# Some results on the Schur multiplier of nilpotent Lie algebras

## TL;DR

This paper investigates bounds on the Schur multiplier of nilpotent Lie algebras, providing new inequalities, characterizing those attaining the bound, and refining the existing upper limit.

## Contribution

It introduces new inequalities for the exterior square and Schur multiplier, characterizes nilpotent Lie algebras reaching the bound, and improves the existing upper bound.

## Key findings

- Characterization of nilpotent Lie algebras attaining the bound
- New inequalities for the exterior square and Schur multiplier
- Refined upper bound on the Schur multiplier

## Abstract

For a non-abelian Lie algebra $L$ of dimension $n$ with the derived subalgebra of dimension $m$ , the first author earlier proved that the dimension of its Schur multiplier is bounded by $\frac{1}{2}(n+m-2)(n-m-1)+1$. In the current work, we give some new inequalities on the exterior square and the Schur multiplier of Lie algebras and then we obtain the class of all nilpotent Lie algebras which attains the above bound. Moreover, we also improve this bound as much as possible.