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SIGMA 4 (2008), 070, 21 pages arXiv:0806.4851
https://doi.org/10.3842/SIGMA.2008.070
Contribution to the Special Issue on Kac-Moody Algebras and Applications
The PBW Filtration, Demazure Modules and Toroidal Current Algebras
Evgeny Feigin a, b
a) Mathematical Institute, University of Cologne, Weyertal 86-90, D-50931,
Cologne, Germany
b) I.E. Tamm Department of Theoretical Physics, Lebedev Physics Institute,
Leninski Prospect 53, Moscow, 119991, Russia
Received July 04, 2008, in final form October 06, 2008; Published online October 14, 2008
Abstract
Let L be the basic (level one vacuum) representation of the affine Kac-Moody Lie algebra
^g. The m-th space Fm of the PBW filtration on L is a linear span of vectors
of the form x1¼xlv0, where l ≤ m, xi Î ^g and v0 is a highest weight
vector of L. In this paper we give two descriptions of the associated graded space Lgr with respect
to the PBW filtration. The ''top-down'' description deals with a structure of Lgr 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
eθ(z)2, which corresponds to the longest root θ. The ''bottom-up'' description
deals with the structure of Lgr as a representation of the current algebra
g Ä C[t]. We prove that each quotient Fm/Fm-1 can be filtered by graded
deformations of the tensor products of m copies of g.
Key words:
affine Kac-Moody algebras; integrable representations; Demazure modules.
pdf (332 kb)
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