Universal central extensions of braided crossed modules in Lie algebras

TL;DR

This paper explores the construction and properties of universal central extensions of braided crossed modules in Lie algebras, establishing conditions under which these extensions exist and are equivalent.

Contribution

It introduces a natural braiding on universal central extensions of crossed modules of Lie algebras and proves their equivalence under certain conditions.

Findings

Existence of a natural braiding on universal central extensions.

Equivalence of two constructions of universal central extensions when one exists.

Unified framework for braided crossed modules in Lie algebras.

Abstract

In this paper, we give a natural braiding on the universal central extension of a crossed module of Lie algebras with a given braiding and construct the universal central extension of a braided crossed module of Lie algebras, showing that, when one of the constructions exists, both exist and coincide.