The PBW Filtration, Demazure Modules and Toroidal Current Algebras

TL;DR

This paper provides two new descriptions of the associated graded space of the basic representation of affine Kac-Moody Lie algebras with respect to the PBW filtration, linking it to abelianized and current algebra structures.

Contribution

It introduces top-down and bottom-up frameworks for understanding the PBW filtration's associated graded space in affine Kac-Moody representations, including explicit relations and deformations.

Findings

Relations generated by coefficients of squared field e_θ(z)^2.

Each quotient F_m/F_{m-1} can be filtered by deformations of tensor products of g.

Provides structural insights into the representation as an abelianized algebra and current algebra.

Abstract

Let $L$ be the basic (level one vacuum) representation of the affine Kac-Moody Lie algebra $\hat{\mathfrak g}$. The $m$-th space $F_m$ of the PBW filtration on $L$ is a linear span of vectors of the form $x_1... x_lv_0$, where $l\le m$, $x_i\in \hat{\mathfrak g}$ and $v_0$ is a highest weight vector of $L$. In this paper we give two descriptions of the associated graded space $L^{\rm gr}$ with respect to the PBW filtration. The "top-down" description deals with a structure of $L^{\rm gr}$ 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 $e_\theta(z)^2$, which corresponds to the longest root $\theta$. The "bottom-up" description deals with the structure of $L^{\rm gr}$ as a representation of the current algebra ${\mathfrak g}\otimes {\mathbb C}[t]$. We prove that each quotient $F_m/F_{m-1}$ can be filtered by graded deformations of the tensor products of $m$ copies of ${\mathfrak g}$.