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The PBW Filtration, Demazure Modules and Toroidal Current Algebras
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2008, том 4
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The PBW Filtration, Demazure Modules and Toroidal Current Algebras
Інший ID:
2000 Mathematics Subject Classification: 17B67
Посилання:
The PBW Filtration, Demazure Modules and Toroidal Current Algebras / E. Feigin // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 26 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue on Kac–Moody Algebras and Applications. EF thanks B. Feigin and P. Littelmann for useful discussions. This work was partially supported by the RFBR Grants 06-01-00037, 07-02-00799 and NSh-3472.2008.2, by Pierre Deligne fund based on his 2004 Balzan prize in mathematics, by Euler foundation and by Alexander von Humboldt Fellowship.
Дата:
2008
Переглядів:
454
Завантажень:
238
Анотація:
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.
