Advertisement

Free Field Realizations of the Date-Jimbo-Kashiwara-Miwa Algebra

  • Ben CoxEmail author
  • Vyacheslav Futorny
  • Renato Alessandro Martins
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 38)

Abstract

We use the description of the universal central extension of the DJKM algebra \(\mathfrak{s}\mathfrak{l}(2,R)\) where \(R = \mathbb{C}[t,t^{-1},u\,\vert \,u^{2} = t^{4} - 2ct^{2} + 1]\) given in [CF11] to construct realizations of the DJKM algebra in terms of sums of partial differential operators.

Key words

Wakimoto modules DJKM algebras Affine Lie algebras Fock spaces 

Mathematics Subject Classification (2010):

17B65 17B69. 

Notes

Acknowledgements

The first author would like to thank the other two authors and the University of São Paulo for hosting him while he visited Brazil in June of 2013 where part of this work was completed. The second author was partially supported by Fapesp (2010/50347-9) and CNPq (301743/2007-0). The third author was supported by FAPESP (2012/02459-8).

References

  1. BCF09.
    André Bueno, Ben Cox, and Vyacheslav Futorny, Free field realizations of the elliptic affine Lie algebra \(\mathfrak{s}\mathfrak{l}(2,\mathbf{R}) \oplus (\varOmega _{R}/d\mathrm{R})\). J. Geom. Phys., 59(9):1258–1270, 2009.Google Scholar
  2. BI82.
    Joaquin Bustoz and Mourad E. H. Ismail, The associated ultraspherical polynomials and their q-analogues. Canad. J. Math., 34(3):718–736, 1982.Google Scholar
  3. Bre94a.
    Murray Bremner, Generalized affine Kac-Moody Lie algebras over localizations of the polynomial ring in one variable. Canad. Math. Bull., 37(1):21–28, 1994.Google Scholar
  4. Bre94b.
    Murray Bremner, Universal central extensions of elliptic affine Lie algebras. J. Math. Phys., 35(12):6685–6692, 1994.Google Scholar
  5. Bre95.
    Murray Bremner, Four-point affine Lie algebras. Proc. Amer. Math. Soc.,123(7):1981–1989, 1995.Google Scholar
  6. BS83.
    N. N. Bogoliubov and D. V. Shirkov, Quantum Fields. Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, MA, 1983. Translated from the Russian by D. B. Pontecorvo.Google Scholar
  7. BT07.
    Georgia Benkart and Paul Terwilliger, The universal central extension of the three-point \(\mathfrak{s}\mathfrak{l}_{2}\) loop algebra. Proc. Amer. Math. Soc., 135(6):1659–1668, 2007.Google Scholar
  8. CF11.
    Ben Cox and Vyacheslav Futorny, DJKM algebras I: their universal central extension. Proc. Amer. Math. Soc., 139(10):3451–3460, 2011.Google Scholar
  9. CFT13.
    Ben Cox, Vyacheslav Futorny, and Juan A. Tirao, DJKM algebras and non-classical orthogonal polynomials. J. Differential Equations, 255(9):2846–2870, 2013.Google Scholar
  10. CJ.
    Ben L. Cox and Elizabeth Jurisich, Realizations of the three point Lie algebra \(\mathfrak{s}\mathfrak{l}(2,R) \oplus (\varOmega _{R}/dR)\). arXiv:1303.6973.Google Scholar
  11. Cox08.
    Ben Cox, Realizations of the four point affine Lie algebra \(\mathfrak{s}\mathfrak{l}(2,R) \oplus (\varOmega _{R}/dR)\). Pacific J. Math., 234(2):261–289, 2008.Google Scholar
  12. DJKM83.
    Etsurō Date, Michio Jimbo, Masaki Kashiwara, and Tetsuji Miwa, Landau-Lifshitz equation: solitons, quasiperiodic solutions and infinite-dimensional Lie algebras. J. Phys. A, 16(2):221–236, 1983.MathSciNetCrossRefzbMATHGoogle Scholar
  13. EFK98.
    Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, Volume 58 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1998.Google Scholar
  14. FBZ01.
    Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, Volume 88, Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001.Google Scholar
  15. FF90.
    Boris L. Feĭgin and Edward V. Frenkel, Affine Kac-Moody algebras and semi-infinite flag manifolds. Comm. Math. Phys., 128(1):161–189, 1990.Google Scholar
  16. FF99.
    Boris Feigin and Edward Frenkel, Integrable hierarchies and Wakimoto modules. In Differential topology, infinite-dimensional Lie algebras, and applications, Volume 194, Amer. Math. Soc. Transl. Ser. 2, pages 27–60. Amer. Math. Soc., Providence, RI, 1999.Google Scholar
  17. Fre05.
    Edward Frenkel, Wakimoto modules, opers and the center at the critical level. Adv. Math., 195(2):297–404, 2005.Google Scholar
  18. Fre07.
    Edward Frenkel, Langlands correspondence for loop groups, Volume 103, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2007.Google Scholar
  19. FS05.
    Alice Fialowski and Martin Schlichenmaier, Global geometric deformations of current algebras as Krichever-Novikov type algebras. Comm. Math. Phys., 260(3):579–612, 2005.Google Scholar
