Some properties on Schur multiplier and cover of a pair of Lie algebras

TL;DR

This paper explores the properties of the Schur multiplier and covers of pairs of Lie algebras, establishing their existence, relationships with crossed modules, and inequalities under certain conditions.

Contribution

It introduces the concept of covering pairs for Lie algebras, proves their existence, and relates them to crossed modules and Schur multipliers.

Findings

Existence of covering pairs for pairs of Lie algebras

Every crossed module is a homomorphic image of a covering pair

Inequalities for the Schur multiplier under specific conditions

Abstract

Similar to works of G. Ellis (1998), the concept of covering pair of Lie algebras is defined. Also, we show the existence of covering pair for the pair of Lie algebras (L,N) and then show that every crossed module is a homomorphic image of a covering pair of (L,N). Finally, under some condition, we present some inequalities for the Schur multiplier of a pair of finite dimensional Lie algebras.