The Schur multiplier of some finite multiplicative Lie algebras

TL;DR

This paper investigates the Schur multiplier and Lie exterior square of certain finite multiplicative Lie algebras, revealing that for non-abelian simple groups with trivial Schur multiplier, these structures are trivial or identical to the original group.

Contribution

It provides explicit calculations of the Schur multiplier and Lie exterior square for specific finite multiplicative Lie algebras, especially those derived from simple groups.

Findings

Schur multiplier of certain finite multiplicative Lie algebras is trivial

Lie exterior square of these algebras is isomorphic to the algebra itself

For simple groups with trivial Schur multiplier, the associated multiplicative Lie algebra has trivial Schur multiplier

Abstract

The main aim of the article is to find the Schur multiplier and the Lie exterior square of some finite multiplicative Lie algebras. For a non abelian simple group $K$ with trivial Schur multiplier, we see that the Schur multiplier of multiplicative Lie algebra $K$ is trivial and the Lie exterior square of $K$ is an improper multiplicative Lie algebra $K.$