which
$[a,[a,a]]\not=0$, where $a\in\mathfrak{a}_{\bar{1}}$, constructed by means of
double extensions, and we called them pre-Lie superalgebras. We denote the
parity of a given non-zero homogeneous element $a\in\mathfrak{a}$ by $|a|$.

Following [36], we say that $\mathfrak{a}$ has a $p|2p$-structure if
$\mathfrak{a}_{\bar{0}}$ is a restricted Lie algebra and

$\mathrm{ad}_{a^{[p]}}(b)=(\mathrm{ad}_{a})^{p}(b)\qquad\text{for all}\
a\in\mathfrak{a}_{\bar{0}}\ \text{and}\ b\in\mathfrak{a}.$

Recall that the bracket of two odd elements of the Lie superalgebra is
polarization of squaring $a\mapsto a^{2}$ (namely,
$[a,b]=(a+b)^{2}-a^{2}-b^{2}$). We set

$[2p]\colon\ \mathfrak{a}_{\bar{1}}\rightarrow\mathfrak{a}_{\bar{0}},\qquad
a\mapsto\bigl{(}a^{2}\bigr{)}^{[p]}\qquad\text{for any}\
a\in\mathfrak{a}_{\bar{1}}.$

The pair $(\mathfrak{a},[p|2p])$ is referred to as a restricted Lie
superalgebra.

For $p=2$, there are several notions of restrictedness, see [13]. We do not
consider them in this paper.

The following theorem is a straightforward superization of Jacobson’s Theorem
2.1.

######  Theorem 2.3.

Let $(e_{j})_{j\in J}$ be a basis of $\mathfrak{a}_{\bar{0}}$, let the
elements $f_{j}\in\mathfrak{a}_{\bar{0}}$ be such that
${(\mathrm{ad}_{e_{j}})^{p}=\mathrm{ad}_{f_{j}}}$. Then, there exists exactly
one $[p|2p]$-mapping $[p|2p]\colon\mathfrak{a}\rightarrow\mathfrak{a}$ such
that

$e_{j}^{[p]}=f_{j}\qquad\text{for all}\ j\in J.$

Additionally, Lemma 2.2 is also valid for restricted Lie superalgebras.

A homogeneous ideal $I=I_{\bar{0}}\oplus I_{\bar{1}}$ of $\mathfrak{a}$ is
called a $p$-ideal if it is closed under the $[p|2p]$-map; namely

$a^{[p]}\in I_{\bar{0}}\qquad\text{for all}\ a\in I_{\bar{0}}.$

As $a^{[2p]}=\big{(}a^{2}\big{)}^{[p]}$, for all $a\in\mathfrak{a}_{\bar{1}}$,
this would imply that

$a^{[2p]}\in I_{\bar{0}}\qquad\text{for all}\ a\in\mathfrak{a}_{\bar{1}}.$

Let $\mathfrak{a}$ be a restricted Lie superalgebra. An $\mathfrak{a}$-module
$M$ is called restricted if

$\underbrace{a\cdots a}_{p\text{~{}~{}times}}\cdot m=a^{[p]}\cdot
m\qquad\text{for all}\ a\in\mathfrak{a}_{\bar{0}}\ \text{and}\ m\in M.$

Consequently,

$\underbrace{a\cdots a}_{2p\text{~{}~{}times}}\cdot m=a^{[2p]}\cdot
m\qquad\text{for all}\ a\in\mathfrak{a}_{\bar{1}}\ \text{and}\ m\in M.$

###  2.3 Restricted cohomology for restricted Lie superalgebras

Hochschild introduced restricted cohomology [29]. By defining the restricted
cohomology groups of restricted Lie algebras with the restricted universal
enveloping algebra introduced by Jacobson [30], Hochschild was able to
establish a connection to the ordinary Chevalley–Eilenberg cohomology of Lie
algebras with a six-term exact sequence [29]. Contributions were also made by
May [32]. Hochschild’s definition is not practical for computation, however.
Fuchs and Evans addressed this issue by providing a new free resolution of the
ground field in terms of modules over the restricted enveloping algebra.
Cohomology groups can be built up to order $p$ if the restricted Lie algebra
is abelian using this approach, which only allows them to describe the first
and second restricted cohomology groups in the non-abelian case due to the Lie
bracket and non-linearity of the $[p]$-map [22]. This cohomology was computed
explicitly and further examined by Evans and Fialowski, see [20, 21] and
references therein. The superization of the restricted cohomology of Lie
algebras was given in [43]. In this section, we review the basic concepts that
are only required for our construction.

Let $\mathfrak{a}$ be a restricted Lie superalgebra, and let $M$ be a
restricted $\mathfrak{a}$-module. Let us review the Chevalley–Eilenberg
cohomology for Lie superalgebras, see [11] for more details.

Let us define the space of cochains $C^{n}_{\text{CE}}(\mathfrak{a};M)$ in the
Chevalley–Eilenberg cohomology. For ${n=0}$, we put
$C^{0}_{\text{CE}}(\mathfrak{a};M):=M$. For $n>0$, the space of cochains
$C^{n}_{\text{CE}}(\mathfrak{a};M)$ is the space of $n$-linear super anti-
symmetric maps. The differential is given as follows:

$\displaystyle d_{\mathrm{CE}}^{0}(m)(x)=(-1)^{|m||x|}x\cdot m\qquad\text{for
any}\ m\in M\ \text{and}\ x\in\mathfrak{a},$ $\displaystyle
d_{\mathrm{CE}}^{n}(\varphi)(x_{1},\dots,x_{n})=\mathop{\sum}\limits_{i<j}(-1)^{(|x_{j}|(|x_{i+1}|+\dots+|x_{j-1}|)+j}$
$\displaystyle\phantom{d_{\mathrm{CE}}^{n}(\varphi)(x_{1},\dots,x_{n})=}{}\times\varphi(x_{1},\dots,x_{i-1},[x_{i},x_{j}],x_{i+1}\dots,\widetilde{x}_{j},\dots,x_{n})$
$\displaystyle\phantom{d_{\mathrm{CE}}^{n}(\varphi)(x_{1},\dots,x_{n})=\times}{}+\mathop{\sum}\limits_{j}(-1)^{(|x_{j}|)(|\varphi|+|x_{1}|+\dots+|x_{j-1}|)+j}x_{j}\varphi(x_{1},\dots,\widetilde{x}_{j},\dots,x_{n})$

for any $\varphi\in C^{n-1}_{\text{CE}}(\mathfrak{a};M)$ with $n>0$, and
$x_{1},\dots,x_{n}\in\mathfrak{a}$.

Let $\varphi\in C^{2}_{\text{CE}}(\mathfrak{a};M)$ and
$\theta\colon\mathfrak{a}_{0}\rightarrow M$ be two maps. We say that $\theta$
has the $*$-property with respect to $\varphi$ if

$\displaystyle\theta(\delta a)=\delta^{p}\theta(a),\qquad\text{for all}\
\delta\in\mathbb{K}\ \text{and}\ a\in\mathfrak{a}_{\bar{0}},$
$\displaystyle\theta(a+b)=\theta(a)+\theta(b)$
$\displaystyle\phantom{\theta(a+b)=}{}+\sum_{\underset{x_{1}=a,~{}x_{2}=b}{x_{i}=a\text{
or }b}}\frac{1}{\pi\\{a\\}}\sum_{k=0}^{p-2}(-1)^{k}x_{p}\dots
x_{p-k+1}\varphi([[\dots[x_{1},x_{2}],x_{3}]\dots,x_{p-k-1}],x_{p-k}),$

with $a,b\in\mathfrak{a}_{\bar{0}}$, and $\pi\\{a\\}$ stands for the number of
factors $x_{i}$ that are equal to $a$.

We are ready now to define the space of restricted cochains as

$\displaystyle C^{1}_{*}(\mathfrak{a};M):=C^{1}_{\text{CE}}(\mathfrak{a};M),$
$\displaystyle
C^{2}_{*}(\mathfrak{a};M):=\big{\\{}(\varphi,\theta),\,\varphi\in
C^{2}_{\text{CE}}(\mathfrak{a};M),\,\theta\colon\mathfrak{a}_{{\bar{0}}}\rightarrow
M\text{ has the $*$-property w.r.t.}\ \varphi\big{\\}}.$

An element $\varphi\in C^{1}(\mathfrak{a};M)$ induces a map

$\displaystyle\mathrm{Ind}^{1}(\varphi)\colon\ \mathfrak{a}$
$\displaystyle\longrightarrow M,$ $\displaystyle x$
$\displaystyle\longmapsto\varphi\bigl{(}x^{[p]}\bigr{)}-x^{p-1}\varphi(x).$

An element $(\alpha;\beta)\in C_{*}^{2}(\mathfrak{a},M)$ induces a map

$\displaystyle\mathrm{Ind}^{2}(\alpha,\beta)\colon\
\mathfrak{a}_{0}\times\mathfrak{a}_{0}$ $\displaystyle\longrightarrow M,$
$\displaystyle(a,b)$
$\displaystyle\longmapsto\alpha\bigl{(}a,b^{[p]}\bigr{)}-\sum_{i+j=p-1}(-1)^{i}b^{i}\alpha\bigl{(}[[\dots[a,\overset{j\text{
terms}}{\overbrace{b],\dots],b}]},b\bigr{)}+a\beta(b).$

A restricted $2$-cocycle is an element $(\alpha;\beta)\in
C_{*}^{2}(\mathfrak{a};M)$ such that

$\bigl{(}d_{\rm CE}^{2}\alpha,\mathrm{Ind}^{2}(\alpha,\beta)\bigr{)}=(0,0).$

We denote by $Z_{*}^{2}(\mathfrak{a};M)$ the space of restricted 2-cocycles.

An even restricted $2$-coboundary is an element $(\alpha;\beta)\in
C_{*}^{2}(\mathfrak{a};M)$ such that there exists $\varphi\in
C_{*}^{1}(\mathfrak{a};M)=C_{\text{CE}}^{1}(\mathfrak{a};M)$ for which

$\displaystyle\alpha(a,b)=\varphi([a,b])-a\varphi(b)+b\varphi(a)\qquad\text{for
all}\ a,b\in\mathfrak{a};$
$\displaystyle\beta(a)=\varphi\bigl{(}a^{[p]}\bigr{)}-a^{p-1}\varphi(a)\qquad\text{for
all}\ a\in\mathfrak{a}_{\bar{0}}.$

We denote by $B_{*}^{2}(\mathfrak{a};M)$ the space of restricted
2-coboundaries.

###  2.4 Quasi-Frobenius Lie superalgebras

As far as we know, the notion of quasi-Frobenius Lie algebra is due to
Seligman. Those Lie algebras have been introduced to answer a question raised
by Jacobson: If $\mathfrak{a}$ is finite-dimensional, what conditions one can
put on $\mathfrak{a}$ in order the enveloping algebra $U(\mathfrak{a})$ admits
an exact simple module? The answer to this question was settled by Ooms in
[35].

Several authors refer to quasi-Frobenius Lie algebras as symplectic Lie
algebras, see [1, 4, 5, 16, 17, 19, 26, 28, 34].

The superization of this notion is immediate. We have preferred to retain the
term ‘quasi-Frobenius’ in the super setting.

Following [3, 14], a Lie superalgebra $\mathfrak{a}$ is called quasi-Frobenius
if it is equipped with a 2-cocycle $\omega\in
Z^{2}_{\text{CE}}(\mathfrak{a};\mathbb{K})$ such that $\omega$ is a non-
degenerate bilinear form. Explicitly, for all $a,b\in\mathfrak{a}$ we have

$\displaystyle(-1)^{|a||c|}\omega(a,[b,c])+(-1)^{|c||b|}\omega(c,[a,b])+(-1)^{|b||a|}\omega(b,[c,a])=0.$

We denote such an algebra by $(\mathfrak{a},\omega)$. For a list of quasi-
Frobenius Lie superalgebras, see [14].

In the case where $\omega\in B^{2}(\mathfrak{a},\mathbb{K})$, the Lie
superalgebra $\mathfrak{a}$ is called a Frobenius Lie superalgebra.

Recall that a bilinear form $\omega$ is called even (resp, odd) if
$\omega(\mathfrak{a}_{\bar{0}},\mathfrak{a}_{\bar{1}})=\omega(\mathfrak{a}_{\bar{1}},\mathfrak{a}_{\bar{0}})=0$
(resp. if
$\omega(\mathfrak{a}_{\bar{0}},\mathfrak{a}_{\bar{0}})=\omega(\mathfrak{a}_{\bar{1}},\mathfrak{a}_{\bar{1}})=0$).

Recall that an even (resp. odd) non-degenerate bilinear form is called
orthosymplectic (resp. periplectic). This leads us to the following
definition:

A quasi-Frobenius Lie superalgebra $(\mathfrak{a},\omega)$ is called
orthosymplectic quasi-Frobenius (resp. periplectic quasi-Frobenius) if the
form $\omega$ is even (resp. odd) on $\mathfrak{a}$.

Let $(\mathfrak{a},\omega)$ be a quasi-Frobenius Lie superalgebra and let
$S\subseteq\mathfrak{a}$ be a subspace. The orthogonal of $S$ in
$(\mathfrak{a},\omega)$ is

$S^{\perp}=\\{v\in\mathfrak{a}\mid\omega(v,w)=0\ \text{for all}\ w\in S\\}.$

The subspace $S$ is called non-degenerate if $S\cap S^{\perp}=\\{0\\}$. It is
called isotropic if ${S\subset S^{\perp}}$. A maximal isotropic subspace is
called Lagrangian if it satisfies $S=S^{\perp}$.

If $I$ is an ideal of $\mathfrak{a}$, then $I^{\perp}$ is an ideal if and only
if $I^{\perp}\subseteq Z_{\mathfrak{a}}(I)$, where $Z_{\mathfrak{a}}(I)$
stands for the centralizer of $I$ in $\mathfrak{a}$ (see [14, Lemma 2.4]).

###  2.5 The adjoint of a derivation

Consider a linear map
${\mathscr{D}}\in\mathrm{End}(\mathfrak{a})=\mathrm{End}(\mathfrak{a})_{\bar{0}}\oplus\mathrm{End}(\mathfrak{a})_{\bar{1}}$.
The map ${\mathscr{D}}$ is called a derivation of $\mathfrak{a}$ if it
satisfies the following condition:

${\mathscr{D}}([a,b])=[{\mathscr{D}}(a),b]+(-1)^{|{\mathscr{D}}||a|}[a,{\mathscr{D}}(b)]\qquad\text{for
all}\ a,b\in\mathfrak{a}_{i},\ \text{where}\ i=\bar{0},\bar{1}.$

Let us denote by $\mathfrak{der}(\mathfrak{a})$ the space of all derivations
on $\mathfrak{a}$.

To each derivation ${\mathscr{D}}\in\mathfrak{der}(\mathfrak{a})$ one can
assign a unique linear map ${\mathscr{D}}^{*}$ with respect to the non-
degenerate form $\omega$, called the adjoint of ${\mathscr{D}}$, that
satisfies the following condition:

$\omega({\mathscr{D}}(a),b)=(-1)^{|a||{\mathscr{D}}|}\omega(a,{\mathscr{D}}^{*}(b)).$

By construction of ${\mathscr{D}}^{*}$, it is easy to see that
${\mathscr{D}}^{*}$ is a derivation as well.

