On the Schur multiplier of nilpotent Lie superalgebra

TL;DR

This paper classifies certain nilpotent Lie superalgebras based on their Schur multiplier and derived subalgebra dimensions, revealing their capability and providing a detailed structural understanding.

Contribution

It provides a classification of nilpotent Lie superalgebras with specific derived subalgebra dimensions and Schur multiplier properties, including their capability.

Findings

Classified nilpotent Lie superalgebras with derived dimension m+n-2

Determined the values of gamma(L) as 0, 1, or 2

Showed all such superalgebras are capable

Abstract

Let $L$ be an $(m\vert n)$-dimensional nilpotent Lie superalgebra where $m + n \geq 4$ and $n \geq 1$. This paper classifies such nilpotent Lie superalgebras $L$ with a derived subsuperalgebra of dimension $m+n-2$ such that $\gamma(L) = m + 2n - 2 - \dim \mathcal{M}(L)$, where $\gamma(L) \in \{0, 1, 2\}$ and $\mathcal{M}(L)$ denotes the Schur multiplier of $L$. Furthermore, we show that all these superalgebras are capable.