Advertisement

Vertex Algebras and Coordinate Rings of Semi-infinite Flags

  • Evgeny Feigin
  • Ievgen MakedonskyiEmail author
Article
  • 10 Downloads

Abstract

The direct sum of irreducible level one integrable representations of affine Kac-Moody Lie algebra of (affine) type ADE carries a structure of P/Q-graded vertex operator algebra. There exists a filtration on this direct sum studied by Kato and Loktev such that the corresponding graded vector space is a direct sum of global Weyl modules. The associated graded space with respect to the dual filtration is isomorphic to the homogenous coordinate ring of semi-infinite flag variety. We describe the ring structure in terms of vertex operators and endow the homogenous coordinate ring with a structure of P/Q-graded vertex operator algebra. We use the vertex algebra approach to derive semi-infinite Plücker-type relations in the homogeneous coordinate ring.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The work on Sections 1, 2, 3 was partially supported by the Russian Academic Excellence Project ‘5-100’. The work on Sections 4,5,6 was partially supported by the grant RSF-DFG 16-41-01013.

References

  1. A.
    Arakawa T.: A remark on the C 2 cofiniteness condition on vertex algebras. Math.Z. 270(1–2), 559–575 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. BF1.
    Braverman A., Finkelberg M.: Weyl modules and q-Whittaker functions. Math. Ann. 359(1), 45–59 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. BF2.
    Braverman A., Finkelberg M.: Semi-infinite Schubert varieties and quantum K-theory of flag manifolds. J. Am. Math. Soc. 27(4), 1147–1168 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. BG.
    Braverman A., Gaitsgory D.: Geometric Eisenstein series. Invent. Math. 150, 287–384 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. BK.
    Bakalov, B., Kac, V.G.: Generalized vertex algebras. In: Dobrev, V.K. et al. (ed.) Lie Theory and Its Applications to Physics VI. Heron Press, Sofia (2006)Google Scholar
  6. C.
    Carter R.: Lie Algebras of Finite and Affine Type. Cambridge University Press, Cambridge (2005)CrossRefzbMATHGoogle Scholar
  7. CFK.
    Chari V., Fourier G., Khandai T.: A categorical approach to Weyl modules. Transform. Groups 15(3), 517–549 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. CI.
    Chari V., Ion B.: BGG reciprocity for current algebras. Compos. Math. 151, 1265–1287 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. CL.
    Chari V., Loktev S.: Weyl, Demazure and fusion modules for the current algebra of \({\mathfrak{sl}_{r+1}}\). Adv. Math. 207, 928–960 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. CP.
    Chari, V., Pressley, A.: Weyl modules for classical and quantum affine algebras. Represent. Theory 5, 191–223 (2001) (electronic)Google Scholar
  11. Ch.
    Cherednik, I.: Double affine Hecke algebras, London Mathematical Society Lecture Note Series, 319, Cambridge University Press, Cambridge (2006)Google Scholar
  12. ChF.
    Cherednik I., Feigin B.: Rogers–Ramanujan type identities and Nil-DAHA. Adv. Math. 248, 1050–1088 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. ChK.
    Cherednik, I., Kato, S.: Nonsymmetric Rogers–Ramanujan sums and thick Demazure modules, arXiv:1802.03819
  14. DL.
    Dong C., Lepowsky J.: Generalized vertex algebras and relative vertex operators. Birkhauser, Basel (1993)CrossRefzbMATHGoogle Scholar
  15. FB.
    Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves, 2nd Edition, Mathematical Survey and Monographs 88, AMS (2004)Google Scholar
  16. FM1.
    Feigin E., Makedonskyi I.: Generalized Weyl modules, alcove paths and Macdonald polynomials. Selecta Math.(N.S.) 23, 2863–2897 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. FM2.
    Feigin, E., Makedonskyi, I.: Semi-infinite Plücker relations and Weyl modules. Int. Math. Res. Not. rny121,  https://doi.org/10.1093/imrn/rny121. arXiv:1709.05674
  18. FMO.
    Feigin, E., Makedonskyi, I., Orr, D.: Generalized Weyl modules and nonsymmetric q-Whittaker functions. arXiv:1605.01560, to appear in Advances in Mathematics
  19. FKM.
    Feigin, E., Kato, S., Makedonskyi, I.: Representation theoretic realization of non-symmetric Macdonald polynomials at infinity. arXiv:1703.04108
  20. FK.
    Frenkel I.B., Kac V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62(1), 23–66 (1980/81)Google Scholar
  21. FL.
    Fourier G., Littelmann P.: Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions. Adv. Math. 211(2), 566–593 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. FLM.
    Frenkel I.B., Lepowsky J., Meurman J.: Vertex operator algebras and the Monster. Academic Press, Cambridge (1988)zbMATHGoogle Scholar
  23. FMi.
    Finkelberg, M., Mirković, I.: Semi-infinite flags I. Case of global curve \({\mathbb{P}^1}\). In: Differential topology, infinite-dimensional Lie algebras, and applications, volume 194 of Am. Math. Soc. Transl. Ser. 2, pp. 81–112. Amer. Math. Soc., Providence, RI, (1999)Google Scholar
  24. I.
    Ion B.: Nonsymmetric Macdonald polynomials and Demazure characters. Duke Math. J. 116(2), 299–318 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Kac1.
    Kac V.: Infinite dimensional Lie algebras, 3rd ed. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  26. Kac2.
    Kac, V.: Vertex Algebras for Beginners, University Lecture Series Vol. 10. American Mathematical Society (1996)Google Scholar
  27. Kat.
    Kato, S.: Demazure character formula for semi-infinite flag varieties, Math. Ann. (2018).  https://doi.org/10.1007/s00208-018-1652-5
  28. KL.
    Kato, S., Loktev, S.: With an appendix by Ryosuke Kodera, A Weyl Module Stratification of Integrable Representations. arXiv:1712.03508
  29. KNS.
    Kato, S., Naito, S., Sagaki, D.: Equivariant K-theory of semi-infinite flag manifolds and Pieri-Chevalley formula. arXiv:1702.02408
  30. Kum.
    Kumar, S.: Kac–Moody groups, their flag varieties and representation theory, Progress in Mathematics, 204. Birkhäuser Boston, Inc., Boston, MA (2002)Google Scholar
  31. Li1.
    Li H.: Vertex algebras and vertex Poisson algebras. Commun. Contemp. Math. 6(1), 61–110 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. Li2.
    Li H.: Abelianizing vertex algebras. Commun. Math. Phys. 259(2), 391–411 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. R.
    Ray U.: Automorphic forms and Lie superalgebras. Springer, Berlin (2006)zbMATHGoogle Scholar
  34. S.
    Scheithauer N.R.: The fake monster superalgebra. Adv. Math. 151(2), 226–269 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Research University Higher School of EconomicsMoscowRussia
  2. 2.Department of MathematicsKyoto UniversityKyotoJapan

Personalised recommendations