Let us assume now that $\mathfrak{a}$ is restricted. A derivation
$\mathscr{D}\in\mathfrak{der}(\mathfrak{a})$ is called restricted if

$\mathscr{D}\bigl{(}a^{[p]}\bigr{)}=(\mathrm{ad}_{a})^{p-1}(\mathscr{D}(a))\qquad\text{for
all}\ a\in\mathfrak{a}_{\bar{0}}.$

Consequently, we have

$\mathscr{D}\bigl{(}a^{[2p]}\bigr{)}=(\mathrm{ad}_{a^{2}})^{p-1}\bigl{(}\mathscr{D}\big{(}a^{2}\big{)}\bigr{)}\qquad\text{for
all}\ a\in\mathfrak{a}_{\bar{1}}.$

The space of restricted derivations is denoted by
$\mathfrak{der}^{p}(\mathfrak{a})$. Outer restricted derivations of nilpotent
restricted Lie algebras has been investigated in [25].

Following [10], a derivation
$\mathscr{D}\in\mathfrak{der}^{p}_{\bar{0}}(\mathfrak{a})$ has $p$-property if
there exist $\gamma\in\mathbb{K}$ and $a_{0}\in\mathfrak{a}_{\bar{0}}$ such
that

$\mathscr{D}^{p}=\gamma\mathscr{D}+\mathrm{ad}_{a_{0}},\qquad\mathscr{D}(a_{0})=0.$
(2.2)

For a list of outer derivations satisfying the $p$-property, see [10].

###  2.6 A few useful cocycles

Let $(\mathfrak{a},\omega)$ be a restricted quasi-Frobenius Lie superalgebra,
and let ${\mathscr{D}}$ be a derivation on $\mathfrak{a}$.

######  Lemma 2.4.

Let us define the map

$\displaystyle C\colon\
\mathfrak{a}\wedge\mathfrak{a}\rightarrow\mathbb{K},\qquad(a,b)\mapsto\omega\bigl{(}\bigl{(}{\mathscr{D}}+{\mathscr{D}}^{*}\bigr{)}(a),b\bigr{)}=\omega({\mathscr{D}}(a),b)+(-1)^{|a||\mathscr{D}|}\omega(a,{\mathscr{D}}(b)).$
(2.3)

Then $C\in Z^{2}_{\mathrm{CE}}(\mathfrak{a};\mathbb{K})$. Moreover, if
$\mathscr{D}$ is inner, then $C\in
B^{2}_{\mathrm{CE}}(\mathfrak{a};\mathbb{K})$.

###### Proof.

Using the fact that $\omega$ is a $2$-cocycle, we get

$\displaystyle\underset{a,b,c}{\circlearrowleft}(-1)^{|a||c|}C(a,[b,c])={}$
$\displaystyle\underset{a,b,c}{\circlearrowleft}(-1)^{|a||c|}\bigl{(}\omega({\mathscr{D}}(a),[b,c])+(-1)^{|{\mathscr{D}}||a|}\omega(a,[{\mathscr{D}}(b),c])$
$\displaystyle+(-1)^{|{\mathscr{D}}||a|+|{\mathscr{D}}||b|}\omega(a,[b,{\mathscr{D}}(c)])\bigr{)}$
$\displaystyle={}$
$\displaystyle\underset{{\mathscr{D}}(a),b,c}{\circlearrowleft}(-1)^{|{\mathscr{D}}(a)||c|}\omega({\mathscr{D}}(a),[b,c])+\underset{a,{\mathscr{D}}(b),c}{\circlearrowleft}\\!(-1)^{|a||c|}\omega(a,[{\mathscr{D}}(b),c])$
$\displaystyle+\underset{a,b,{\mathscr{D}}(c)}{\circlearrowleft}(-1)^{p({\mathscr{D}}(c))|a|}\omega(a,[b,{\mathscr{D}}(c)])=0.$

Suppose now that $\mathscr{D}$ is inner; namely,
$\mathscr{D}=\mathrm{ad}_{X}$, for some $X\in\mathfrak{a}$. Using the fact
that $\omega$ is closed, a direct computation shows that

$C(a,b)=\omega(X,[a,b]).$

The following definition is essential to us. Denote by
$\sigma_{i}^{\mathfrak{a}}(a,b)$ the expression that appears in the following
equation:

$\omega\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(\mu
a+b),\bigl{(}\mathrm{ad}^{\mathfrak{a}}_{\mu
a+b}\bigr{)}^{p-2}(a)\bigr{)}=\sum_{1\leq i\leq
p-1}i\sigma_{i}^{\mathfrak{a}}(a,b)\mu^{i-1}.$ (2.4)

  * •

If $p=2$, then
$\sigma_{1}^{\mathfrak{a}}(a,b)=\omega(({\mathscr{D}}+{\mathscr{D}}^{*})(b),a).$

  * •

If $p=3$, then
$\sigma_{1}^{\mathfrak{a}}(a,b)=\omega(({\mathscr{D}}+{\mathscr{D}}^{*})(b),[b,a])$
and
$\sigma_{2}^{\mathfrak{a}}(a,b)=2\omega(({\mathscr{D}}+{\mathscr{D}}^{*})(a),[b,a])$.

  * •

If $p$ is any prime, then
$\sigma_{1}^{\mathfrak{a}}(a,b)=\omega\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(b),(\mathrm{ad}^{\mathfrak{a}}_{b})^{p-2}(a)\bigr{)}$.

We will need the following lemmas.

######  Lemma 2.5.

  * $(i)$

For all $a,b\in\mathfrak{a}_{\bar{0}}$, we have

$\displaystyle
s_{i}^{\mathfrak{g}}(a,b)=\begin{cases}s_{i}^{\mathfrak{a}}(a,b)+\sigma_{i}^{\mathfrak{a}}(a,b)x&\text{if}\
|{\mathscr{D}}|+|\omega|={\bar{0}},\\\ s_{i}^{\mathfrak{a}}(a,b)&\text{if}\
|{\mathscr{D}}|+|\omega|={\bar{1}}.\end{cases}$

  * $(ii)$

For every $a,b\in\mathfrak{a}_{\bar{0}}$, we have

$\displaystyle\sigma_{1}^{\mathfrak{a}}({\mathscr{D}}(a),a)=\omega\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(a),\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-2}\circ{\mathscr{D}}(a)\bigr{)},$
$\displaystyle\sigma_{1}^{\mathfrak{a}}([a,b],a)=\omega\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(a),\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-1}(b)\bigr{)}.$

###### Proof.

For part (i), since

$\big{(}\mathrm{ad}_{\mu
a+b}^{\mathfrak{g}}\big{)}^{p-1}(a)=\big{(}\mathrm{ad}_{\mu
a+b}^{\mathfrak{a}}\big{)}^{p-1}(a)+\omega\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(\mu
a+b),\big{(}\mathrm{ad}_{\mu a+b}^{\mathfrak{a}}\big{)}^{p-2}(a)\bigr{)}x,$

the result follows immediately. Part (ii) follows from the very definition of
$\sigma^{\mathfrak{a}}$ given in equation (2.4), by taking $\mu=0$ and choose
$a$ and $b$ appropriately. ∎

######  Lemma 2.6.

For all $a,b\in\mathfrak{a}_{\bar{0}}$, we have

$\displaystyle\sum_{i=1}^{p-1}\sigma_{i}^{\mathfrak{a}}(a,b)=\sum_{\underset{x_{p-1}=b,~{}x_{p}=a}{x_{k}=a\text{
or
}b}}\frac{1}{\pi\\{a\\}}\omega(({\mathscr{D}}+{\mathscr{D}}^{*})(x_{1}),[x_{2},[\dots,[x_{p-1},x_{p}]_{\mathfrak{a}}\dots]_{\mathfrak{a}}]_{\mathfrak{a}}).$

###### Proof.

Similar to that of Lemma 2.2. ∎

We are now ready to introduce another map $P$ satisfying certain conditions.
The goal here is to construct a 2-cocycle $(C,P)\in
Z^{2}_{*}(\mathfrak{a};\mathbb{K})$, as we have already shown in Lemma 2.4
that $C\in Z^{2}_{\text{CE}}(\mathfrak{a};\mathbb{K})$.This construction will
be used on Section 3 when we study symplectic double extensions. Let us define
a map

$P\colon\ \mathfrak{a}_{\bar{0}}\rightarrow\mathbb{K},\qquad a\mapsto P(a)$
(2.5)

satisfying the two conditions

$\displaystyle P(\delta a)=\delta^{p}P(a)\qquad\text{for all}\
a\in\mathfrak{a}_{\bar{0}}\ \text{and}\ \delta\in\mathbb{F},$ (2.6)
$\displaystyle
P(a+b)=P(a)+P(b)+\sum_{i=1}^{p-1}\sigma_{i}^{\mathfrak{a}}(a,b)\qquad\text{for
all}\ a,b\in\mathfrak{a}_{\bar{0}}.$ (2.7)

######  Lemma 2.7.

The map $P$ has the $*$-property with respect to the $2$-cochain $C$ defined
as in equation (2.3).

###### Proof.

Using Lemma 2.6, it is enough to show that (for all
$a,b\in\mathfrak{a}_{\bar{0}}$)

$\displaystyle\sum_{\underset{x_{p-1}=b,~{}x_{p}=a}{x_{k}=a\text{ or
}b}}\frac{1}{\pi\\{a\\}}\omega(({\mathscr{D}}+{\mathscr{D}}^{*})(x_{1}),[x_{2},[\dots,[x_{p-1},x_{p}]_{\mathfrak{a}}\dots]_{\mathfrak{a}}]_{\mathfrak{a}})$
$\displaystyle=\sum_{\underset{x_{1}=a,~{}x_{2}=b}{x_{i}=a\text{ or
}b}}\frac{1}{\pi\\{a\\}}\sum_{k=0}^{p-2}(-1)^{k}x_{p}\dots
x_{p-k+1}\omega(({\mathscr{D}}+{\mathscr{D}}^{*})([[\dots[x_{1},x_{2}],x_{3}]\dots,x_{p-k-1}]_{\mathfrak{a}}),x_{p-k}).$

Equivalently,

$\displaystyle\sum_{i=1}^{p-1}\sigma_{i}^{\mathfrak{a}}(a,b)=$
$\displaystyle\sum_{x_{k}=a\text{ or
}b}\frac{1}{\pi\\{a\\}}\omega(({\mathscr{D}}+{\mathscr{D}}^{*})(x_{1}),[x_{2},[x_{3},[\dots,[b,a]_{\mathfrak{a}}\dots]_{\mathfrak{a}}]_{\mathfrak{a}})$
$\displaystyle=$ $\displaystyle\sum_{x_{k}=a\text{ or
}b}\frac{1}{\pi\\{a\\}}\omega(({\mathscr{D}}+{\mathscr{D}}^{*})(x_{1}),[x_{2},[x_{3},[\dots,[b,a]_{\mathfrak{a}}\dots]_{\mathfrak{a}}]_{\mathfrak{a}}]_{\mathfrak{a}})$
$\displaystyle=$ $\displaystyle\sum_{x_{k}=a\text{ or
}b}\frac{1}{\pi\\{a\\}}\omega(({\mathscr{D}}+{\mathscr{D}}^{*})(x_{p}),[x_{p-1},[x_{p-2},[\dots,[b,a]_{\mathfrak{a}}\dots]_{\mathfrak{a}}]_{\mathfrak{a}}]_{\mathfrak{a}})$
$\displaystyle=$ $\displaystyle(-1)^{p-2}\sum_{x_{k}=a\text{ or
}b}\frac{1}{\pi\\{a\\}}\omega(({\mathscr{D}}+{\mathscr{D}}^{*})(x_{p}),[[\dots[a,b]_{\mathfrak{a}},x_{3}]_{\mathfrak{a}},\dots,x_{p-1}]_{\mathfrak{a}})$
$\displaystyle=$ $\displaystyle\sum_{x_{k}=a\text{ or
}b}\frac{1}{\pi\\{a\\}}\omega([[\dots[a,b]_{\mathfrak{a}},x_{3}]_{\mathfrak{a}},\dots,x_{p-1}]_{\mathfrak{a}},({\mathscr{D}}+{\mathscr{D}}^{*})(x_{p}))$
$\displaystyle=$ $\displaystyle\sum_{x_{k}=a\text{ or
}b}\frac{1}{\pi\\{a\\}}\omega(({\mathscr{D}}+{\mathscr{D}}^{*})([[\dots[a,b]_{\mathfrak{a}},x_{3}]_{\mathfrak{a}},\dots,x_{p-1}]_{\mathfrak{a}}),x_{p})$
$\displaystyle=$
$\displaystyle\sum_{\underset{x_{1}=a,~{}x_{2}=b}{x_{i}=a\text{ or
}b}}\frac{1}{\pi\\{a\\}}\times$
$\displaystyle\quad\sum_{k=1}^{p-2}(-1)^{k}x_{p}\dots
x_{p-k+1}\omega(({\mathscr{D}}+{\mathscr{D}}^{*})([[\dots[x_{1},x_{2}]_{\mathfrak{a}},x_{3}]_{\mathfrak{a}}\dots,x_{p-k-1}]_{\mathfrak{a}}),x_{p-k})$
$\displaystyle\qquad+\sum_{\underset{x_{1}=a,~{}x_{2}=b}{x_{i}=a\text{ or
}b}}\frac{1}{\pi\\{a\\}}\omega(({\mathscr{D}}+{\mathscr{D}}^{*})([[\dots[x_{1},x_{2}]_{\mathfrak{a}},x_{3}]_{\mathfrak{a}}\dots,x_{p-1}]_{\mathfrak{a}}),x_{p}).$

The last equality holds as every term in the inner sum $\sum_{k=1}^{p-2}$
vanishes, since $\mathbb{K}$ is a trivial $\mathfrak{a}$-module. ∎

######  Proposition 2.8.

Let $C$ be defined as in (2.3), and $P$ defines as in (2.5). The $2$-cochain
$(C,P)\in C^{2}_{*}(\mathfrak{a};\mathbb{K})$ is a restricted $2$-cocycle if
and only if

$\omega(({\mathscr{D}}+{\mathscr{D}}^{*})\bigl{(}b^{[p]},a\bigr{)}=\omega\bigl{(}\bigl{(}{\mathscr{D}}+{\mathscr{D}}^{*}\bigr{)}(b),\mathrm{ad}_{b}^{p-1}(a)\bigr{)}\qquad\text{
for all}\ a,b\in\mathfrak{a}_{\bar{0}}.$

###### Proof.

We have already proved in Lemma 2.4 that $C\in
Z^{2}_{\mathrm{CE}}(\mathfrak{a};\mathbb{K})$, and that $P$ satisfies the
$*$-property with respect to $C$ in Lemma 2.7. It remains to check that
$\mathrm{Ind}^{2}(C,P)(a,b)=0$, for every $a,b\in\mathfrak{a}_{0}.$ Indeed,

$\displaystyle\mathrm{Ind}^{2}(C,P)(a,b)$
$\displaystyle=C\bigl{(}a,b^{[p]}\bigr{)}-\sum_{i+j=p-1}(-1)^{i}b^{i}C\bigl{(}[[\dots[a,\overset{j\text{
terms}}{\overbrace{b],\dots],b}]},b\bigr{)}+aP(b)$
$\displaystyle=C\bigl{(}a,b^{[p]}\bigr{)}-C\bigl{(}[[\dots[a,\overset{p-1\text{
terms}}{\overbrace{b],\dots],b}]},b\bigr{)}$
$\displaystyle=\omega\bigl{(}\bigl{(}{\mathscr{D}}+{\mathscr{D}}^{*}\bigr{)}(a),b^{[p]}\bigr{)}-\omega(({\mathscr{D}}+{\mathscr{D}}^{*})([[\dots[a,\overset{p-1\text{
terms}}{\overbrace{b],\dots],b}]},b)$
$\displaystyle=-\omega\bigl{(}\bigl{(}{\mathscr{D}}+{\mathscr{D}}^{*}\bigr{)}\bigl{(}b^{[p]}\bigr{)},a\bigr{)}+\omega(({\mathscr{D}}+{\mathscr{D}}^{*})(b),\mathrm{ad}_{b}^{p-1}(a)).$
∎

##  3 Restricted orthosymplectic double extensions

###  3.1 $\boldsymbol{{\mathscr{D}}_{\bar{0}}}$-extensions

Let $(\mathfrak{a},\omega_{\mathfrak{a}})$ be a restricted orthosymplectic
quasi-Frobenius Lie superalgebra, and let
${\mathscr{D}}\in\mathfrak{der}_{\bar{0}}^{p}(\mathfrak{a})$ be a restricted
derivation.

