SEMINAIRE Mulhousien de MATHEMATIQUES
résumé/abstract
Karl Herrmann Neeb (Darmstadt)
Central extensions of current algebras -old and new aspects
According to a classical result of Kassel, the universal
central extension of a Lie algebra of the form $A \otimes
\g$, $\g$ complex simple, can be described quite directly and explicitly in
terms of a cocycle with values in $\Omega^1(A)/d A$. If
$\g$ is not simple, or even infinite-dimensional, the classifications of central
extensions is much more intricate. In this talk we report on recent work with
F. Wagemann on an analysis of central extensions of
general current Lie algebras $A \otimes \g$ which
applies in particular to the topological context of locally convex Lie algebras
and continuous cocycles. For $A = C^\infty(M)$,
this leads to three types of cocycles with values in
$C^\infty(M)$, $\Omega^1(M)$ and $\Omega^1(M)/dC^\infty(M)$. Similar cocycles
exist for Lie algebras of smooth sections of Lie algebra bundles, but in this
generality a classification still seems out of reach.
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Dernières modifications / Last modifications : 18 Avril 2008