A note on the Schur multiplier of a nilpotent Lie algebra

TL;DR

This paper establishes an upper bound for the dimension of the Schur multiplier of a nilpotent Lie algebra based on its dimension and the dimension of its derived algebra, with specific characterization when the derived algebra has dimension one.

Contribution

It provides a new upper bound for the Schur multiplier of nilpotent Lie algebras and characterizes the case of equality for derived algebra of dimension one.

Findings

Upper bound formula for dim(M(L)) involving n and m

Equality condition when m=1, L is a direct sum involving the Heisenberg algebra

Characterization of the structure of L when the bound is tight

Abstract

For a nilpotent Lie algebra $L$ of dimension $n$ and dim$(L^2)=m$, we find the upper bound dim$(M(L))\leq {1/2}(n+m-2)(n-m-1)+1$, where $M(L)$ denotes the Schur multiplier of $L$. In case $m=1$ the equality holds if and only if $L\cong H(1)\oplus A$, where $A$ is an abelian Lie algebra of dimension $n-3$ and H(1) is the Heisenberg algebra of dimension 3.