Non-abelian extensions of infinite-dimensional Lie groups

Karl-Hermann Neeb[1]

  • [1] Technische Universität Darmstadt Schlossgartenstrasse 7 64289 Darmstadt(Deutschland)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 1, page 209-271
  • ISSN: 0373-0956

Abstract

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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 Ext ( G , N ) S of extension classes is a principal homogeneous space of the locally smooth cohomology group H s s 2 ( G , Z ( N ) ) S . To each S a locally smooth obstruction class χ ( S ) in a suitably defined cohomology group H s s 3 ( 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 α : H → G , which we view as a central extension of a normal subgroup of G .

How to cite

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Neeb, Karl-Hermann. "Non-abelian extensions of infinite-dimensional Lie groups." Annales de l’institut Fourier 57.1 (2007): 209-271. <http://eudml.org/doc/10220>.

@article{Neeb2007,
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 $\{\rm Ext\}(G,N)_S$ 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 $\alpha \colon H \rightarrow G$, which we view as a central extension of a normal subgroup of $G$.},
affiliation = {Technische Universität Darmstadt Schlossgartenstrasse 7 64289 Darmstadt(Deutschland)},
author = {Neeb, Karl-Hermann},
journal = {Annales de l’institut Fourier},
keywords = {Lie group extension; smooth outer action; crossed module; Lie group cohomology; automorphisms of group extension},
language = {eng},
number = {1},
pages = {209-271},
publisher = {Association des Annales de l’institut Fourier},
title = {Non-abelian extensions of infinite-dimensional Lie groups},
url = {http://eudml.org/doc/10220},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Neeb, Karl-Hermann
TI - Non-abelian extensions of infinite-dimensional Lie groups
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 1
SP - 209
EP - 271
AB - 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 ${\rm Ext}(G,N)_S$ 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 $\alpha \colon H \rightarrow G$, which we view as a central extension of a normal subgroup of $G$.
LA - eng
KW - Lie group extension; smooth outer action; crossed module; Lie group cohomology; automorphisms of group extension
UR - http://eudml.org/doc/10220
ER -

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