The PBW Filtration, Demazure Modules and Toroidal Current Algebras

Abstract

Abstract. Let L be the basic (level one vacuum) representation of the affine Kac-Moody Lie algebra g. The m-th space F m of the PBW filtration on L is a linear span of vectors of the form x 1 · · · x l v 0 , where l ≤ m, x i ∈ g and v 0 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 θ (z) 2 , which corresponds to the longest root θ. 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 F m /F m−1 can be filtered by graded deformations of the tensor products of m copies of g.

Demazure Modules, Chari-Venkatesh Modules and Fusion Products

“…The Corollary 2 generalizes a result of Feigin (see [6,Corollary 2]), where he only considers the case m = 0. Theorem 1, Corollaries 1 and 2 are proved in Section 4.…”

Untitled

“…Remark. Theorem 3.5 generalizes a result of [Feigin 2008], where the theorem was proved for = 1, λ 0 = 0 and λ 1 = θ . Unfortunately, the proof in that paper has a gap (personal communication by the author), which is now fixed by the proof above.…”

“…To be consistent with the notation in [Feigin 2008], we refer to it as the t N -filtration. Let gr t N T(N ) and gr t N T( , N ), respectively, be the associated graded spaces…”

Fusion products and toroidal algebras

“…To be consistent with the notation in [10], we refer to it as the t N -filtration. Let gr t N T(N ) respectively gr t N T(ℓ, N ) be the associated graded space: N ) is a module for gr t N T(N ).…”