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The PBW filtration, Demazure modules and toroidal current algebras. (English) Zbl 1215.17015
Summary: Let \(L\) be the basic (level one vacuum) representation of the affine Kac-Moody Lie algebra \(\widehat {\mathfrak g}\). The \(m\)-th space \(F_m\) of the PBW filtration on \(L\) is a linear span of vectors of the form \(x_{1}\cdots x_lv_{0}\), where \(l \leq m\), \(x_i\in \widehat {\mathfrak g}\) and \(v_0\) is a highest weight vector of \(L\). In this paper we give two descriptions of the associated graded space \(L^{\text{gr}}\) with respect to the PBW filtration. The “top-down” description deals with a structure of \(L^{\text{gr}}\) 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 \(L^{\text{gr}}\) 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 \(\mathfrak g\).

MSC:
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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