The PBW Filtration, Demazure Modules and Toroidal Current Algebras⋆

Let L be the basic (level one vacuum) representation of the affine Kac–Moody Lie algebra ĝ. 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 ∈ ĝ and v0 is a highest weight vector of L. In this paper we give two descriptions of the associated graded space L with respect to the PBW filtration. The “top-down” description deals with a structure of L 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) , which corresponds to the longest root θ. The “bottom-up” description deals with the structure of L 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.