  20. FS06.
    Alice Fialowski and Martin Schlichenmaier, Global geometric deformations of the virasoro algebra, current and affine algebras by Krichever-Novikov type algebra. math.QA/0610851, 2006.Google Scholar
  21. Hua98.
    Kerson Huang, Quantum field theory. John Wiley & Sons Inc., New York, 1998. From operators to path integrals.Google Scholar
  22. Ism05.
    Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Volume 98, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche, with a foreword by Richard A. Askey.Google Scholar
  23. JK85.
    H. P. Jakobsen and V. G. Kac, A new class of unitarizable highest weight representations of infinite-dimensional Lie algebras. In Nonlinear equations in classical and quantum field theory (Meudon/Paris, 1983/1984), pages 1–20. Springer, Berlin, 1985.Google Scholar
  24. Kac98.
    Victor Kac, Vertex Algebras for Beginners. American Mathematical Society, Providence, RI, second edition, 1998.Google Scholar
  25. Kas84.
    Christian Kassel, Kähler differentials and coverings of complex simple Lie algebras extended over a commutative algebra. In Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983), Volume 34, pp 265–275, 1984.Google Scholar
  26. KL82.
    C. Kassel and J.-L. Loday, Extensions centrales d’algèbres de Lie. Ann. Inst. Fourier (Grenoble), 32(4):119–142 (1983), 1982.Google Scholar
  27. KL91.
    David Kazhdan and George Lusztig, Affine Lie algebras and quantum groups. Internat. Math. Res. Notices, (2):21–29, 1991.Google Scholar
  28. KL93.
    D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. I, II. J. Amer. Math. Soc., 6(4):905–947, 949–1011, 1993.Google Scholar
  29. KN87a.
    Igor Moiseevich Krichever and S. P. Novikov, Algebras of Virasoro type, Riemann surfaces and strings in Minkowski space. Funktsional. Anal. i Prilozhen., 21(4):47–61, 96, 1987.Google Scholar
  30. KN87b.
    Igor Moiseevich Krichever and S. P. Novikov, Algebras of Virasoro type, Riemann surfaces and the structures of soliton theory. Funktsional. Anal. i Prilozhen., 21(2):46–63, 1987.Google Scholar
  31. KN89.
    Igor Moiseevich Krichever and S. P. Novikov, Algebras of Virasoro type, the energy-momentum tensor, and operator expansions on Riemann surfaces. Funktsional. Anal. i Prilozhen., 23(1):24–40, 1989.Google Scholar
  32. MN99.
    Atsushi Matsuo and Kiyokazu Nagatomo, Axioms for a vertex algebra and the locality of quantum fields, Volume 4, MSJ Memoirs. Mathematical Society of Japan, Tokyo, 1999.Google Scholar
  33. Sch03a.
    Martin Schlichenmaier, Higher genus affine algebras of Krichever-Novikov type. Mosc. Math. J., 3(4):1395–1427, 2003.Google Scholar
  34. Sch03b.
    Martin Schlichenmaier, Local cocycles and central extensions for multipoint algebras of Krichever-Novikov type. J. Reine Angew. Math., 559:53–94, 2003.Google Scholar
  35. She03.
    O. K. Sheinman, Second-order Casimirs for the affine Krichever-Novikov algebras \(\widehat{\mathfrak{g}\mathfrak{l}}_{g,2}\) and \(\widehat{\mathfrak{s}\mathfrak{l}}_{g,2}\). In Fundamental mathematics today (Russian), pp 372–404. Nezavis. Mosk. Univ., Moscow, 2003.Google Scholar
  36. She05.
    O. K. Sheinman, Highest-weight representations of Krichever-Novikov algebras and integrable systems. Uspekhi Mat. Nauk, 60(2(362)):177–178, 2005.Google Scholar
  37. SS98.
    M. Schlichenmaier and O. K. Scheinman, The Sugawara construction and Casimir operators for Krichever-Novikov algebras. J. Math. Sci. (New York), 92(2):3807–3834, 1998. Complex analysis and representation theory, 1.Google Scholar
  38. SS99.
    M. Shlikhenmaier and O. K. Sheinman, The Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras. Uspekhi Mat. Nauk, 54(1(325)):213–250, 1999.Google Scholar
  39. SV90.
    V. V. Schechtman and A. N. Varchenko, Hypergeometric solutions of Knizhnik-Zamolodchikov equations. Lett. Math. Phys., 20(4):279–283, 1990.MathSciNetCrossRefzbMATHGoogle Scholar
  40. Wak86.
    Minoru Wakimoto, Fock representations of the affine Lie algebra \(A_{1}^{(1)}\). Comm. Math. Phys., 104(4):605–609, 1986.Google Scholar
  41. Wim87.
    Jet Wimp, Explicit formulas for the associated Jacobi polynomials and some applications. Canad. J. Math., 39(4):983–1000, 1987.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Ben Cox
    • 1
    Email author
  • Vyacheslav Futorny
    • 2
  • Renato Alessandro Martins
    • 2
  1. 1.Department of MathematicsThe College of CharlestonCharlestonUSA
  2. 2.Departamento de MatemáticaInstituto de Matemática e EstatísticaSão PauloBrasil

Personalised recommendations