Universal Central Extensions of Lie Crossed Modules Over a Fixed Lie Algebra

Abstract

Let L be a Lie algebra over a field of arbitrary characteristic. In this paper, we give a necessary and sufficient condition for the existence of universal central extensions in the category of crossed modules of Lie algebras over L. Also, we determine the structure of the universal central extension of a crossed L-module and show that the kernel of this extension is related to the first non-abelian homology of L.