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The Second Cohomology of Current Algebras of General Lie Algebras

Published online by Cambridge University Press:  20 November 2018

Karl-Hermann Neeb  and
Friedrich Wagemann
Karl-Hermann Neeb
Affiliation:
Fachbereich Mathematik, Technische Universität Darmstadt, 64285 Darmstadt, Germany e-mail:[email protected]
Friedrich Wagemann
Affiliation:
Laboratoire de Mathématiques Jean Leray, Faculté des Sciences et Techniques, Université de Nantes, 44322 Nantes cedex 3, France e-mail:[email protected]
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Abstract

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Let $A$ be a unital commutative associative algebra over a field of characteristic zero, $\mathfrak{k}$ a Lie algebra, and $\mathfrak{z}$ a vector space, considered as a trivial module of the Lie algebra $\mathfrak{g}:=A\otimes \mathfrak{k}$. In this paper, we give a description of the cohomology space ${{H}^{2}}(\mathfrak{g},\mathfrak{z})$ in terms of easily accessible data associated with $A$ and $\mathfrak{k}$. We also discuss the topological situation, where $A$ and $\mathfrak{k}$ are locally convex algebras.

Keywords

17B5617B65current algebraLie algebra cohomologyLie algebra homologyinvariant bilinear formcentral extension
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 4 , 01 August 2008 , pp. 892 - 922
Copyright
Copyright © Canadian Mathematical Society 2008

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