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Representation Theory

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The PBW filtration


Author: Evgeny Feigin
Journal: Represent. Theory 13 (2009), 165-181
MSC (2000): Primary 17B67, 81R10
DOI: https://doi.org/10.1090/S1088-4165-09-00349-5
Published electronically: May 1, 2009
MathSciNet review: 2506263
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Abstract: In this paper we study the PBW filtration on irreducible integrable highest weight representations of affine Kac-Moody algebras $\widehat {\mathfrak {g}}$. The $n$-th space of this filtration is spanned by the vectors $x_1\dots x_s v$, where $x_i\in \widehat {\mathfrak {g}}$, $s\le n$, and $v$ is a highest weight vector. For the vacuum module we give a conjectural description of the corresponding adjoint graded space in terms of generators and relations. For $\mathfrak {g}$ of the type $A_1$ we prove our conjecture and derive the fermionic formula for the graded character.


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Additional Information

Evgeny Feigin
Affiliation: Tamm Theory Division, Lebedev Physics Institute, Russian Academy of Sciences, Russia, 119991, Moscow, Leninski prospect, 53 – and – Independent University of Moscow, Russia, Moscow, 119002, Bol’shoi Vlas’evskii, 11
Email: evgfeig@gmail.com

Received by editor(s): November 15, 2007
Received by editor(s) in revised form: February 4, 2009
Published electronically: May 1, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.