Multipliers of nilpotent Lie superalgebras

TL;DR

This paper classifies finite-dimensional special Heisenberg Lie superalgebras with even centers and establishes bounds for the Schur multiplier of nilpotent Lie superalgebras, identifying cases of equality and their structural implications.

Contribution

It provides a classification of special Heisenberg Lie superalgebras with even centers and derives an explicit upper bound for the Schur multiplier of nilpotent Lie superalgebras, including conditions for equality.

Findings

All finite-dimensional special Heisenberg Lie superalgebras with even center have the same dimension.

An explicit upper bound for the Schur multiplier of nilpotent Lie superalgebras is established.

Equality cases characterize specific direct sum decompositions involving Heisenberg 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 $\dim (L') = (r\mid s)$ with $r+s \geq 1$, we find the upper bound $\dim \mathcal{M}(L)\leq \frac{1}{2}\left[(m + n + r + s - 2)(m + n - r -s -1) \right] + n + 1$, 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.