Lie triple system central extensions of Lie algebras

TL;DR

This paper explores the relationships between universal central extensions of perfect Lie algebras when viewed as Lie triple systems, Leibniz algebras, and Lie algebras, revealing isomorphisms among these extensions.

Contribution

It demonstrates that the universal central extension of a perfect Lie algebra as a Lie triple system aligns with its extensions as Lie or Leibniz algebras, establishing a unifying perspective.

Findings

Lie triple system universal central extension is isomorphic to Lie algebra extension

Lie triple system universal central extension is isomorphic to Leibniz algebra extension

Extensions are equivalent when viewed through appropriate algebraic frameworks

Abstract

Three kinds of universal central extension are considered for a perfect Lie algebra. More precisely, one can consider such a Lie algebra as a Lie triple system, or a Leibniz algebra and construct appropriate central extensions. We show that Lie triple system universal central extension is isomorphic either to Lie or Leibniz algebra universal central extension considered as a Lie triple system in appropriate way.