Abelian extensions of infinite-dimensional Lie groups

TL;DR

This paper investigates abelian extensions of infinite-dimensional Lie groups modeled on locally convex spaces, using cohomology to classify extensions and analyze obstructions to integrability, with applications to diffeomorphism groups.

Contribution

It introduces a cohomology framework for classifying abelian extensions of infinite-dimensional Lie groups and describes obstructions to their integrability.

Findings

Parametrization of extension classes via a cohomology group $H^2_s(G,A)$.

Description of obstructions using period and flux homomorphisms.

Characterization of extensions with global smooth sections.

Abstract

In the present paper we study abelian extensions of connected Lie groups $G$ modeled on locally convex spaces by smooth $G$-modules $A$. We parametrize the extension classes by a suitable cohomology group $H^2_s(G,A)$ defined by locally smooth cochains and construct an exact sequence that describes the difference between $H^2_s(G,A)$ and the corresponding continuous Lie algebra cohomology space $H^2_c(\g,\a)$. The obstructions for the integrability of a Lie algebra extensions to a Lie group extension are described in terms of period and flux homomorphisms. We also characterize the extensions with global smooth sections resp. those given by global smooth cocycles. Finally we apply the general theory to extensions of several types of diffeomorphism groups.