Neeb KH (2006)
Publication Type: Journal article, Original article
Publication year: 2006
Publisher: Taylor & Francis: STM, Behavioural Science and Public Health Titles / Taylor & Francis
Book Volume: 34
Pages Range: 991-1041
Journal Issue: 3
DOI: 10.1080/00927870500441973
In this paper 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 the set of equivalence classes of extensions of the Lie algebra by the Lie algebra a disjoint union of affine spaces with translation group H^2(\g,\z(\n))_{[S]}, where [S] denotes the equivalence class of the continuous outer action S \: \g \to \der \n. We also discuss topological crossed modules and explain how they are related to extensions of Lie algebras by showing that any continuous outer action gives rise to a crossed module whose obstruction class in H^3(\g,\z(\n))_S is the characteristic class of the corresponding crossed module. The correspondence between crossed modules and extensions further leads to a description of \n-extensions of \g in terms of certain \z(\n)-extensions of a Lie algebra which is an extension of \g by \n/\z(\n). We discuss several types of examples, describe applications to Lie algebras of vector fields on principal bundles, and in two appendices we describe the set of automorphisms and derivations of topological Lie algebra extensions.
APA:
Neeb, K.H. (2006). Non-abelian extensions of topological Lie algebras. Communications in Algebra, 34(3), 991-1041. https://dx.doi.org/10.1080/00927870500441973
MLA:
Neeb, Karl Hermann. "Non-abelian extensions of topological Lie algebras." Communications in Algebra 34.3 (2006): 991-1041.
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