Consider the map $P\colon\mathfrak{a}_{\bar{0}}\mapsto\mathbb{K}$ defined as
in (2.5) and satisfying the conditions (2.6) and (2.7). We have shown in Lemma
2.7 that $P$ has the $*$-property with respect to the cocycle $C$ given by
(2.3).

Consider now the following two maps

$\displaystyle\Omega\colon\
\mathfrak{a}\wedge\mathfrak{a}\rightarrow\mathbb{K},\qquad(a,b)\mapsto\omega_{\mathfrak{a}}({\mathscr{D}}\circ{\mathscr{D}}(a)+2{\mathscr{D}}^{*}\circ{\mathscr{D}}(a)+{\mathscr{D}}^{*}\circ{\mathscr{D}}^{*}(a)+\lambda({\mathscr{D}}+{\mathscr{D}}^{*})(a),b),$
$\displaystyle T\colon\ \mathfrak{a}_{\bar{0}}\rightarrow\mathbb{K},\qquad
a\mapsto\omega_{\mathfrak{a}}(({\mathscr{D}}+{\mathscr{D}}^{*})(a),\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-2}({\mathscr{D}}(a)))+\lambda
P(a).$

Let us suppose that $T$ satisfies the $*$-property with respect to $\Omega$;
and that $(\Omega,T)$ is in $B^{2}_{*}(\mathfrak{a};\mathbb{K})$. Let us write
$(\Omega,T)=\bigl{(}d^{1}_{\mathrm{CE}}(\chi),\mathrm{ind}^{1}(\chi)\bigr{)}$
for some $\chi\in C^{1}_{*}(\mathfrak{a};\mathbb{K})$; namely,
$\Omega(a,b)=\chi\bigl{(}[a,b]_{\mathfrak{a}}\bigr{)}$ for every
$a,b\in\mathfrak{a}$ and $T(a)=\chi(a^{[p]})$ for every
$a\in\mathfrak{a}_{\bar{0}}$. Since $\omega_{\mathfrak{a}}$ is non-degenerate,
there exists $Z_{\Omega}\in\mathfrak{a}$ such that

$\Omega(a,b)=\omega_{\mathfrak{a}}(Z_{\Omega},[a,b]_{\mathfrak{a}}),\qquad\forall
a,b\in\mathfrak{a}\qquad\text{and}\qquad
T(a)=\omega_{\mathfrak{a}}\bigl{(}Z_{\Omega},a^{[p]}\bigr{)},\qquad\forall
a\in\mathfrak{a}_{\bar{0}}.$ (3.1)

######  Theorem 3.1 (${\mathscr{D}}_{\bar{0}}$-extension – the case where
$\omega$ is orthosymplectic).

Let $\mathfrak{a}$ be a restricted orthosymplectic quasi-Frobenius Lie
superalgebra. Let ${\mathscr{D}}\in\mathfrak{der}_{\bar{0}}^{p}(\mathfrak{a})$
be a restricted derivation satisfying the $p$-property (2.2). Suppose further
that

$(\Omega,T)\in
B^{2}_{*}(\mathfrak{a};\mathbb{K})\qquad\text{and}\qquad(C,P)\in
Z^{2}_{*}(\mathfrak{a};\mathbb{K}).$

  * $(i)$

There exists a Lie superalgebra structure on
$\mathfrak{g}:=\mathscr{K}\oplus\mathfrak{a}\oplus\mathscr{K}^{*}$, where
$\mathscr{K}:=\mathrm{Span}\\{x\\}$ for $x$ even, defined as follows $($for
any $a,b\in\mathfrak{a})$:

$\displaystyle[x,x^{*}]_{\mathfrak{g}}=\lambda
x,\qquad[a,b]_{\mathfrak{g}}:=[a,b]_{\mathfrak{a}}+\omega_{\mathfrak{a}}({\mathscr{D}}(a)+{\mathscr{D}}^{*}(a),b)x,$
$\displaystyle[x^{*},a]_{\mathfrak{g}}:={\mathscr{D}}(a)+\omega_{\mathfrak{a}}(Z_{\Omega},a)x,$

where $\lambda\in\mathbb{K}$ and $Z_{\Omega}\in\mathfrak{a}$ is as in equation
(3.1). There exists a closed anti-symmetric orthosymplectic form
$\omega_{\mathfrak{g}}$ on $\mathfrak{g}$ defined as follows:

$\displaystyle{\omega_{\mathfrak{g}}}|_{\mathfrak{a}\times\mathfrak{a}}:=\omega_{\mathfrak{a}},\qquad\omega_{\mathfrak{g}}(\mathfrak{a},\mathscr{K}):=\omega_{\mathfrak{g}}(\mathfrak{a},\mathscr{K}^{*}):=0,$
$\displaystyle\omega_{\mathfrak{g}}(x^{*},x):=1,\qquad\omega_{\mathfrak{g}}(x,x):=\omega_{\mathfrak{g}}(x^{*},x^{*}):=0.$

  * $(ii)$

There exists a $p|2p$-map on the double extension $\mathfrak{g}$ of
$\mathfrak{a}$ given by

$\displaystyle
a^{[p]_{\mathfrak{g}}}=a^{[p]_{\mathfrak{a}}}+P(a)x,\qquad(x^{*})^{[p]_{\mathfrak{g}}}=\gamma
x^{*}+a_{0}+\tilde{\lambda}x,\qquad x^{[p]_{\mathfrak{g}}}=b_{0}+\sigma
x+\delta x^{*},$

where

    * $(a)$

The case where $\lambda\not=0$:

$\displaystyle{\mathscr{D}}(a_{0})=0,\qquad\tilde{\lambda}=\frac{1}{\lambda}\omega(Z_{\Omega},a_{0}),\qquad\gamma=\lambda^{p-1},\qquad\delta=0,$
$\displaystyle{\mathscr{D}}(b_{0})=0,\qquad\sigma=\frac{1}{\lambda}\omega(Z_{\Omega},b_{0}),\qquad{\mathscr{D}}^{*}(b_{0})=0,\qquad
b_{0}\in{\mathfrak{z}}(\mathfrak{a}),$

and

${\mathscr{D}}^{*}(a_{0})=\sum_{1\leq i\leq
p-1}(-1)^{p-1-i}\lambda^{p-1-i}{{\mathscr{D}}^{*}}^{i}(Z_{\Omega}).$ (3.2)

    * $(b)$

The case where $\lambda=0$ and
${\mathscr{D}}\not=-\delta^{-1}\mathrm{ad}_{b_{0}}$:

$\displaystyle{\mathscr{D}}(a_{0})=0,\qquad\omega(Z_{\Omega},a_{0})=0,\qquad\delta=0,$
$\displaystyle{\mathscr{D}}(b_{0})=0,\qquad\omega(Z_{\Omega},b_{0})=0,\qquad{\mathscr{D}}^{*}(b_{0})=0,\qquad
b_{0}\in{\mathfrak{z}}(\mathfrak{a}),$

and

${\mathscr{D}}^{*}(a_{0})+\gamma
Z_{\Omega}={{\mathscr{D}}^{*}}^{p-1}(Z_{\Omega}).$ (3.3)

    * $(c)$

The case where $\lambda=0$ and ${\mathscr{D}}=-\delta^{-1}\mathrm{ad}_{b_{0}}$
is inner:

$\displaystyle{\mathscr{D}}(a_{0})=0,\qquad\omega(Z_{\Omega},a_{0})=0,$
$\displaystyle{\mathscr{D}}(b_{0})=0,\qquad\omega(Z_{\Omega},b_{0})=0,\qquad{\mathscr{D}}^{*}(b_{0})=-\delta
Z_{\Omega},$

and

${\mathscr{D}}^{*}(a_{0})+\gamma
Z_{\Omega}={{\mathscr{D}}^{*}}^{p-1}(Z_{\Omega}).$ (3.4)

The orthosymplectic quasi-Frobenius Lie superalgebra
$(\mathfrak{g},\omega_{\mathfrak{g}})$ will be called the symplectic double
extension by means of a 1-dimensional space of the orthosymplectic quasi-
Frobenius Lie superalgebra $(\mathfrak{a},\omega_{\mathfrak{a}})$. In the
particular case where $x$ is central in $\mathfrak{g}$, namely $\lambda=0$,
then the double extension was called classical in [3, 17].

###### Proof.

The proof of part (i) is in [14], so we omit it. Let us prove part (ii), using
Jacobson’s theorem. We have

$\displaystyle\mathrm{ad}_{(x^{*})^{[p]_{\mathfrak{g}}}}^{\mathfrak{g}}(x)-\bigl{(}\mathrm{ad}_{x^{*}}^{\mathfrak{g}}\bigr{)}^{p}(x)$
$\displaystyle=\big{[}a_{0}+\gamma
x^{*}+\tilde{\lambda}x,x\big{]}_{\mathfrak{g}}-\bigl{(}\mathrm{ad}_{x^{*}}^{\mathfrak{g}}\bigr{)}^{p-1}(-\lambda
x)$ $\displaystyle=-\lambda\gamma x-(-1)^{p}\lambda^{p}x=0.$

Additionally,

$\displaystyle\bigl{(}\mathrm{ad}_{(x^{*})^{[p]_{\mathfrak{g}}}}^{\mathfrak{g}}\bigr{)}(x^{*})-\bigl{(}\mathrm{ad}_{x^{*}}^{\mathfrak{g}}\bigr{)}^{p}(x^{*})$
$\displaystyle=\big{[}a_{0}+\gamma
x^{*}+\tilde{\lambda}x,x^{*}\big{]}_{\mathfrak{g}}-(\mathrm{ad}_{x^{*}}^{\mathfrak{g}})^{p-1}\bigl{(}[x^{*},x^{*}]_{\mathfrak{g}}\bigr{)}$
$\displaystyle=-(\mathscr{D}(a_{0})+\omega_{\mathfrak{a}}(Z_{\Omega},a_{0})xx)+\tilde{\lambda}\lambda
x$
$\displaystyle=-\mathscr{D}(a_{0})+\bigl{(}\tilde{\lambda}\lambda-\omega_{\mathfrak{a}}(Z_{\Omega},a_{0})\bigr{)}x$
$\displaystyle=0.$

Moreover,

$\displaystyle\mathrm{ad}_{(x^{*})^{[p]_{\mathfrak{g}}}}^{\mathfrak{g}}(a)-\bigl{(}\mathrm{ad}_{x^{*}}^{\mathfrak{g}}\bigr{)}^{p}(a)={}$
$\displaystyle\big{[}a_{0}+\gamma
x^{*}+\tilde{\lambda}x,a\big{]}_{\mathfrak{g}}-\bigl{(}\mathrm{ad}_{x^{*}}^{\mathfrak{g}}\bigr{)}^{p-1}\bigl{(}\big{[}x^{*},a\big{]}_{\mathfrak{g}}\bigr{)}$
$\displaystyle={}$
$\displaystyle[a_{0},a]_{\mathfrak{g}}+\gamma\big{[}x^{*},a\big{]}_{\mathfrak{a}}+-\bigl{(}\mathrm{ad}_{x^{*}}^{\mathfrak{g}}\bigr{)}^{p-1}\bigl{(}\mathscr{D}(a)+\omega_{\mathfrak{a}}(Z_{\Omega},a)\bigr{)}$
$\displaystyle={}$
$\displaystyle[a_{0},a]_{\mathfrak{a}}+\omega_{\mathfrak{a}}\bigl{(}\bigl{(}{\mathscr{D}}+{\mathscr{D}}^{*}\bigr{)}(a_{0}),a\bigr{)}x+\gamma\bigl{(}{\mathscr{D}}(a)+\omega_{\mathfrak{a}}(a_{0},a)x\bigr{)}$
$\displaystyle-\bigg{(}\mathscr{D}^{p}(a)-\omega_{\mathfrak{a}}\bigg{(}Z_{\Omega},\sum_{i=0}^{p-1}(-1)^{p-1-i}\lambda^{p-1-i}{\mathscr{D}}^{i}(a)\bigg{)}\bigg{)}$
$\displaystyle={}$ $\displaystyle 0,$

as ${\mathscr{D}}$ satisfies the $p$-property and (3.2), or (3.3) or (3.4).

On the other hand,

$\mathrm{ad}_{a^{[p]_{\mathfrak{g}}}}^{\mathfrak{g}}(x)-\big{(}\mathrm{ad}_{a}^{\mathfrak{g}}\big{)}^{p}(x)=0.$

Additionally,

$\displaystyle\mathrm{ad}_{a^{[p]_{\mathfrak{g}}}}^{\mathfrak{g}}(x^{*})-\bigl{(}\mathrm{ad}_{a}^{\mathfrak{g}}\bigr{)}^{p}(x^{*})={}$
$\displaystyle\big{[}a^{[p]_{\mathfrak{a}}}+P(a)x,x^{*}\big{]}_{\mathfrak{g}}-(\mathrm{ad}_{a}^{\mathfrak{g}})^{p-1}\bigl{(}-\mathscr{D}(a)-\omega_{\mathfrak{a}}(Z_{\Omega},a)x\bigr{)}$
$\displaystyle={}$
$\displaystyle-{\mathscr{D}}\bigl{(}a^{[p]_{\mathfrak{a}}}\bigr{)}-\omega_{\mathfrak{a}}\bigl{(}Z_{\Omega},a^{[p]_{\mathfrak{a}}}\bigr{)}x+\lambda
P(a)x$
$\displaystyle{}-\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p}\circ{\mathscr{D}}(a)-\omega_{\mathfrak{a}}\bigl{(}\bigl{(}{\mathscr{D}}+{\mathscr{D}}^{*}\bigr{)}(a),\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-2}\circ{\mathscr{D}}(a)\bigr{)}x$
$\displaystyle={}$ $\displaystyle 0,$

as ${\mathscr{D}}$ is a restricted derivation, and $(\Omega,T)$ is a
coboundary.

Moreover,

$\displaystyle\mathrm{ad}_{a^{[p]_{\mathfrak{g}}}}^{\mathfrak{g}}(b)-(\mathrm{ad}_{a}^{\mathfrak{g}})^{p}(b)={}$
$\displaystyle\big{[}a^{[p]_{\mathfrak{a}}}+P(a)x,b\big{]}_{\mathfrak{g}}-\bigl{(}\mathrm{ad}_{a}^{\mathfrak{g}}\bigr{)}^{p-1}([a,b]_{\mathfrak{g}})$
$\displaystyle={}$
$\displaystyle\big{[}a^{[p]_{\mathfrak{a}}},b\big{]}_{\mathfrak{a}}+\omega_{\mathfrak{a}}\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})\bigl{(}a^{[p]_{\mathfrak{a}}}\bigr{)},b\bigr{)}x-\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-1}(b)$
$\displaystyle-\omega_{\mathfrak{a}}\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(a),\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-1}(b)\bigr{)}x$
$\displaystyle={}$ $\displaystyle 0,$

as the Lie superalgebra $\mathfrak{a}$ is restricted, $(C,P)$ is a 2-cocycle
and hence Proposition 2.8 can be applied.

