The theory of Lie superalgebras have many applications in various areas of Mathematics and Physics. Kac gives a comprehensive description of mathematical theory of Lie superalgebras, and establishes the classification of all finite dimensional simple Lie superalgebras over an algebraically closed field of characteristic zero. In the last few years the theory of Lie superalgebras has evolved remarkably, obtaining many results in representation theory and classification. Most of the results are extension of well known facts of Lie algebras. But the classification of all finite dimensional nilpotent Lie superalgebras is still an open problem like that of finite dimensional nilpotent Lie algebras. Till today nilpotent Lie superalgebras $L$ of $\dim L \leq 5$ over real and complex fields are known.

Batten introduced and studied Schur multiplier and cover of Lie algebras
and later on studied by several authors. We have extended these notation
to Lie superalgebra case. Given a free presentation $ 0 \longrightarrow R
\longrightarrow F \longrightarrow L \longrightarrow 0 $ of Lie
superalgebra $L$ we define the *multiplier* of $L$ as $\mathcal{M}(L) =
\frac{[F,F]\cap R}{[F, R]}$. In this talk we prove that for nilpotent Lie
superalgebra $L = L_{\bar{0}} \oplus L_{\bar{1}}$ of dimension $(m\mid
n)$ and $\dim L^2= (r\mid s)$ with $r+s \geq 1$,
\begin{equation}
\dim \mathcal{M}(L)\leq \frac{1}{2}\left[(m + n + r + s - 2)(m + n - r -s -1) \right] + n + 1.
\end{equation}
Moreover, if $r+s = 1$, then the equality holds if and only if $ L \cong
H(1, 0) \oplus A(m-3 \mid n)$
where $A(m-3 \mid n)$ is an abelian Lie superalgebra of dimension $(m-3
\mid n)$, and $H(1, 0)$ is special Heisenberg Lie superalgebra of
dimension $(3 \mid 0)$. Then we define the function $s(L)$ as
\begin{equation}
s(L)= \frac{1}{2}(m+n-2)(m+n-1)+n+1-\dim \mathcal{M}(L).
\end{equation}
Clearly $s(L) \geq 0$ and structure of $L$ with $s(L)=0$ is known. We
obtain classification all finite dimensional nilpotent Lie superalgebras
with $s(L) \leq 2$.

We hope, this leads to a complete classification of the finite dimensional nilpotent Lie superalgebras of dimension $6,7$.

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Last updated: 23 Jun 2024