Non-abelian extensions of infinite-dimensional Lie groups

@article{Neeb2005NonabelianEO,
  title={Non-abelian extensions of infinite-dimensional Lie groups},
  author={Karl-Hermann Neeb},
  journal={arXiv: Group Theory},
  year={2005}
}
  • K. Neeb
  • Published 2005
  • Mathematics
  • arXiv: Group Theory
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… Expand
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References

SHOWING 1-10 OF 44 REFERENCES
Abelian extensions of infinite-dimensional Lie groups
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 groupExpand
Non-Abelian Extensions of Topological Lie Algebras
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 theExpand
Extensions and low dimensional cohomology theory of locally compact groups. II
Introduction. In a previous paper [8], we have defined a sequence of cohomology groups Hr(G, A) defined when G is a locally compact group and A is an abelian locally compact group on which G actsExpand
Automorphisms of group extensions
If 1 G I> E X-4 I -> 1 is a group extension, with t an inclusion, any automorphism T of E which takes G onto itself induces automorphisms T on G and a on 11. However, for a pair (a, T) ofExpand
An introduction to homological algebra
Preface 1. Generalities concerning modules 2. Tensor products and groups of homomorphisms 3. Categories and functors 4. Homology functors 5. Projective and injective modules 6. Derived functors 7.Expand
Central extensions of infinite-dimensional Lie groups
Le principal resultat de cet article est une suite exacte pour le groupe abelien des extensions centrales d'un groupe de Lie connexe G de dimension infinie par un groupe abelien de Lie Z pour lequelExpand
Higher-dimensional loop algebras, non-abelian extensions and p-branes
We postulate a new type of operator algebra with a non-abelian extension. This algebra generalizes the Kac-Moody algebra in string theory and the Mickelsson-Faddeev algebra in three dimensions toExpand
The Convenient Setting of Global Analysis
Introduction Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly realcompact spaces Extensions and liftings of mappings Infinite dimensionalExpand
Twisted Actions and Obstructions in Group Cohomology
This article is intended to answer the question “Why do you guys always want to twist everything?” We review the various ways in which twists, twisted actions and twisted crossed products arise, andExpand
...
1
2
3
4
5
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