Finally, using the same techniques as before we can show that
$\mathrm{ad}_{x^{[p]_{\mathfrak{g}}}}^{\mathfrak{g}}=\big{(}\mathrm{ad}_{x}^{\mathfrak{g}}\big{)}^{p}$.
∎

######  Theorem 3.2 (converse of Theorem 3.1).

Let $(\mathfrak{g},\omega_{\mathfrak{g}})$ be a restricted orthosymplectic
quasi-Frobenius Lie superalgebra. Suppose there exists an even non-zero
$x\in([\mathfrak{g},\mathfrak{g}]_{\mathfrak{g}})^{\perp}$ such that
$\mathscr{K}:=\mathrm{Span}\\{x\\}$ is an ideal, and $\mathscr{K}^{\perp}$ is
a $p$-ideal. Then, $(\mathfrak{g},\omega_{\mathfrak{g}})$ is obtained as a
symplectic ${\mathscr{D}}_{\bar{0}}$-extension by a 1-dimensional space from a
restricted orthosymplectic quasi-Frobenius Lie superalgebra
$(\mathfrak{a},\omega_{\mathfrak{a}})$. Moreover, if
$\mathfrak{z}_{\bar{0}}(\mathfrak{g})\not=0$, then we can choose
$x\in\mathfrak{z}_{\bar{0}}(\mathfrak{g})$, so the double extension is
classical.

###### Proof.

It has been proved in [14] that the space $\mathscr{K}^{\perp}$ is an ideal in
$(\mathfrak{g},\omega_{\mathfrak{g}})$, and that there exists
$x^{*}\in\mathfrak{g}_{\bar{0}}$ (since
$\bigl{(}\mathscr{K}^{\perp}\bigr{)}_{\bar{1}}=\mathfrak{g}_{\bar{1}}$) such
that

$\mathfrak{g}=\mathscr{K}^{\perp}\oplus\mathscr{K}^{*},\qquad\text{where}\
\mathscr{K}^{*}:=\mathrm{Span}\\{x^{*}\\}.$

We can normalize $x^{*}$ so that $\omega_{\mathfrak{g}}(x^{*},x)=1$.

Let us define $\mathfrak{a}:=(\mathscr{K}+\mathscr{K}^{*})^{\perp}$. We then
have a decomposition
$\mathfrak{g}=\mathscr{K}\oplus\mathfrak{a}\oplus\mathscr{K}^{*}$.

Let us define an orthosymplectic form on $\mathfrak{a}$ by setting

$\omega_{\mathfrak{a}}={\omega_{\mathfrak{g}}}|_{\mathfrak{a}\times\mathfrak{a}}.$

It has been proved in [14] that the vector space $\mathfrak{a}$ can be endowed
with a Lie superalgebra structure, and there exists an orthosymplectic
structure on $\mathfrak{a}$ for which $\mathfrak{g}$ is its symplectic double
extension by means of the form $\omega_{\mathfrak{a}}$, a derivation
$\mathscr{D}$ and $Z_{\Omega}\in\mathfrak{a}$ as in part (i) of Theorem 3.1.
In particular, it has been shown that the map

$\Omega\colon\
\mathfrak{a}\wedge\mathfrak{a}\rightarrow\mathbb{K},\qquad(a,b)\mapsto\omega_{\mathfrak{a}}({\mathscr{D}}\circ{\mathscr{D}}(a)+2{\mathscr{D}}^{*}\circ{\mathscr{D}}(a)+{\mathscr{D}}^{*}\circ{\mathscr{D}}^{*}(a)+\lambda({\mathscr{D}}+{\mathscr{D}}^{*})(a),b)$

is in $B^{2}_{\text{CE}}(\mathfrak{a};\mathbb{K})$, which implies that

$\Omega(a,b)=\omega_{\mathfrak{a}}(Z_{\Omega},[a,b]_{\mathfrak{a}})\qquad\text{for
some}\ Z_{\Omega}\in\mathfrak{a}.$ (3.5)

Moreover, the map

$C\colon\
\mathfrak{a}\wedge\mathfrak{a}\rightarrow\mathbb{K},\qquad(a,b)\mapsto\omega_{\mathfrak{a}}(({\mathscr{D}}+{\mathscr{D}}^{*})(a),b)$

is in $Z_{\mathrm{CE}}^{2}(\mathfrak{a};\mathbb{K})$ by Lemma 2.4.

It remains to show that there is a $[p|2p]$-map on $\mathfrak{a}$. Since
$\mathfrak{a}\subset\mathscr{K}^{\perp}$ and ${\mathscr{K}}^{\perp}$ is a
$p$-ideal, then

$a^{[p]_{\mathfrak{g}}}\in\mathscr{K}^{\perp}=\mathscr{K}\oplus\mathfrak{a}\qquad\text{
for any}\ a\in\mathfrak{a}_{\bar{0}}.$

It follows that

$a^{[p]_{\mathfrak{g}}}=s(a)+P(a)x.$

We will show that the map

$s\colon\ \mathfrak{a}_{\bar{0}}\rightarrow\mathfrak{a}_{\bar{0}},\qquad
a\mapsto s(a)$

is a $[p|2p]$-mapping on $\mathfrak{a}$. Since $(\delta
a)^{[p]_{\mathfrak{g}}}=\delta^{p}\bigl{(}a^{[p]_{\mathfrak{g}}}\bigr{)}$, for
all $\delta\in\mathbb{K}$ and for all $a\in\mathfrak{a}_{\bar{0}}$, it follows
that

$\displaystyle s(\delta a)=\delta^{p}s(a),$ (3.6) $\displaystyle P(\delta
a)=\delta^{p}P(a).$ (3.7)

Besides, for all $a\in\mathfrak{a}_{\bar{0}}$ and for all $b\in\mathfrak{a}$,
we have

$\displaystyle\begin{split}0&=\big{[}a^{[p]_{\mathfrak{g}}},b\big{]}_{\mathfrak{g}}-\bigl{(}\mathrm{ad}_{a}^{\mathfrak{g}}\bigr{)}^{p}(b)\\\
&=[s(a),b]_{\mathfrak{a}}+\omega_{\mathfrak{a}}(({\mathscr{D}}+{\mathscr{D}}^{*})(s(a)),b)x-\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p}(b)-\omega_{\mathfrak{a}}\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(a),\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-1}(b)\bigr{)}x.\end{split}$

Therefore,

$\displaystyle\omega_{\mathfrak{a}}(({\mathscr{D}}+{\mathscr{D}}^{*})(s(a)),b)=\omega_{\mathfrak{a}}\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(a),\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-1}(b)\bigr{)},$
(3.8)
$\displaystyle[s(a),b]_{\mathfrak{a}}=\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p}(b).$
(3.9)

Now, since

$\displaystyle\sum_{1\leq i\leq p-1}is_{i}^{\mathfrak{g}}(a,b)\mu^{i-1}$
$\displaystyle=(\mathrm{ad}_{\mu a+b}^{\mathfrak{g}})^{p-1}(a)$
$\displaystyle=(\mathrm{ad}_{\mu
a+b}^{\mathfrak{a}})^{p-1}(a)+\omega_{\mathfrak{a}}\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(\mu
a+b),(\mathrm{ad}_{\mu a+b}^{\mathfrak{a}})^{p-2}(a)\bigr{)}x,$

it follows that

$s^{\mathfrak{g}}_{i}(a,b)=s_{i}^{\mathfrak{a}}(a,b)+\sigma_{i}^{\mathfrak{a}}(a,b)x.$

Moreover,

$\displaystyle 0={}$
$\displaystyle(a+b)^{[p]_{\mathfrak{g}}}-a^{[p]_{\mathfrak{g}}}-b^{[p]_{\mathfrak{g}}}-\sum_{1\leq
i\leq p-1}s_{i}^{\mathfrak{g}}(a,b)$ $\displaystyle={}$
$\displaystyle\bigg{(}P(a+b)-P(a)-P(b)-\sum_{1\leq i\leq
p-1}\sigma_{i}^{\mathfrak{a}}(a,b)\bigg{)}x$
$\displaystyle{}+s(a+b)-s(a)-s(b)-\sum_{1\leq i\leq
p-1}s_{i}^{\mathfrak{a}}(a,b).$

Consequently,

$\displaystyle s(a+b)=s(a)+s(b)+\sum_{1\leq i\leq
p-1}s_{i}^{\mathfrak{a}}(a,b),$ (3.10) $\displaystyle
P(a+b)-P(a)-P(b)=\sum_{1\leq i\leq p-1}\sigma_{i}^{\mathfrak{a}}(a,b).$

Equations (3.6), (3.9), (3.10) imply that $s$ defines a $[p|2p]$-map on
$\mathfrak{a}$.

Now, since
$\big{[}a^{[p]_{\mathfrak{g}}},x^{*}\big{]}_{\mathfrak{g}}=\bigl{(}\mathrm{ad}_{a}^{\mathfrak{g}}\bigr{)}^{p}(x^{*})$,
then

$\displaystyle P(a)\lambda
x-\mathscr{D}(s(a))-\omega_{\mathfrak{a}}(Z_{\Omega},s(a))x$
$\displaystyle\qquad{}=-\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-1}\circ\mathscr{D}(a)-\omega_{\mathfrak{a}}(({\mathscr{D}}+{\mathscr{D}}^{*})(a),\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-2}\circ{\mathscr{D}}(a))x.$

It follows that

$\displaystyle\mathscr{D}(s(a))=\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-1}\circ\mathscr{D}(a),$
(3.11) $\displaystyle
P(a)\lambda=\omega_{\mathfrak{a}}(Z_{\Omega},s(a))-\omega_{\mathfrak{a}}(({\mathscr{D}}+{\mathscr{D}}^{*})(a),\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-2}\circ{\mathscr{D}}(a)).$
(3.12)

Equation (3.11) implies that ${\mathscr{D}}$ is a restricted derivation of
$\mathfrak{a}$ (relative to the $[p|2p]$-map $s$). Let us defined the map

$T\colon\ a\rightarrow\mathbb{K},\qquad
a\mapsto\omega_{\mathfrak{a}}(({\mathscr{D}}+{\mathscr{D}}^{*})(a),\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-2}\circ{\mathscr{D}}(a))+\lambda
P(a).$

We need the following lemma.

######  Lemma 3.3.

$(C,P)\in Z^{2}_{*}(\mathfrak{a},\mathbb{K})\qquad\text{ and
}\qquad(\Omega,T)\in B^{2}_{*}(\mathfrak{a},\mathbb{K}).$

###### Proof.

Equations (3.7) and (3.10) and Lemma 2.7, imply that $P$ satisfies the
$*$-property with respect to the cocycle $C$. Moreover, (3.8) and Proposition
2.8 imply that $(C,P)\in Z^{2}_{*}(\mathfrak{a};\mathbb{K})$.

We have already showed in [14] that $\Omega\in
B^{2}_{\text{CE}}(\mathfrak{a};\mathbb{K})$. Let us show that the map $T$
satisfies the $*$-property with respect to $\Omega$. Indeed, for all
$\delta\in\mathbb{K}$ and for all $a\in\mathfrak{a}_{\bar{0}}$, we have

$\displaystyle T(\delta a)$
$\displaystyle=\omega_{\mathfrak{a}}(({\mathscr{D}}+{\mathscr{D}}^{*})(\delta
a),(\mathrm{ad}_{\delta a}^{\mathfrak{a}})^{p-2}\circ{\mathscr{D}}(\delta
a))+\lambda P(\delta a)$
$\displaystyle=\delta^{p}\omega(({\mathscr{D}}+{\mathscr{D}}^{*})(a),(\mathrm{ad}_{a}^{\mathfrak{a}})^{p-2}\circ{\mathscr{D}}(a))+\lambda\delta^{p}P(a)$
$\displaystyle=\delta^{p}T(a).$

Moreover, for all $a,b\in\mathfrak{a}_{\bar{0}}$, we have

$\displaystyle
T(a+b)\overset{\text{(by~{}\eqref{Pcbound})}}{=}\omega_{\mathfrak{a}}\bigl{(}Z_{\Omega},(a+b)^{[p]_{\mathfrak{a}}}\bigr{)}$
$\displaystyle\qquad{}\overset{\text{(by definition)
}}{=}\omega_{\mathfrak{a}}\bigl{(}Z_{\Omega},a^{[p]_{\mathfrak{a}}}\bigr{)}+\omega_{\mathfrak{a}}\bigl{(}Z_{\Omega},b^{[p]_{\mathfrak{a}}}\bigr{)}+\omega_{\mathfrak{a}}\bigg{(}Z_{\Omega},\sum_{1\leq
i\leq p-1}s_{i}(a,b)\bigg{)}$ $\displaystyle\qquad{}\overset{\text{(by Lemma
\ref{1lemma})}}{=}{}T(a)+T(b)+\sum_{\underset{x_{p-1}=b,~{}x_{p}=a}{x_{k}=a\text{
or
}b}}\frac{1}{\pi(a)}\omega_{\mathfrak{a}}(Z_{\Omega},[x_{1},[x_{2},[\dots,[x_{p-1},x_{p}]_{\mathfrak{a}}\dots]_{\mathfrak{a}}]_{\mathfrak{a}}]_{\mathfrak{a}})$
$\displaystyle\qquad{}\overset{\text{(by~{}\eqref{Omegatr})}}{=}T(a)+T(b)+\sum_{\underset{x_{p-1}=b,~{}x_{p}=a}{x_{k}=a\text{
or
}b}}\frac{1}{\pi(a)}\Omega(x_{1},[x_{2},[\dots,[x_{p-1},x_{p}]_{\mathfrak{a}}\dots]_{\mathfrak{a}}]_{\mathfrak{a}})$
$\displaystyle\qquad{}=T(a)+T(b)+\sum_{\underset{x_{p-1}=b,~{}x_{p}=a}{x_{k}=a\text{
or
}b}}\frac{1}{\pi(a)}\Omega([[[x_{p},x_{p-1}]_{\mathfrak{a}},x_{p-2}]_{\mathfrak{a}},\dots,x_{2}]_{\mathfrak{a}},x_{1}).$

Now, equations (3.5), (3.12) imply that $(\Omega,T)\in
B^{2}_{*}(\mathfrak{a};\mathbb{Z})$. ∎

Now, suppose that

$\displaystyle(x^{*})^{[p]_{\mathfrak{g}}}=a_{0}+\beta x+\gamma
x^{*},\qquad\text{where}\ a_{0}\in\mathfrak{a}\ \text{and}\
\beta,\gamma\in\mathbb{K}.$

We have

$0=\big{[}(x^{*})^{[p]},x\big{]}_{\mathfrak{g}}-\bigl{(}\mathrm{ad}_{x^{*}}^{\mathfrak{g}}\bigr{)}^{p}(x)=-\gamma\lambda
x+\lambda^{p}x.$

Therefore, $\gamma\lambda=\lambda^{p}$. Additionally,

$0=\big{[}(x^{*})^{[p]},x^{*}\big{]}_{\mathfrak{g}}-\bigl{(}\mathrm{ad}_{x^{*}}^{\mathfrak{g}}\bigr{)}^{p}(x^{*})=\beta\lambda
x-({\mathscr{D}}(a_{0})+\omega_{\mathfrak{a}}(Z_{\Omega},a_{0})x).$

Therefore, ${\mathscr{D}}(a_{0})=0$ and
$\lambda\beta=\omega_{\mathfrak{a}}(Z_{\Omega},a_{0})$.

