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Title: Colloquium: Schur Multipliers and Classification of finite dimensional nilpotent Lie superalgebras
Speaker: Saudamini Nayak (Kalinga Institute of Industrial Technology, Bhubaneswar)
Date: 12 December 2022
Time: 4 pm
Venue: Microsoft Teams (Online)

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: 25 Jan 2023