The PBW Filtration, Demazure Modules and Toroidal Current Algebras

Abstract

Let L be the basic (level one vacuum) representation of the affine Kac-Moody Lie algebra ^g. The m-th space Fm of the PBW filtration on L is a linear span of vectors of the form x1¼xlv0, where l ≤ m, xi Î ^g and v0 is a highest weight vector of L. In this paper we give two descriptions of the associated graded space Lgr with respect to the PBW filtration. The ''top-down'' description deals with a structure of Lgr as a representation of the abelianized algebra of generating operators. We prove that the ideal of relations is generated by the coefficients of the squared field eθ(z)2, which corresponds to the longest root θ. The ''bottom-up'' description deals with the structure of Lgr as a representation of the current algebra g Ä C[t]. We prove that each quotient Fm/Fm-1 can be filtered by graded deformations of the tensor products of m copies of g.