Relative Lie central extension and Schur multiplier of pairs of multiplicative Lie algebras

TL;DR

This paper introduces the concept of relative Lie central extensions for pairs of multiplicative Lie algebras, explores isoclinism, defines the Frattini subalgebra, and investigates the Schur multiplier and covering pairs.

Contribution

It develops the theory of relative Lie central extensions and Schur multipliers specifically for pairs of multiplicative Lie algebras, including new definitions and properties.

Findings

Defined relative Lie central extension for pairs of multiplicative Lie algebras

Established properties of the Frattini subalgebra in this context

Proved existence of multiplicative covering pairs under certain conditions

Abstract

In this paper, we introduce the concept of relative Lie central extension for pair of multiplicative Lie algebras. Then, we discuss the concept of isoclinism for relative Lie central extensions and prove some related results. We also define the Frattini subalgebra for multiplicative Lie algebras and discuss its properties, finally the Schur multiplier for pair of multiplicative Lie algebras is introduced and under certain conditions prove the existence of multiplicative covering pair.