An attempt to relate area-preserving diffeomorphisms to Kac-Moody algebras (Q1174727)

The affine Kac-Moody algebra \(\hat{\mathfrak g}\) is essentially the central extension of the Lie algebra of maps from the circle \(S^ 1\) to the Lie algebra \({\mathfrak g}\). The Virasoro algebra acts on \(\hat{\mathfrak g}\) as derivations. The present work considers the case of super Kac-Moody like algebras on the 2-dimensional compact surfaces \(\mathcal M\) with one or two supersymmetries, i.e., \({\mathcal M}\times\hbox{Gr}(1)\) and \({\mathcal M}\times \hbox{Gr}(2)\), where \(\hbox{Gr}(n)\) is the Grassmann algebra of dimension \(2^ n\). It is proved that there exist compatible pairs of Virasoro and Kac-Moody like algebras in these cases. The paper also makes an attempt to generalize the Sugawara operators. The original Sugawara construction, roughly speaking, amounts to construct Virasoro operators in terms of the affine Kac-Moody operators in certain representations.