The Second Cohomology of Current Algebras of General Lie Algebras

http://dx.doi.org/10.4153/CJM-2008-038-6
Canad. J. Math. 60(2008), 892-922

  Published:2008-08-01
 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.