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# The Second Cohomology of Current Algebras of General Lie Algebras

Published:2008-08-01
Printed: Aug 2008
• Karl-Hermann Neeb
• Friedrich Wagemann
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## Abstract

Let $A$ be a unital commutative associative algebra over a field of characteristic zero, $\k$ a Lie algebra, and $\zf$ a vector space, considered as a trivial module of the Lie algebra $\gf := A \otimes \kf$. In this paper, we give a description of the cohomology space $H^2(\gf,\zf)$ in terms of easily accessible data associated with $A$ and $\kf$. We also discuss the topological situation, where $A$ and $\kf$ are locally convex algebras.
 Keywords: current algebra, Lie algebra cohomology, Lie algebra homology, invariant bilinear form, central extension
 MSC Classifications: 17B56 - Cohomology of Lie (super)algebras 17B65 - Infinite-dimensional Lie (super)algebras [See also 22E65]