For all $a\in\mathfrak{a}$, we have

$\displaystyle 0={}$
$\displaystyle\big{[}(x^{*})^{[p]_{\mathfrak{g}}},a\big{]}_{\mathfrak{g}}-\bigl{(}\mathrm{ad}_{x^{*}}^{\mathfrak{g}}\bigr{)}^{p}(a)$
$\displaystyle={}$
$\displaystyle[a_{0},a]+\omega_{\mathfrak{a}}(({\mathscr{D}}+{\mathscr{D}}^{*})(a_{0}),a)x+\gamma{\mathscr{D}}(a)+\gamma\omega(Z_{\Omega},a)x-{\mathscr{D}}^{p}(a)$
$\displaystyle-\sum_{1\leq i\leq
p-1}\omega_{\mathfrak{a}}\bigl{(}(-\lambda)^{p-1-i}{\mathscr{D}^{*}}^{i}(Z_{\Omega}),a\bigr{)}x-(-\lambda)^{p-1}\omega(Z_{\Omega},a)x.$

It follows that ${\mathscr{D}}^{p}=\gamma{\mathscr{D}}+\mathrm{ad}_{a_{0}}$
and ${\mathscr{D}}^{*}(a_{0})+\gamma Z_{\Omega}=\sum_{0\leq i\leq
p-1}(-\lambda)^{p-1-i}{\mathscr{D}^{*}}^{i}(Z_{\Omega})$
($\omega_{\mathfrak{g}}$ is non-degenerate).

Suppose now that

$x^{[p]_{\mathfrak{g}}}=b_{0}+\sigma x+\delta x^{*},\qquad\text{where}\
\sigma,\delta\in\mathbb{K}\ \text{and}\ b_{0}\in\mathfrak{a}_{\bar{0}}.$

We have

$0=\big{[}x^{[p]},x\big{]}_{\mathfrak{g}}-\bigl{(}\mathrm{ad}_{x}^{\mathfrak{g}}\bigr{)}^{p}(x)=-\delta\lambda
x.$

Therefore, $\lambda\delta=0$. Additionally,

$0=\big{[}x^{[p]},x^{*}\big{]}_{\mathfrak{g}}-\bigl{(}\mathrm{ad}_{x}^{\mathfrak{g}}\bigr{)}^{p}(x^{*})=\sigma\lambda
x-({\mathscr{D}}(b_{0})+\omega_{\mathfrak{a}}(Z_{\Omega},b_{0})x).$

Therefore, ${\mathscr{D}}(b_{0})=0$ and
$\lambda\sigma-\omega(Z_{\Omega},b_{0})=0$.

Now, for any $b\in\mathfrak{a}$, we have

$0=\big{[}x^{[p]_{\mathfrak{g}}},b\big{]}_{\mathfrak{g}}-\bigl{(}\mathrm{ad}_{x}^{\mathfrak{g}}\bigr{)}^{p}(b)=[b_{0},b]_{\mathfrak{a}}+\omega_{\mathfrak{a}}(({\mathscr{D}}+{\mathscr{D}}^{*})(b_{0}),b)x+\delta({\mathscr{D}}(b)+\omega_{\mathfrak{a}}(Z_{\Omega},b)x).$

It follows that $\delta{\mathscr{D}}(b)+[b_{0},b]_{\mathfrak{a}}=0$ and
${\mathscr{D}}^{*}(b_{0})+\delta Z_{\Omega}=0$ (since $\omega_{\mathfrak{a}}$
is non-degenerate).

The case where $\lambda\not=0$. It follows that $\delta=0$,
${\mathscr{D}}^{*}(b_{0})=0$ and $b_{0}\in\mathfrak{z}(\mathfrak{a})$.
Therefore, $\mathfrak{g}$ can be obtained from the restricted Lie superalgebra
algebra $\mathfrak{a}$ as in Theorem 3.1.

The case where $\lambda=0$. It follows that $\omega(Z_{\Omega},b_{0})=0$. Here
if $\delta=0$ then $b_{0}\in\mathfrak{z}(\mathfrak{a})$. Therefore,
$\mathfrak{g}$ can be obtained from the restricted Lie superalgebra algebra
$\mathfrak{a}$ as in Theorem 3.1. However, if $\delta\not=0$, then it follows
that ${\mathscr{D}}=-\delta^{-1}\mathrm{ad}_{b_{0}}$ and
${\mathscr{D}}^{*}(b_{0})=-\delta Z_{\Omega}$. ∎

###  3.2 $\boldsymbol{{\mathscr{D}}_{\bar{1}}}$-extensions

Let $(\mathfrak{a},\omega_{\mathfrak{a}})$ be a restricted orthosymplectic
quasi-Frobenius Lie superalgebra, and let
${\mathscr{D}}\in\mathfrak{der}_{\bar{1}}(\mathfrak{a})$ be a derivation.

Consider the following two maps

$\displaystyle\Omega\colon\
\mathfrak{a}\wedge\mathfrak{a}\rightarrow\mathbb{K},\qquad(a,b)\mapsto\omega_{\mathfrak{a}}({\mathscr{D}}\circ{\mathscr{D}}(a)-{\mathscr{D}}^{*}\circ{\mathscr{D}}^{*}(a),b),$
$\displaystyle\Delta\colon\ \mathfrak{a}_{\bar{0}}\rightarrow\mathbb{K},\qquad
a\mapsto\omega_{\mathfrak{a}}(({\mathscr{D}}+{\mathscr{D}}^{*})(a),\mathrm{ad}_{a}^{p-2}({\mathscr{D}}(a))).$

Let us suppose that $(\Omega,\Delta)$ is in
$B^{2}_{*}(\mathfrak{g};\mathbb{K})$ (i.e., a 2-coboundary in the restricted
cohomology). Let us write
$(\Omega,\Delta)=\bigl{(}d^{1}_{\mathrm{CE}}(\chi),\mathrm{ind}^{1}(\chi)\bigr{)}$
for some $\chi\in C^{1}_{*}(\mathfrak{a};\mathbb{K})$; namely,

$\Omega(a,b)=\chi([a,b]_{\mathfrak{a}})\qquad\text{ and
}\qquad\Delta(a)=\chi\bigl{(}a^{[p]}\bigr{)}.$

Since $\omega_{\mathfrak{a}}$ is non-degenerate, there exists
$a_{0}\in\mathfrak{a}$ such that

$\Omega(a,b)=\omega_{\mathfrak{a}}(a_{0},[a,b]_{\mathfrak{a}}),\qquad\forall
a,b\in\mathfrak{a}\qquad\text{and}\qquad\Delta(a)=\omega_{\mathfrak{a}}\bigl{(}a_{0},a^{[p]}\bigr{)},\qquad\forall
a\in\mathfrak{a}_{\bar{0}}.$ (3.13)

######  Theorem 3.4 (${\mathscr{D}}_{\bar{1}}$-extension – the case where
$\omega$ is orthosymplectic).

Let $(\mathfrak{a},\omega_{\mathfrak{a}})$ be a restricted orthosymplectic
quasi-Frobenius Lie superalgebra. Let
${\mathscr{D}}\in\mathfrak{der}_{\bar{1}}(\mathfrak{a})$ be a restricted
derivation such that the following conditions are satisfied

$(\Omega,\Delta)\in B^{2}_{*}(\mathfrak{a};\mathbb{K}),$

together with

${\mathscr{D}}^{2}=\mathrm{ad}_{a_{0}}\qquad\text{ and
}\qquad{\mathscr{D}}(a_{0})=0,$

where $a_{0}$ is as in (3.13).

  * $(i)$

There exists a Lie superalgebra structure on
$\mathfrak{g}:=\mathscr{K}\oplus\mathfrak{a}\oplus\mathscr{K}^{*}$, where
$\mathscr{K}:=\mathrm{Span}\\{x\\}$ for $x$ odd, defined as follows: $($for
any $a,b\in\mathfrak{a})$

$\displaystyle[\mathscr{K},\mathfrak{g}]_{\mathfrak{g}}=0,\qquad[a,b]_{\mathfrak{g}}:=[a,b]_{\mathfrak{a}}+\bigl{(}\omega_{\mathfrak{a}}({\mathscr{D}}(a),b)+(-1)^{|a|}\omega_{\mathfrak{a}}(a,{\mathscr{D}}(b))\bigr{)}x,$
$\displaystyle[x^{*},x^{*}]_{\mathfrak{g}}=2a_{0},\qquad[x^{*},a]_{\mathfrak{g}}:={\mathscr{D}}(a)-\omega_{\mathfrak{a}}(a,a_{0})x.$

There exists a closed anti-symmetric orthosymplectic form
$\omega_{\mathfrak{g}}$ on $\mathfrak{g}$ defined as follows:

$\displaystyle{\omega_{\mathfrak{g}}}|_{\mathfrak{a}\times\mathfrak{a}}:=\omega_{\mathfrak{a}},\qquad\omega_{\mathfrak{g}}(\mathfrak{a},\mathscr{K}):=\omega_{\mathfrak{g}}(\mathfrak{a},\mathscr{K}^{*}):=0,$
$\displaystyle\omega_{\mathfrak{g}}(x^{*},x):=1,\qquad\omega_{\mathfrak{g}}(x,x):=\omega_{\mathfrak{g}}(x^{*},x^{*}):=0.$

  * $(ii)$

There exists a $[p|2p]$-map on the double extension $\mathfrak{g}$ of
$\mathfrak{a}$ given by

$a^{[p]_{\mathfrak{g}}}=a^{[p]_{\mathfrak{a}}}.$

###### Proof.

Similar to that of Theorem 3.1. ∎

######  Theorem 3.5 (converse of Theorem 3.4).

Let $(\mathfrak{g},\omega_{\mathfrak{g}})$ be a restricted orthosymplectic
quasi-Frobenius Lie superalgebra. Suppose there exists a non-zero
$x\in\mathfrak{z}(\mathfrak{g})_{\bar{1}}$ such that $\omega(x,x)=~{}0$. Then,
$(\mathfrak{g},\omega_{\mathfrak{g}})$ is a
${\mathscr{D}}_{\bar{1}}$-extension of a restricted orthosymplectic quasi-
Frobenius Lie superalgebra $(\mathfrak{a},\omega_{\mathfrak{a}})$.

######  Remark 3.6 (the condition $\omega(x,x)=0$).

The condition $\omega(x,x)=0$ is actually necessary. A counterexample is the
Lie superalgebra $C^{1}_{1/2}+A$, see [14].

###### Proof of Theorem 3.5.

Let $x$ be a non-zero element in $\mathfrak{z}(\mathfrak{g})_{\bar{1}}$. It
has been proved in [14] that the subspaces $\mathscr{K}:=\mathrm{Span}\\{x\\}$
and $\mathscr{K}^{\perp}$ are ideals in
$(\mathfrak{g},\omega_{\mathfrak{g}})$. Since $\mathscr{K}$ is 1-dimensional
and $\omega(x,x)=0$, it follows that $\mathscr{K}\subset\mathscr{K}^{\perp}$
and $\dim\bigl{(}\mathscr{K}^{\perp}\bigr{)}=\dim(\mathfrak{g})-1$. Therefore,
there exists $x^{*}\in\mathfrak{g}_{\bar{1}}$ such that

$\mathfrak{g}=\mathscr{K}^{\perp}\oplus\mathscr{K}^{*},\qquad\text{where}\
\mathscr{K}^{*}:=\mathrm{Span}\\{x^{*}\\}.$

This $x^{*}$ can be normalized to have $\omega_{\mathfrak{g}}(x^{*},x)=1$.

Let us define $\mathfrak{a}:=(\mathscr{K}+\mathscr{K}^{*})^{\perp}$. We then
have a decomposition
$\mathfrak{g}=\mathscr{K}\oplus\mathfrak{a}\oplus\mathscr{K}^{*}$.

Let us define an orthosymplectic form on $\mathfrak{a}$ by setting

$\omega_{\mathfrak{a}}={\omega_{\mathfrak{g}}}|_{\mathfrak{a}\times\mathfrak{a}}.$

It has been proved in [14] that the form $\omega_{\mathfrak{a}}$ is non-
degenerate on $\mathfrak{a}$, and that $\mathfrak{g}$ is a symplectic double
extension of $\mathfrak{a}$ by means of the form $\omega_{\mathfrak{a}}$ and a
derivation ${\mathscr{D}}$ satisfying ${\mathscr{D}}^{2}=\mathrm{ad}_{a_{0}}$
and ${\mathscr{D}}(a_{0})=0$. Moreover, it has been proved that the map
$\Omega\in B^{2}_{\text{CE}}(\mathfrak{a};\mathbb{K})$, and that the map $C\in
Z^{2}_{\text{CE}}(\mathfrak{a};\mathbb{K})$.

It remains to show that there is a $[p|2p]$-map on $\mathfrak{a}$. Since
$\mathfrak{a}\subset\mathscr{K}^{\perp}$ and since $x$ and $x^{*}$ are both
odd, then

$a^{[p]_{\mathfrak{g}}}\in\mathfrak{a}\qquad\text{for any}\
a\in\mathfrak{a}_{\bar{0}}.$

It follows that

$a^{[p]_{\mathfrak{g}}}=s(a).$

We will show that the map

$s\colon\ \mathfrak{a}_{\bar{0}}\rightarrow\mathfrak{a}_{\bar{0}},\qquad
a\mapsto s(a),$

is a $[p|2p]$-map on $\mathfrak{a}$. The fact that $(\delta
a)^{[p]_{\mathfrak{g}}}=\delta^{p}\bigl{(}a^{[p]_{\mathfrak{g}}}\bigr{)}$
implies that $s(\delta a)=\delta^{p}s(a)$, for all $\delta\in\mathbb{K}$ and
$a\in\mathfrak{a}_{\bar{0}}$. Besides,

$\displaystyle 0$
$\displaystyle=\big{[}a^{[p]_{\mathfrak{g}}},b\big{]}_{\mathfrak{g}}-\bigl{(}\mathrm{ad}_{a}^{\mathfrak{g}}\bigr{)}^{p}(b)$
$\displaystyle=[s(a),b]_{\mathfrak{a}}+\omega_{\mathfrak{a}}(({\mathscr{D}}+{\mathscr{D}}^{*})(s(a)),b)x-\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p}(b)-\omega_{\mathfrak{a}}\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(a),\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-1}(b)\bigr{)}x.$

Therefore,

$\displaystyle\omega_{\mathfrak{a}}(({\mathscr{D}}+{\mathscr{D}}^{*})(s(a)),b)=\omega_{\mathfrak{a}}\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(a),\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-1}(b)\bigr{)},$
$\displaystyle[s(a),b]_{\mathfrak{a}}=\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p}(b).$

Now, since

$\displaystyle\sum_{1\leq i\leq p-1}is_{i}^{\mathfrak{g}}(a,b)\lambda^{i-1}$
$\displaystyle=\bigl{(}\mathrm{ad}_{\lambda
a+b}^{\mathfrak{g}}\bigr{)}^{p-1}(a)$
$\displaystyle=\bigl{(}\mathrm{ad}_{\lambda
a+b}^{\mathfrak{a}}\bigr{)}^{p-1}(a)+\omega_{\mathfrak{a}}\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(\lambda
a+b),\bigl{(}\mathrm{ad}_{\lambda
a+b}^{\mathfrak{a}}\bigr{)}^{p-2}(a)\bigr{)}x$
$\displaystyle=\bigl{(}\mathrm{ad}_{\lambda
a+b}^{\mathfrak{a}}\bigr{)}^{p-1}(a)$

it follows that

$s^{\mathfrak{g}}_{i}(a,b)=s_{i}^{\mathfrak{a}}(a,b).$

Moreover,

$\displaystyle 0$
$\displaystyle=(a+b)^{[p]_{\mathfrak{g}}}-a^{[p]_{\mathfrak{g}}}-b^{[p]_{\mathfrak{g}}}-\sum_{1\leq
i\leq p-1}s_{i}^{\mathfrak{g}}(a,b)$
$\displaystyle=s(a+b)-s(a)-s(b)-\sum_{1\leq i\leq
p-1}s_{i}^{\mathfrak{a}}(a,b).$

