Relative Lie central extension and Schur multiplier of pairs of multiplicative Lie algebras
Dev Karan Singh, Shiv Datt Kumar
TL;DR
This work develops a theory of relative Lie central extensions for pairs of multiplicative Lie algebras, extending central extension concepts to a setting with compatible actions encoded by pairs $ $ (H,G) $. It introduces isoclinism for these relative extensions and the Frattini subalgebra, and defines the Schur multiplier for pairs, denoted $\tilde{M}(H,G)$, including a free-presentation perspective. A central result is a four-term exact sequence relating multipliers of a pair and its quotients, along with the construction and existence criteria for multiplicative covering pairs under suitable hypotheses (e.g., $L$ perfect with $\tilde{\Phi}(L) \neq L$). The framework establishes cohomology-like invariants for pairs of multiplicative Lie algebras and offers potential applications in representation theory and K-theory analogues.
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.