The Second Cohomology of Current Algebras of General Lie Algebras
Printed: Aug 2008
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.
current algebra, Lie algebra cohomology, Lie algebra homology, invariant bilinear form, central extension
17B56 - Cohomology of Lie (super)algebras
17B65 - Infinite-dimensional Lie (super)algebras [See also 22E65]