Consequently,

$s(a+b)=s(a)+s(b)+\sum_{1\leq i\leq p-1}s_{i}^{\mathfrak{a}}(a,b).$

It follows that $s$ defines a $[p|2p]$-map on $\mathfrak{a}$. Now, since
$\big{[}a^{[p]_{\mathfrak{g}}},x^{*}\big{]}_{\mathfrak{g}}=\bigl{(}\mathrm{ad}_{a}^{\mathfrak{g}}\bigr{)}^{p}(x^{*})$,
then

$-\mathscr{D}(s(a))+\omega(s(a),a_{0})x=-\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-1}\circ\mathscr{D}(a)-\omega\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(a),\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-2}\circ{\mathscr{D}}(a)\bigr{)}x.$

It follows that

$\mathscr{D}(s(a))=\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-1}\circ\mathscr{D}(a),\qquad\text{
and
}\qquad\omega(a_{0},s(a))=\omega\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(a),\bigl{(}\mathrm{ad}_{a}^{\mathfrak{a}}\bigr{)}^{p-2}\circ{\mathscr{D}}(a)\bigr{)}.$

The first equation shows that ${\mathscr{D}}$ is a restricted derivation of
$\mathfrak{a}$ (relative to the $[p|2p]$-map $s$). The second equation
completes the proof that the map $(\Omega,\Delta)\in
B^{2}_{*}(\mathfrak{a};\mathbb{K})$, where $\Omega$ and $\Delta$ are given by

$\displaystyle\Omega\colon\
\mathfrak{a}\wedge\mathfrak{a}\rightarrow\mathbb{K},\qquad(a,b)\mapsto\omega_{\mathfrak{a}}({\mathscr{D}}\circ{\mathscr{D}}(a)-{\mathscr{D}}^{*}\circ{\mathscr{D}}^{*}(a),b),$
$\displaystyle\Delta\colon\ \mathfrak{a}_{\bar{0}}\rightarrow\mathbb{K},\qquad
a\mapsto\omega_{\mathfrak{a}}\bigl{(}({\mathscr{D}}+{\mathscr{D}}^{*})(a),\mathrm{ad}_{a}^{p-2}({\mathscr{D}}(a))\bigr{)}.$

Therefore, $\mathfrak{g}$ is a symplectic double extension of $\mathfrak{a}$.
∎

##  4 Restricted periplectic double extensions

###  4.1 $\boldsymbol{{\mathscr{D}}_{\bar{0}}}$-extensions

Let $(\mathfrak{a},\omega_{\mathfrak{a}})$ be a restricted periplectic quasi-
Frobenius Lie superalgebra, and let
${\mathscr{D}}\in\mathfrak{der}_{\bar{0}}(\mathfrak{a})$ be a derivation.

Consider the following map

$\Omega\colon\
\mathfrak{a}\wedge\mathfrak{a}\rightarrow\mathbb{K},\qquad(a,b)\mapsto\omega_{\mathfrak{a}}\bigl{(}({\mathscr{D}}\circ{\mathscr{D}}+2{\mathscr{D}}^{*}\circ{\mathscr{D}}+{\mathscr{D}}^{*}\circ{\mathscr{D}}^{*}+\lambda{\mathscr{D}}+\lambda{\mathscr{D}}^{*})(a),b\bigr{)}.$

Let us suppose that $(\Omega,0)$ is in $B^{2}_{*}(\mathfrak{a};\mathbb{K})$
(i.e., a 2-coboundary in the restricted cohomology). Let us write
$(\Omega,0)=\bigl{(}d^{1}_{\mathrm{CE}}(\chi),\mathrm{ind}^{1}(\chi)\bigr{)}$
for some $\chi\in C^{1}_{*}(\mathfrak{a};\mathbb{K})$; namely,

$\Omega(a,b)=\chi([a,b]_{\mathfrak{a}})\qquad\text{ and
}\qquad\chi\bigl{(}a^{[p]}\bigr{)}=0.$

Since $\omega_{\mathfrak{a}}$ is non-degenerate, there exists
$Z_{\Omega}\in\mathfrak{a}$ such that

$\Omega(a,b)=\omega_{\mathfrak{a}}(Z_{\Omega},[a,b]_{\mathfrak{a}})\qquad\text{and}\qquad\omega_{\mathfrak{a}}\bigl{(}Z_{\Omega},a^{[p]}\bigr{)}=0.$

######  Theorem 4.1 (${\mathscr{D}}_{\bar{0}}$-extension – the case where
$\omega$ is periplectic).

Let $(\mathfrak{a},\omega_{\mathfrak{a}})$ be a restricted periplectic quasi-
Frobenius Lie superalgebra. Let
${\mathscr{D}}\in\mathfrak{der}_{\bar{0}}(\mathfrak{a})$ be a restricted
derivation satisfying the $p$-property (2.2). Suppose that the following
condition is satisfied

$(\Omega,0)\in B^{2}_{*}(\mathfrak{a};\mathbb{K}).$

  * $(i)$

There exists a Lie superalgebra structure on
$\mathfrak{g}:=\mathscr{K}\oplus\mathfrak{a}\oplus\mathscr{K}^{*}$, where
$\mathscr{K}:=\mathrm{Span}\\{x\\}$ for $x$ odd and
$\mathscr{K}^{*}:=\mathrm{Span}\\{e\\}$ for $e$ even, defined as follows:
$($for any $a,b\in\mathfrak{a})$

$\displaystyle[x,e]_{\mathfrak{g}}=\lambda
x,\qquad[a,b]_{\mathfrak{g}}:=[a,b]_{\mathfrak{a}}+(\omega_{\mathfrak{a}}({\mathscr{D}}(a),b)+\omega_{\mathfrak{a}}(a,{\mathscr{D}}(b)))x,$
$\displaystyle[e,a]_{\mathfrak{g}}:={\mathscr{D}}(a)+\omega_{\mathfrak{a}}(Z_{\Omega},a)x.$

There exists a closed anti-symmetric periplectic form $\omega_{\mathfrak{g}}$
on $\mathfrak{g}$ defined as follows:

$\displaystyle{\omega_{\mathfrak{g}}}|_{\mathfrak{a}\times\mathfrak{a}}:=\omega_{\mathfrak{a}},\qquad\omega_{\mathfrak{g}}(\mathfrak{a},\mathscr{K}):=\omega_{\mathfrak{g}}(\mathfrak{a},\mathscr{K}^{*}):=0,$
$\displaystyle\omega_{\mathfrak{g}}(e,x):=1,\qquad\omega_{\mathfrak{g}}(x,x):=\omega_{\mathfrak{g}}(e,e):=0.$

  * $(ii)$

There exists a $[p|2p]$-map on symplectic the double extension $\mathfrak{g}$
of $\mathfrak{a}$ given by

$a^{[p]_{\mathfrak{g}}}=a^{[p]_{\mathfrak{a}}},\qquad
e^{[p]_{\mathfrak{g}}}=a_{0}+\gamma e,$

where $a_{0}\in\mathfrak{a}$ and $\gamma\in\mathbb{K}$ are as in (2.2), and
the following conditions must be satisfied:

$\displaystyle\lambda\gamma=\lambda^{p},\qquad\omega(Z_{\Omega},a_{0})=0,$
$\displaystyle\sum_{i+j=p-1}(-\lambda)^{j}({\mathscr{D}}^{*})^{i}(Z_{\Omega})={\mathscr{D}}^{*}(a_{0})+\gamma
Z_{\Omega}.$

###### Proof.

Similar to that of Theorem 3.1. ∎

######  Theorem 4.2 (converse of Theorem 4.1).

Let $(\mathfrak{g},\omega_{\mathfrak{g}})$ be a restricted periplectic quasi-
Frobenius Lie superalgebra. Suppose that there exists
$0\not=x\in([\mathfrak{g},\mathfrak{g}])^{\perp}_{\bar{1}}$ such that
$\mathscr{K}:=\mathrm{Span}\\{x\\}$ is an ideal and $K^{\perp}$ is a
$p$-ideal. Then $(\mathfrak{g},\omega_{\mathfrak{g}})$ is obtained a
symplectic ${\mathscr{D}}_{\bar{0}}$-extension from a restricted periplectic
quasi-Frobenius Lie superalgebra $(\mathfrak{a},\omega_{\mathfrak{a}})$.

###### Proof.

It has been proved in [14] that $\mathscr{K}^{\perp}$ is also an ideal in
$(\mathfrak{g},\omega_{\mathfrak{g}})$. Since $\mathscr{K}$ is 1-dimensional
and $\omega_{\mathfrak{g}}(x,x)=0$, it follows that
$\mathscr{K}\subset\mathscr{K}^{\perp}$ and
$\dim\bigl{(}\mathscr{K}^{\perp}\bigr{)}=\dim(\mathfrak{g})-1$. Therefore,
there exists $e\in\mathfrak{g}_{\bar{0}}$ (since
$\bigl{(}\mathscr{K}^{\perp}\bigr{)}_{\bar{1}}=\mathfrak{g}_{\bar{1}}$) such
that

$\mathfrak{g}=\mathscr{K}^{\perp}\oplus\mathscr{K}^{*},\qquad\text{ where}\
\mathscr{K}^{*}:=\mathrm{Span}\\{x^{*}\\}.$

We can normalize $e$ so that $\omega_{\mathfrak{g}}(e,x)=1$.

Let us define $\mathfrak{a}:=(\mathscr{K}+\mathscr{K}^{*})^{\perp}$. We then
have a decomposition
$\mathfrak{g}=\mathscr{K}\oplus\mathfrak{a}\oplus\mathscr{K}^{*}$. It has been
proved in [14] that $\mathfrak{g}$ is a symplectic double extension of
$\mathfrak{a}$.

Now, it remains to show that there is a $[p|2p]$-map on $\mathfrak{a}$. Since
$\mathfrak{a}\subset\mathscr{K}^{\perp}$ and $\mathscr{K}^{\perp}$ is a
$p$-ideal, then

$a^{[p]_{\mathfrak{g}}}\in\mathscr{K}^{\perp}=\mathscr{K}\oplus\mathfrak{a}\qquad\text{
for any}\ a\in\mathfrak{a}_{\bar{0}}.$

It follows that $a^{[p]_{\mathfrak{g}}}=s(a).$ We will show that the map

$s\colon\ \mathfrak{a}_{\bar{0}}\rightarrow\mathfrak{a}_{\bar{0}},\qquad
a\mapsto s(a),$

is a $[p|2p]$-map on $\mathfrak{a}$. The proof follows as in Theorem 3.2. ∎

###  4.2 $\boldsymbol{{\mathscr{D}}_{\bar{1}}}$-extensions

Let $(\mathfrak{a},\omega_{\mathfrak{a}})$ be a restricted periplectic quasi-
Frobenius Lie superalgebra, and let
${\mathscr{D}}\in\mathfrak{der}_{\bar{1}}(\mathfrak{a})$ be a derivation.

Consider the following map

$\Omega\colon\
\mathfrak{a}\wedge\mathfrak{a}\rightarrow\mathbb{K},\qquad(a,b)\mapsto\omega_{\mathfrak{a}}({\mathscr{D}}\circ{\mathscr{D}}(a)-{\mathscr{D}}^{*}\circ{\mathscr{D}}^{*}(a),b).$

Let us suppose that $(\Omega,0)$ is in $B^{2}_{*}(\mathfrak{g};\mathbb{K})$
(i.e., a 2-coboundary in the restricted cohomology). Let us write
$(\Omega,0)=\bigl{(}d^{1}_{\mathrm{CE}}(\chi),\mathrm{ind}^{1}(\chi)\bigr{)}$
for some $\chi\in C^{1}_{*}(\mathfrak{a};\mathbb{K})$; namely,

$\Omega(a,b)=\chi([a,b]_{\mathfrak{a}}),\qquad\forall
a,b\in\mathfrak{a}\qquad\text{ and
}\qquad\chi\bigl{(}a^{[p]}\bigr{)}=0,\qquad\forall
a\in\mathfrak{a}_{\bar{0}}.$

Since $\omega_{\mathfrak{a}}$ is non-degenerate, there exists
$a_{0}\in\mathfrak{a}$ such that

$\Omega(a,b)=\omega_{\mathfrak{a}}(a_{0},[a,b]_{\mathfrak{a}})\qquad\text{and}\qquad\omega_{\mathfrak{a}}\bigl{(}a_{0},a^{[p]}\bigr{)}=0.$
(4.1)

Consider now a map $P\colon\mathfrak{a}_{\bar{0}}\mapsto\mathbb{K}$ satisfying
the conditions (2.6) and (2.7). We have shown in Lemma 2.7 that $P$ has the
$*$-property with respect to the cocycle $C$ given by equation (2.3).

######  Theorem 4.3 (${\mathscr{D}}_{\bar{1}}$-extension – the case where
$\omega$ is periplectic).

Let $(\mathfrak{a},\omega_{\mathfrak{a}})$ be a restricted periplectic quasi-
Frobenius Lie superalgebra, and let
${\mathscr{D}}\in\mathfrak{der}_{\bar{1}}(\mathfrak{a})$ be a derivation.
Suppose that the following conditions are satisfied:

$(\Omega,0)\in
B^{2}_{*}(\mathfrak{a},\mathbb{K})\qquad\text{and}\qquad(C,P)\in
Z^{2}_{*}(\mathfrak{a},\mathbb{K}).$

Suppose further that $(a_{0}$ is given as in (4.1)$)$

${\mathscr{D}}^{2}=\mathrm{ad}_{a_{0}},\qquad{\mathscr{D}}(a_{0})={\mathscr{D}}^{*}(a_{0})=0.$

  * $(i)$

There exists a Lie superalgebra structure on
$\mathfrak{g}:=\mathscr{K}\oplus\mathfrak{a}\oplus\mathscr{K}^{*}$, where
$\mathscr{K}:=\mathrm{Span}\\{x\\}$ for $x$ even and
$\mathscr{K}^{*}:=\mathrm{Span}\\{e\\}$ for $e$ odd, defined as follows:
$($for any $a,b\in\mathfrak{a})$

$\displaystyle[\mathscr{K},\mathfrak{g}]_{\mathfrak{g}}=0,\qquad[a,b]_{\mathfrak{g}}:=[a,b]_{\mathfrak{a}}+\bigl{(}\omega_{\mathfrak{a}}({\mathscr{D}}(a),b)+(-1)^{|a|}\omega_{\mathfrak{a}}(a,{\mathscr{D}}(b))\bigr{)}x,$
$\displaystyle[e,e]_{\mathfrak{g}}=2a_{0},\qquad[e,a]_{\mathfrak{g}}:={\mathscr{D}}(a)+\omega_{\mathfrak{a}}(a_{0},a)x.$

There exists a closed anti-symmetric periplectic form $\omega_{\mathfrak{g}}$
on $\mathfrak{g}$ defined as follows:

$\displaystyle{\omega_{\mathfrak{g}}}|_{\mathfrak{a}\times\mathfrak{a}}:=\omega_{\mathfrak{a}},\qquad\omega_{\mathfrak{g}}(\mathfrak{a},\mathscr{K}):=\omega_{\mathfrak{g}}(\mathfrak{a},\mathscr{K}^{*}):=0,$
$\displaystyle\omega_{\mathfrak{g}}(e,x):=1,\qquad\omega_{\mathfrak{g}}(x,x):=\omega_{\mathfrak{g}}(e,e):=0.$

  * $(ii)$

Suppose there exists $b_{0}\in\mathfrak{z}(\mathfrak{a})$ such that

${\mathscr{D}}(b_{0})={\mathscr{D}}^{*}(b_{0})=\omega(a_{0},b_{0})=0.$

The $[p|2p]$-map on $\mathfrak{a}$ can be extended to $\mathfrak{g}$ as
follows $(\mu$ is arbitrary$)$:

$\begin{array}[]{lcllcl}a^{[p]_{\mathfrak{g}}}&=&a^{[p]_{\mathfrak{a}}}+P(a)x,\qquad
x^{[p]_{\mathfrak{g}}}&=&b_{0}+\mu x.\end{array}$

###### Proof.

Similar to that of Theorem 3.1. ∎

The converse of Theorem 4.3 is given by the following theorem.

