Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 070, 21 pages      arXiv:0806.4851
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

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 x1xlv0, where lm, xi ^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 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.

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