Non-abelian extensions of infinite-dimensional Lie groups

TL;DR

This paper investigates non-abelian extensions of infinite-dimensional Lie groups, establishing a cohomological framework that reduces complex extension problems to abelian cases and introduces smooth crossed modules as a key tool.

Contribution

It develops a cohomological classification of non-abelian Lie group extensions using smooth outer actions and introduces the concept of smooth crossed modules for analysis.

Findings

Extension classes form a principal homogeneous space over a cohomology group.

Obstruction classes determine the existence of extensions.

Reduction of non-abelian extension problems to abelian cases.

Abstract

In this paper we study non-abelian extensions of a Lie group $G$ modeled on a locally convex space by a Lie group $N$. The equivalence classes of such extension are grouped into those corresponding to a class of so-called smooth outer actions $S$ of $G$ on $N$. If $S$ is given, we show that the corresponding set $\Ext(G,N)_S$ of extension classes is a principal homogeneous space of the locally smooth cohomology group $H^2_{ss}(G,Z(N))_S$. To each $S$ a locally smooth obstruction class $\chi(S)$ in a suitably defined cohomology group $H^3_{ss}(G,Z(N))_S$ is defined. It vanishes if and only if there is a corresponding extension of $G$ by $N$. A central point is that we reduce many problems concerning extensions by non-abelian groups to questions on extensions by abelian groups, which have been dealt with in previous work. An important tool is a Lie theoretic concept of a smooth crossed module $\alpha : H \to G$, which we view as a central extension of a normal subgroup of $G$.