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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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