Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The PBW Filtration, Demazure Modules and Toroidal Current Algebras   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 ${\mathfrak g}$.

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The categories of this paper are: hep-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: comp-gas
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The categories of this paper are: cond-mat
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The categories of this paper are: cs
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-th
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The categories of this paper are: patt-sol
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The categories of this paper are: plasm-ph