Non-Abelian Extensions of Topological Lie Algebras

Abstract

ABSTRACT In this article we extend and adapt several results on extensions of Lie algebras to topological Lie algebras over topological fields of characteristic zero. In particular, we describe the set of equivalence classes of extensions of the Lie algebra 𝔤 by the Lie algebra 𝔫 as a disjoint union of affine spaces with translation group H 2(𝔤, 𝔷(𝔫))[S], where [S] denotes the equivalence class of the continuous outer action S : 𝔤 → der sp;𝔫. We also discuss topological crossed modules and explain how they are related to extensions of Lie algebras by showing that any continuous outer action gives rise to a crossed module whose obstruction class in H 3(𝔤, 𝔷(𝔫)) S is the characteristic class of the corresponding crossed module. The correspondence between crossed modules and extensions further leads to a description of 𝔫-extensions of 𝔤 in terms of certain 𝔷(𝔫)-extensions of a Lie algebra which is an extension of 𝔤 by 𝔫/𝔷(𝔫). We discuss several types of examples, describe applications to Lie algebras of vector fields on principal bundles, and in two appendices we describe the set of automorphisms and derivations of topological Lie algebra extensions.