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 $\hat{\mathfrak{g}}$. The $m$-th space $F_m$ of the PBW filtration on $L$ is a linear span of vectors of the form $x_1...x_l{v}_0$, 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 $L^{\mathrm{gr}}$ with respect to the PBW filtration. The "top-down" description deals with a structure of $L^{\mathrm{gr}}$ 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 $L^{\mathrm{gr}}$ 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 $\mathfrak{g}$.

This is a contribution to the Special Issue on Kac-Moody Algebras and Applications, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

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• Fusion products, Kostka polynomials, and fermionic characters of su(r+1)_k
• Standard Bases for Affine SL(n)-Modules