A Weyl module filtration for the vertex algebra of differential operators

Zhu, Minxian

doi:10.1090/S1088-4165-2011-00381-2

A Weyl module filtration for the vertex algebra of differential operators
HTML articles powered by AMS MathViewer

by Minxian Zhu

Represent. Theory 15 (2011), 370-384

DOI: https://doi.org/10.1090/S1088-4165-2011-00381-2

Published electronically: April 29, 2011

PDF | Request permission

Abstract:

The aim of this paper is to give a proof of a conjecture stated in a previous paper by the author. We prove the existence of certain filtrations, in the case of rational central charges, for the vertex algebras of differential operators on a Lie group.

References

S. M. Arkhipov, Semiinfinite cohomology of quantum groups, Comm. Math. Phys. 188 (1997), no. 2, 379–405. MR 1471820, DOI 10.1007/s002200050170
S. M. Arkhipov, Semi-infinite cohomology of associative algebras and bar duality, Internat. Math. Res. Notices 17 (1997), 833–863. MR 1474841, DOI 10.1155/S1073792897000548
S. Arkhipov and D. Gaitsgory, Differential operators on the loop group via chiral algebras, Int. Math. Res. Not. 4 (2002), 165–210. MR 1876958, DOI 10.1155/S1073792802102078
B. L. Feĭgin, Semi-infinite homology of Lie, Kac-Moody and Virasoro algebras, Uspekhi Mat. Nauk 39 (1984), no. 2(236), 195–196 (Russian). MR 740035
I. B. Frenkel, H. Garland, and G. J. Zuckerman, Semi-infinite cohomology and string theory, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 22, 8442–8446. MR 865483, DOI 10.1073/pnas.83.22.8442
Igor B. Frenkel and Konstantin Styrkas, Modified regular representations of affine and Virasoro algebras, VOA structure and semi-infinite cohomology, Adv. Math. 206 (2006), no. 1, 57–111. MR 2261751, DOI 10.1016/j.aim.2005.08.005
Vassily Gorbounov, Fyodor Malikov, and Vadim Schechtman, Gerbes of chiral differential operators. II. Vertex algebroids, Invent. Math. 155 (2004), no. 3, 605–680. MR 2038198, DOI 10.1007/s00222-003-0333-4
Vassily Gorbounov, Fyodor Malikov, and Vadim Schechtman, On chiral differential operators over homogeneous spaces, Int. J. Math. Math. Sci. 26 (2001), no. 2, 83–106. MR 1836785, DOI 10.1155/S0161171201020051
D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. I, II, J. Amer. Math. Soc. 6 (1993), no. 4, 905–947, 949–1011. MR 1186962, DOI 10.1090/S0894-0347-1993-99999-X
D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. I, II, J. Amer. Math. Soc. 6 (1993), no. 4, 905–947, 949–1011. MR 1186962, DOI 10.1090/S0894-0347-1993-99999-X
D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. III, J. Amer. Math. Soc. 7 (1994), no. 2, 335–381. MR 1239506, DOI 10.1090/S0894-0347-1994-1239506-X
D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. IV, J. Amer. Math. Soc. 7 (1994), no. 2, 383–453. MR 1239507, DOI 10.1090/S0894-0347-1994-1239507-1
Wolfgang Soergel, Character formulas for tilting modules over Kac-Moody algebras, Represent. Theory 2 (1998), 432–448. MR 1663141, DOI 10.1090/S1088-4165-98-00057-0
Alexander A. Voronov, Semi-infinite homological algebra, Invent. Math. 113 (1993), no. 1, 103–146. MR 1223226, DOI 10.1007/BF01244304
Minxian Zhu, Vertex operator algebras associated to modified regular representations of affine Lie algebras, Adv. Math. 219 (2008), no. 5, 1513–1547. MR 2458145, DOI 10.1016/j.aim.2008.07.005
Minxian Zhu, Regular representations of quantum groups at roots of unity, Int. Math. Res. Not. IMRN 15 (2010), 3039–3065. MR 2673718, DOI 10.1093/imrn/rnp236