######  Theorem 4.4 (converse of Theorem 4.3).

Let $(\mathfrak{g},\omega_{\mathfrak{g}})$ be a restricted periplectic quasi-
Frobenius Lie superalgebra. Suppose that
$\mathfrak{z}(\mathfrak{g})_{\bar{0}}\not=\\{0\\}.$ Then,
$(\mathfrak{g},\omega_{\mathfrak{g}})$ is obtained as an
${\mathscr{D}}_{\bar{1}}$-extension of a restricted periplectic quasi-
Frobenius Lie superalgebra $(\mathfrak{a},\omega_{\mathfrak{a}})$.

###### Proof.

Let $x$ be a non-zero element in $\mathfrak{z}(\mathfrak{g})_{\bar{0}}$. It
has been showed in [14] that both the subspaces
$\mathscr{K}:=\mathrm{Span}\\{x\\}$ and $\mathscr{K}^{\perp}$ are ideals in
$(\mathfrak{g},\omega_{\mathfrak{g}})$. Since $\omega_{\mathfrak{g}}(x,x)=0$,
it follows that $\mathscr{K}\subset\mathscr{K}^{\perp}$ and
$\dim\bigl{(}\mathscr{K}^{\perp}\bigr{)}=\dim(\mathfrak{g})-1$. Therefore,
there exists $e\in\mathfrak{g}_{\bar{1}}$ (since
$\bigl{(}\mathscr{K}^{\perp}\bigr{)}_{\bar{0}}=\mathfrak{g}_{\bar{0}}$) such
that

$\mathfrak{g}=\mathscr{K}^{\perp}\oplus\mathscr{K}^{*},\qquad\text{ where}\
\mathscr{K}^{*}:=\mathrm{Span}\\{e\\}.$

We can normalize $e$ so that $\omega_{\mathfrak{g}}(e,x)=1$.

Let us define $\mathfrak{a}:=(\mathscr{K}+\mathscr{K}^{*})^{\perp}$. We then
have a decomposition
$\mathfrak{g}=\mathscr{K}\oplus\mathfrak{a}\oplus\mathscr{K}^{*}$.

We define a periplectic form on $\mathfrak{a}$ by setting
$\omega_{\mathfrak{a}}={\omega_{\mathfrak{g}}}|_{\mathfrak{a}\times\mathfrak{a}}.$
The form $\omega_{\mathfrak{a}}$ is non-degenerate on $\mathfrak{a}$.

It has been showed in [14] that the vector space $\mathfrak{a}$ can be endowed
with a Lie superalgebra structure, and there exists a periplectic structure on
$\mathfrak{a}$ for which $\mathfrak{g}$ is its symplectic double extension.
The proof now follows as in Theorem 3.4.∎

##  5 Examples of symplectic double extensions

###  5.1 The Lie superalgebra $\boldsymbol{D^{7}_{q,-q}}$ (for
$\boldsymbol{q\not=0,1}$), see [2]

Consider the Lie superalgebra $D^{7}_{q,-q}$, for $q\not=0,1$, where the non-
zero brackets are given in the basis $e_{1},e_{2}\mid e_{3},e_{4}$ as follows:

$[e_{1},e_{2}]=e_{2},\qquad[e_{1},e_{3}]=qe_{3},\qquad[e_{1},e_{4}]=-qe_{4}.$

It has been shown in [14] that this Lie superalgebra is orthosymplectic quasi-
Frobenius, where the form is given by111As in [14], we adopt the following
convention: $\langle e_{i}^{*},e_{j}\rangle=\delta_{ij}$ and $\langle
e_{i}^{*}\otimes e_{j}^{*},e_{k}\otimes
e_{l}\rangle=(-1)^{|e_{k}||e_{j}|}\langle e_{i}^{*},e_{k}\rangle\langle
e_{j}^{*},e_{l}\rangle$.

$\omega=e_{1}^{*}\wedge e_{2}^{*}+e_{3}^{*}\wedge e_{4}^{*}.$

There exists a $[p|2p]$-map given by $e_{1}^{[p]}=e_{1}$ and $e_{2}^{[p]}=0.$
Let us consider the even outer derivations

${\mathscr{D}}_{1}=e_{3}\otimes e_{3}^{*},\qquad{\mathscr{D}}_{2}=e_{4}\otimes
e_{4}^{*}.$

Obviously, both of them are restricted derivations. An easy computation shows
that ${\mathscr{D}}_{1}^{*}={\mathscr{D}}_{2}$. Besides, we have

${\mathscr{D}}_{1}^{2}+2{\mathscr{D}}_{1}\circ{\mathscr{D}}_{1}^{*}+{\mathscr{D}}_{1}^{*}\circ{\mathscr{D}}_{1}^{*}+\lambda({\mathscr{D}}_{1}+{\mathscr{D}}_{1}^{*})=(\lambda+1)(e_{3}\otimes
e_{3}^{*}+e_{4}\otimes e_{4}^{*}).$

The cocycle $(\Omega,T)$ (see Theorem 3.1) is a coboundary if and only if
$\lambda=-1$. In this case, $\Omega\equiv 0$ and $C=e_{3}^{*}\wedge
e_{4}^{*}$. Let us choose $Z_{\Omega}=ue_{2}$, where $u\in\mathbb{K}$. It
follows that $P(e_{1})=u$ and $P(e_{2})=0$. Additionally,

$a_{0}=0,\qquad\tilde{\lambda}=0,\qquad\gamma=1,\qquad
b_{0}=0,\qquad\sigma=0.$

###  5.2 The Lie superalgebra $\boldsymbol{C_{1}^{1}+A}$, see [2]

Consider the Lie superalgebra $C_{1}^{1}+A$ with the brackets given in the
basis $e_{1},e_{2}\mid e_{3},e_{4}$ as follows:

$[e_{1},e_{2}]=e_{2},\qquad[e_{1},e_{3}]=e_{3},\qquad[e_{3},e_{4}]=e_{2}.$

It has been shown in [14] that this Lie superalgebra is orthosymplectic quasi-
Frobenius with a form given by

$\omega=e_{1}^{*}\wedge e_{2}^{*}-e_{3}^{*}\wedge e_{4}^{*}.$

The $[p|2p]$-map is given by $e_{1}^{[p]}=e_{1}$, $e_{2}^{[p]}=0$. There
exists an odd outer derivation given by

${\mathscr{D}}=e_{1}\otimes e_{4}^{*}+e_{3}\otimes e_{2}^{*}.$

A direct computation shows that ${\mathscr{D}}^{*}=-{\mathscr{D}}$. As a
result, the cocycle $(C,\Delta)=(0,0)$ (see Theorem 3.4). Additionally, since
${\mathfrak{z}}(\mathfrak{g})=0$, the condition
${\mathscr{D}}^{2}=\mathrm{ad}_{a_{0}}=0$ implies that $a_{0}=0$.

###  5.3 The Lie superalgebra $\boldsymbol{D^{5}}$, see [2]

Consider the Lie superalgebra $D^{5}$ with the brackets given in the basis
$e_{1},e_{2},\mid e_{3},e_{4}$ as follows

$[e_{1},e_{3}]=e_{3},\qquad[e_{1},e_{4}]=e_{4},\qquad[e_{2},e_{4}]=e_{3}.$

It has been shown in [14] that Lie superalgebra is periplectic quasi-Frobenius
with a form given by

$\omega=e_{1}^{*}\wedge e_{3}^{*}+e_{2}^{*}\wedge e_{4}^{*}.$

The $[p|2p]$-map is given by $e_{1}^{[p]}=e_{1}$ and $e_{2}^{[p]}=0$. There
exists an even outer derivative on $D^{5}$ given by

${\mathscr{D}}=e_{2}\otimes e_{2}^{*}-e_{4}\otimes e_{4}^{*}.$

A direct computation shows that ${\mathscr{D}}^{*}=-{\mathscr{D}}$, and
${\mathscr{D}}^{p}={\mathscr{D}}$ so $\gamma=1$ and $a_{0}=0$ (see Theorem 4.1
for the definitions). As a result $\Omega=C=0$. Now, choose $\lambda$ any
scalar that is a root to the equation $\lambda^{p}-\lambda=0$, and choose
$Z_{\Omega}=0$.

###  5.4 The Lie superalgebra $\boldsymbol{(2A_{1,1}+2A)^{2}_{1/2}}$, see [2]

Consider the Lie superalgebra $(2A_{1,1}+2A)^{2}_{1/2}$ with the following
brackets given in the basis $(e_{1},e_{2}\mid e_{3},e_{4})$:

$[e_{3},e_{3}]=e_{1},\qquad[e_{4},e_{4}]=e_{2},\qquad[e_{3},e_{4}]=\tfrac{1}{2}(e_{1}+e_{2}).$

It has been shown in [14] that Lie superalgebra is periplectic quasi-Frobenius
with a form given by

$\omega=e_{2}^{*}\wedge e_{3}^{*}-e_{1}^{*}\wedge e_{4}^{*}.$

A $[p|2p]$-map can be defined as follows: $e_{i}^{[p]}=u_{i}e_{1}+v_{i}e_{2}$,
where $u_{i}$ and $v_{i}$ are scalar, for $i=1,2$.

There exists an outer odd derivation of this Lie superalgebra given by
${\mathscr{D}}=e_{2}\otimes e_{3}^{*}$. A direct computation shows that
${\mathscr{D}}^{*}={\mathscr{D}}$. It follows that $\Omega=\Delta=C=P=0$ (see
Theorem 4.3 for the definitions). In addition, $a_{0}=0$ and
$b_{0}=\lambda_{1}e_{1}+\lambda_{2}e_{2}$ is arbitrary.

###  5.5 The Witt algebra $\boldsymbol{W(1)}$ for $\boldsymbol{p>3}$

The Witt algebra, denoted by $W(1)$, is spanned by the generators
$e_{-1},e_{0},\dots,e_{p-2}$ and endowed with the bracket

$\displaystyle[e_{i},e_{j}]=\begin{cases}(j-i)e_{i+j}&\text{if}\
i+j\in\\{-1,\dots,p-2\\},\\\ 0&\text{otherwise.}\end{cases}$

Let $\omega$ be an anti-symmetric bilinear form on $W(1)$, given by
$\omega=\sum_{i,j=-1}^{p-2}\lambda_{i,j}e_{i}^{*}\wedge e_{j}^{*}$,
$\lambda_{i,j}\in\mathbb{K}$. Let us suppose further that $\omega$ is a
$2$-cocycle, namely

$\omega(e_{k},[e_{i},e_{j}])+\omega(e_{j},[e_{k},e_{i}])+\omega(e_{i},[e_{j},e_{k}])=0.$
(5.1)

We will show that any anti-symmetric bilinear form on $W(1)$ satisfying (5.1)
is degenerate. Suppose that $i$, $j$, $k$ are such that
$i+j,i+k,j+k\in\\{-1,\dots,p-2\\}.$ It follows

$\displaystyle(\ref{wittcocycle})$
$\displaystyle\Longleftrightarrow\omega(e_{k},(j-i)e_{i+j})+\omega(e_{j},(i-k)e_{i+k})+\omega(e_{i},(k-j)e_{j+k})=0$
$\displaystyle\Longleftrightarrow(j-i)\lambda_{k,i+j}+(i-k)\lambda_{j,i+k}+(k-j)\lambda_{i,j+k}=0.$

If we take $k=0$ in the above equation, we obtain

$(j-i)\lambda_{0,i+j}+(i+j)\lambda_{i,j}=0.$

By taking $i=0$, the last equation reduces to

$\lambda_{0,j}=0,\qquad\forall j\in\\{-1,\dots,p-2\\}.$

That is to say, there is a row in the matrix of $\omega$ that contains only
zeros. As a result, $\omega$ is degenerate.

###  5.6 The Lie superalgebra $\boldsymbol{K^{2,m}}$, $\boldsymbol{m}$ odd,
see [27]

The Lie superalgebra $K^{2,m}$ is spanned by the generators $x_{0},x_{1}\mid
y_{1},\ldots y_{m}$ (even $|$ odd), with non-zero brackets given by

$\displaystyle[x_{0},y_{i}]=-[y_{i},x_{0}]=y_{i+1},\qquad i\leq m-1,$
$\displaystyle[y_{i},y_{m+1-i}]=[y_{m+1-i},y_{i}]=(-1)^{i+1}x_{1},\qquad 1\leq
i\leq\frac{m+1}{2}.$

It has been shown in [27] that a Lie superalgebra of superdimension $n|m$ has
a maximal nilindex $n+m-1$ only when $n=2$ and $m$ odd. Moreover, for any odd
$m$, there is only one Lie superalgebra with this maximal nilindex and that is
$K^{2,m}$.

######  Claim 5.1.

  * $(i)$

For all $m\geq 3$, the Lie superalgebra $K^{2,m}$ is not periplectic quasi-
Frobenius.

  * $(ii)$

$K^{2,m}$ is orthosymplectic quasi-Frobenius if and only if $m=0\bmod(p)$.
Moreover, if ${m=0}\bmod(p)$, the form is given by

$x_{0}^{*}\wedge x_{1}^{*}-\frac{1}{2}\,y_{1}^{*}\wedge
y_{1}^{*}-\frac{1}{2}\,(-1)^{\frac{m+3}{2}}y_{\frac{m+1}{2}}^{*}\wedge
y_{\frac{m+3}{2}}^{*}-\sum_{1\leq
i\leq\frac{m-3}{2}}i\,(-1)^{i+1}\;y_{i+1}^{*}\wedge y_{m+1-i}^{*}.$

###### Proof.

Let us prove part (i). Suppose that $K^{2,m}$ is periplectic quasi-Frobenius.
The 2-cocycle condition applied to the periplectic form $\omega$ with respect
to $\\{x_{0},x_{1},y_{i}$} gives

$\displaystyle 0$
$\displaystyle=\omega(x_{0},[x_{1},y_{i}])+\omega(y_{i},[x_{0},x_{1}])+\omega(x_{1},[y_{i},x_{0}])$
$\displaystyle=0+0-\omega(x_{1},y_{i+1}),\qquad\text{ for}\qquad 1\leq i\leq
m-1.$

Similarly, the 2-cocycle condition with respect to $\\{y_{1},y_{2},y_{m-1}\\}$
gives $\omega(x_{1},y_{1})=0$. Now, since $\omega$ is odd it follows that
$\omega(x_{1},v)=0$ for all $v\in K^{2,m}$. Thus, $\omega$ is degenerate which
is a contradiction.

