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Non-abelian extensions of infinite-dimensional Lie groups

Neeb KH (2007)

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Publication Type: Journal article, Original article

Publication year: 2007

Journal

Publisher: Association des Annales de l'Institute Fourier; 1999

Book Volume: 57

Pages Range: 209-271

Journal Issue: 01

Abstract

In this article 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 $\mathrm{Ext}{(G,N)}_S$ of extension classes is a principal homogeneous space of the locally smooth cohomology group $H_{ss}^2{(G,Z\left(N\right))}_S$. To each $S$ a locally smooth obstruction class $\chi \left(S\right)$ in a suitably defined cohomology group $H_{ss}^3{(G,Z\left(N\right))}_S$ 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 $G$.

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How to cite

APA:

Neeb, K.H. (2007). Non-abelian extensions of infinite-dimensional Lie groups. Annales de l'Institut Fourier, 57(01), 209-271.

MLA:

Neeb, Karl Hermann. "Non-abelian extensions of infinite-dimensional Lie groups." Annales de l'Institut Fourier 57.01 (2007): 209-271.

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