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 b. The m-th space Fm of the PBW filtration on L is a linear span of vectors of the form x1 ···xlv0, where l m, xi 2bg and v0 is a highest weight vector of L. In this paper we give two descriptions of the associated graded space L gr with respect to the PBW filtration. The top-down description deals with a structure of L gr 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 e (z) 2 , which corresponds to the longest root . The bottom-up description deals with the structure of L gr as a representation of the current algebra g 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.