Let us proof part (ii). Since $\omega$ is even we will check the 2-cocycle
condition only for the triple $\\{x_{0},y_{i},y_{i}\\}$ and
$\\{x_{0},y_{i},y_{j}\\}$ for $i<j$. Let us write

$\omega=\alpha\,x^{*}_{0}\wedge
x^{*}_{1}-\sum_{i,j}\alpha_{i,j}\,y^{*}_{i}\wedge y^{*}_{j}.$

The case of $\\{x_{0},y_{i},y_{i}\\}$. For $i\leq\frac{m+1}{2}$, the 2-cocycle
condition is equivalent to

$\displaystyle\alpha=2(-1)^{\frac{m+3}{2}}\alpha_{\frac{m+1}{2},\frac{m+3}{2}}\qquad\text{if}\
i=\frac{m+1}{2},\qquad\alpha_{i,i+1}=0\qquad\text{if}\ i<\frac{m+1}{2}.$ (5.2)

Moreover, if $i>\frac{m+1}{2}$ the 2-cocycle condition is equivalent to

$\alpha_{i,i+1}=0\qquad\text{ for }\ \frac{m+1}{2}<i<m.$

The case of $\\{x_{0},y_{i},y_{j}\\}$, where $i<j$. In the case of $1\leq
i\leq\frac{m+1}{2}$ and $j=m+1-i$, the 2-cocycle condition implies that

$\displaystyle\alpha=\alpha_{m,2}\qquad\text{if}\ i=1,$
$\displaystyle(-1)^{i+1}\alpha-\alpha_{m+1-i,i+1}-\alpha_{i,m+2-i}=0\qquad\text{if}\
1<i<\frac{m+1}{2}.$ (5.3)

On the other hand, if $j\neq m+1-i$, the 2-cocycle condition is equivalent to

$\displaystyle\alpha_{m,i+1}=0\qquad\text{if}\ j=m,\ \text{and}\ i\not=1,$
$\displaystyle\alpha_{j,i+1}+\alpha_{i,j+1}=0\qquad\text{if}\ 1\leq
i\leq\frac{m+1}{2},\ \text{ and }\ j\not=m.$

Now, in the case where $i>\frac{m+1}{2}$, the 2-cocycle condition is
equivalent to

$\displaystyle\alpha_{m,i}=0\qquad\text{if}\ j=m,$
$\displaystyle\alpha_{j,i+1}+\alpha_{i,j+1}=0\qquad\text{if}\
\frac{m+1}{2}<i<j<m.$ (5.4)

Combining equations (5.2) and (5.3) we get $m\alpha=0$. If $m\not=0\bmod(p)$
then $\alpha=0$ and the form $\omega$ will be degenerate. If $m=0\bmod(p)$,
then the system of equations given by (5.2)–(5.4) is consistent and is not
unique. Let us choose
$\alpha_{\frac{m+1}{2},\frac{m+3}{2}}=\frac{1}{2}(-1)^{\frac{m+3}{2}}$ which
would imply that $\alpha=1$. Now, equations (5.3) imply that
$\alpha_{m+1-i,i+1}=i(-1)^{i+1}$ for $i=1,\ldots\frac{m-1}{2}$. Choosing
$\alpha_{1,1}=\frac{1}{2}$ and the remaining coefficients to be trivial
produces a non-degenerate form.

######  Claim 5.2.

$K^{2,m}$ is restricted if and only if $m\leq p$. Explicitly, the $[p|2p]$-map
is given by

$x_{0}^{[p]}=s_{1}x_{1},\qquad x_{1}^{[p]}=s_{2}x_{1},\qquad\text{ where}\
s_{1},s_{2}\in\mathbb{K}.$

###### Proof.

First, observe that the even part of $K^{2,m}$ is abelian. Let us write
${x_{1}^{[p]}=s_{2}x_{1}+\tilde{s}_{2}x_{0}}$. Now, the condition
$\big{[}x_{1}^{[p]},y_{i}\big{]}=\mathrm{ad}_{x_{1}}^{p}(y_{i})$ implies that
$\tilde{s}_{2}=0$. Therefore, $x_{1}^{[p]}=s_{2}x_{1}$. Additionally, let us
write $x_{0}^{[p]}=s_{1}x_{1}+\tilde{s}_{1}x_{0}$. The condition
$[x_{0}^{[p]},y_{i}]=\mathrm{ad}_{x_{0}}^{p}(y_{i})$ implies that
$\tilde{s}_{1}y_{i+1}=y_{p+i}$ for $i<m$. This condition implies that $p+i>m$,
for all $i<m$, and $\tilde{s}_{1}=0$.

Hereafter, suppose that the $[p|2p]$-map is given by

$x_{0}^{[p]}=0,\qquad x_{1}^{[p]}=x_{1}.$

We will list a few outer derivations and study symplectic double extensions
using those derivations. ∎

####  5.6.1 A non-suitable derivation

Consider the outer derivation given by

${\mathscr{D}}=x_{1}\otimes x_{1}^{*}-\frac{p-1}{2}\sum_{1\leq i\leq
p}y_{i}\otimes y_{i}^{*}.$

Now,
${\mathscr{D}}\bigl{(}x_{1}^{[p]}\bigr{)}={\mathscr{D}}(x_{1})=x_{1}\not=\mathrm{ad}_{x_{1}}^{p-1}\circ{\mathscr{D}}(x_{1})=0.$
Hence, this derivation is not suitable for a double extension because it is
not a restricted derivation.

####  5.6.2 A derivation yielding a trivial extension

Consider the outer derivation given by

${\mathscr{D}}=x_{1}\otimes x_{0}^{*}.$

This is obviously a restricted derivation. A direct computation shows that
${\mathscr{D}}^{*}=-{\mathscr{D}}$. It follows that the cocycle $C$ of Lemma
2.4 as well as the map $\Omega$ in Section 3.1 are identically zero.

The conditions of Theorem 3.1 are all satisfied. We can choose

$a_{0}=0,\qquad\lambda=0,\qquad\gamma=0,\qquad b_{0}=0,\qquad\sigma=0,\qquad
Z_{\Omega}=0,\qquad P\equiv 0.$

####  5.6.3 A derivation yielding a non-trivial extension

Consider the outer derivation given by

${\mathscr{D}}=y_{p-1}\otimes y_{1}^{*}+y_{p}\otimes y_{2}^{*}.$

Obviously, this is a restricted derivation. A direct computation shows that

${\mathscr{D}}^{*}=y_{p}\otimes y_{2}^{*}-2y_{1}\otimes y_{3}^{*}.$

It follows that

${\mathscr{D}}^{2}+2{\mathscr{D}}^{*}\circ{\mathscr{D}}+{{\mathscr{D}}^{*}}^{2}+\lambda({\mathscr{D}}+{\mathscr{D}}^{*})=\lambda
y_{p-1}\otimes y_{1}^{*}+2\lambda y_{p}\otimes y^{*}_{2}-2\lambda y_{1}\otimes
y^{*}_{3}.$

A direct computation shows that the map $\Omega$, defined in Section 3.1, has
the following form (the zero terms are omitted):

$\Omega(y_{1},y_{3})=-2\lambda,\qquad\Omega(y_{2},y_{2})=2\lambda.$

The case where $p=3$. In this case, the map $C$ is given by

$C=\frac{1}{2}\;y_{1}^{*}\wedge y_{3}^{*}+y_{2}^{*}\wedge y_{2}^{*}.$

The map $\Omega$ is a coboundary in the usual cohomology, namely
$\Omega=\partial\phi$ with $\phi=\lambda\,x_{1}^{*}$. Let us choose (see
Theorem 3.1, for the definitions)

$Z_{\Omega}=\lambda x_{0},\qquad\text{ and }\qquad
P(a)=x^{*}_{1}\bigl{(}a^{[p]}\bigr{)}\text{ for $a\in K^{2,3}_{\bar{0}}$}.$

On the other hand (see Theorem 3.1, for the definitions),

$\gamma=\lambda^{p-1},\qquad a_{0}=-\gamma
x_{0},\qquad\tilde{\lambda}=0,\qquad b_{0}=x_{1},\qquad\sigma=1.$

The case where $p>3$. In this case, the map $C$ is given by

$C=2\,y_{1}^{*}\wedge y_{3}^{*}-2\,y_{2}^{*}\wedge y_{2}^{*}.$

The map $\Omega$ cannot be a coboundary, except for $\lambda=0$ where it
becomes identically trivial. In this case, we can choose

$\gamma=0,\qquad b_{0}=0,\qquad a_{0}=x_{1},\qquad Z_{\Omega}=x_{1}.$

###  5.7 The Lie superalgebra $\boldsymbol{K^{2,m}}$, $\boldsymbol{m}$ even,
see [27]

For $m$ even, the Lie superalgebra $K^{2,m}$ is spanned by the generators
$x_{0},x_{1}\mid y_{1},\ldots y_{m}$ (even $|$ odd), with non-zero brackets
given by

$\displaystyle[x_{0},y_{i}]=-[y_{i},x_{0}]=y_{i+1},\qquad 1\leq i\leq m-1,$
$\displaystyle[y_{i},y_{m-i}]=[y_{m-i},y_{i}]=(-1)^{(m-2i)/2}x_{1},\qquad
1\leq i\leq\frac{m}{2}.$

It has been shown in [27] that the superalgebra $K^{2,m}$ has maximal nilindex
$m$.

The superalgebra $K^{2,m},$ where $m$ is even, is restricted if and only if
$m\leq p$. In that case, the $[p|2p]$-map is given by

$x_{0}^{[p]}=s_{1}x_{1},\qquad x_{1}^{[p]}=s_{2}x_{1},\qquad\text{ where}\
s_{1},s_{2}\in\mathbb{K}.$

The proof is similar to the one for $m$ odd.

######  Claim 5.3.

  * $(i)$

For all $m\geq 2$, the Lie superalgebra $K^{2,m}$ is not periplectic quasi-
Frobenius, except when ${p=3}$. In this case, the form is given by

$\omega=x_{0}^{*}\wedge y_{2}^{*}+x_{1}^{*}\wedge y_{1}^{*}.$

  * $(ii)$

For all $m\geq 2$, the Lie superalgebra $K^{2,m}$ is orthosymplectic quasi-
Frobenius if and only if $m<p$. If $m<p$, the form $\omega$ is given by

$\omega=\sum_{i=1}^{m/2}\alpha_{i}\;y_{i}^{*}\wedge
y_{m-i+1}^{*}+2\,x_{0}^{*}\wedge x_{1}^{*},$

where $\alpha_{i}=(-1)^{\frac{m}{2}-i+1}(m-2i+1)$.

####  5.7.1 A derivation yielding a non-trivial extension

Consider the outer derivation given by

${\mathscr{D}}=x_{1}\otimes x_{0}^{*}.$

This derivation is even and restricted. Moreover, it has the $p$-property (see
(2.2) for definitions) with $\gamma=0$ and
$a_{0}=l_{1}x_{1},~{}l_{1}\in\mathbb{K}$. Because $\gamma$ is zero, it follows
that we only have to consider the case $(b)$ of Theorem 3.1. Direct
computations show that ${\mathscr{D}}^{2}=0$ and
${\mathscr{D}}^{*}=-{\mathscr{D}}$. It follows that
$\sum_{i=1}^{p-1}\sigma_{i}^{\mathfrak{a}}(a,b)=0$ (see (2.7)), that $(C,P)$
is a restricted $2$-cocycle, and that the maps $\Omega$ and $T$ in Section 3.1
are identically zero. Because $(\Omega,T)$ has to be a restricted coboundary,
we can choose $Z_{\Omega}=tx_{1}+sy_{m},~{}t,s\in\mathbb{K},$ to obtain

$\displaystyle 0=\omega(Z_{\Omega},[a,b])\qquad\text{ and }\qquad
0=\omega\bigl{(}Z_{\Omega},\tilde{a}^{[p]}\bigr{)},$ $\displaystyle\text{ for
all }\qquad a,b\in K^{2,m}\qquad\text{ and }\qquad\tilde{a}\in
K^{2,m}_{\bar{0}}.$

The conditions of Theorem 3.1 are satisfied and we can choose
$b_{0}=l_{2}x_{1},~{}l_{2}\in\mathbb{K}$ and $\sigma\in\mathbb{K}$ arbitrary
to build a non-trivial double extension.

##  6 Conclusions and outlook

  1. (1)

In [1], Novikov Lie algebras appearing within symplectic structures are
discussed, and a classification is offered in low dimensions. It is
interesting to superize the results of [1].

  2. (2)

In [19], a certain class of free nilpotent Lie algebras possessing a
symplectic structure is studied. The investigation is based on the fact that
for a quasi-Frobenius Lie algebra we must have (see [4])

$\mathrm{dim}(\mathfrak{z}(\mathfrak{a}))\leq\mathrm{dim}(\mathfrak{a}/[\mathfrak{a},\mathfrak{a}]).$

It is easy to supersize this formula, and the proof is similar to that given
in [4]. It seems to be possible to construct a super-analog of the Lie
algebras in [19], and their results can be superized. However, it is not clear
yet whether these Lie (super)algebras are restricted or not. On the other
hand, the classification of restricted $p$-nilpotent Lie algebras in low
dimension have been obtained in [18, 31]. It is interesting to superize this
classification, and select among those Lie algebras that are quasi-Frobenius.

  3. (3)

A nilpotent orthosymplectic quasi-Frobenius Lie superalgebra cannot be
Frobenius. To see this, take the restriction of the form on the even part
${\mathfrak{g}}_{{\bar{0}}}$ and use the result [28, Proposition 6]. Engel’s
theorem is used to prove the result, which is also valid over a field with
characteristic non-zero. It would be interesting to examine the case where the
form is periplectic.

  4. (4)

Simple modular Lie (super)algebras over a field of positive characteristic may
have a degenerate Killing form and some of them do not admit a Cartan matrix,
for instance, those of Cartan type. Hence, the techniques used in [28] to show
that real or complex semi-simple Lie algebras cannot be Frobenius will not
work in characteristic $p$. Furthermore, we only know the classification of
modular Lie algebras over a field of characteristic $p>3$, see [39]. The
problem of classification remains open in the super setting. It is interesting
to investigate which modular Lie superalgebras are Frobenius.

  5. (5)

If the Lie superalgebra $\mathfrak{a}$ is Frobenius (i.e.,
$\omega_{\mathfrak{a}}=d_{\text{CE}}(f_{\mathfrak{a}})$), can its double
extension $\mathfrak{g}$ be Frobenius as well? In the case where $\omega$ is
odd, then $\mathfrak{g}$ cannot be Frobenius. Indeed, suppose that
$\omega_{\mathfrak{g}}=d_{\text{CE}}(f_{\mathfrak{g}})$. It follows that

$\displaystyle
0=\omega_{\mathfrak{g}}(x^{*},x^{*})=f_{\mathfrak{g}}([x^{*},x^{*}])=f_{\mathfrak{g}}(2a_{0})$
$\displaystyle\text{while}\
1=\omega_{\mathfrak{g}}(x^{*},x)=f_{\mathfrak{g}}([x^{*},x])=f_{\mathfrak{g}}(2a_{0}),$

a contradiction. In the case where $\omega_{\mathfrak{a}}$ is even, the linear
map $f_{\mathfrak{g}}$ must satisfy:

$\displaystyle f_{\mathfrak{g}}(x)=-\lambda^{-1},\qquad
f_{\mathfrak{g}}({\mathscr{D}}(a))=\lambda^{-1}\omega(Z_{\Omega},a),$
$\displaystyle
f_{\mathfrak{g}}([a,b]_{\mathfrak{a}})=f_{\mathfrak{a}}([a,b]_{\mathfrak{a}})+\lambda^{-1}\omega_{\mathfrak{a}}({\mathscr{D}}(a)+{\mathscr{D}}^{*}(a),b).$

One could describe all Lie superalgebras $\mathfrak{a}$ for which
$f_{\mathfrak{g}}$ exists.

### Acknowledgements

We are grateful for the valuable feedback we received from the referees, which
enabled us to significantly improve the presentation of this paper.

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