Multipliers of nilpotent Lie superalgebras

Abstract

In this paper, first we prove that all finite dimensional special Heisenberg Lie superalgebras with even center have same dimension, say $(2m+1\mid n)$ for some non-negative integers $m,n$ and are isomorphism with them. Further, for a nilpotent Lie superalgebra $L$ of dimension $(m\mid n)$ and $\dim (L^{\prime })=(r\mid s)$ with $r+s\ge 1$, we find the upper bound $\dim \mathcal{M}(L)\le \frac{1}{2}\left[(m+n+r+s-2)(m+n-r-s-1)\right]+n+1$, where $\mathcal{M}(L)$ denotes the Schur multiplier of $L$. Moreover, if $(r,s)=(1,0)(\mathrm{respectively}(r,s)=(0,1))$, then the equality holds if and only if $L\cong H(1,0)\oplus A_1(\mathrm{respectively}H(0,1)\oplus A_2)$, where $A_1$ and $A_2$ are abelian Lie superalgebras with $\dim A_1=(m-3\mid n),\dim A_2=(m-1\mid n-1)$ and $H(1,0),H(0,1)$ are special Heisenberg Lie superalgebras of dimension $3$ and $2$ respectively.

16 pages