• E. Horozov
Conference paper
Part of the NATO Science Series book series (NAII, volume 201)


We introduce a Lie algebra which we call adelic W-algebra. It is a central extension of the Lie algebra of the differential operators on the complex line with rational coefficients. We construct its natural bosonic representation similar to highest weight representation. Then we show that the rank one algebras of bispectral operators are in 1:1 correspondence with the tau-functions in this representation.


Differential Operator Vertex Operator Central Extension Darboux Transformation Weyl Algebra 
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  1. 1.
    Adler, M. and Moser, J. (1978) On a class of polynomials connected with the Korteweg-de Vries equation, Commun. Math. Phys. 61, pp. 1–30.MATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Krichever, I. M. (1978) On rational solutions of Kadomtsev-Petviashvily equation and integrable systems of N particles on the line, Funct. Anal. Appl. 12(1), pp. 76–78 (Russian), 59–61 (English).MATHGoogle Scholar
  3. 3.
    Wilson, G. (1998) Collisions of Calogero-Moser particles and an adelic Grassmannian (with an appendix by I.G. Macdonald), Invent. Math. 133, pp. 1–41.MATHCrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Cannings, R. C. and Holland, M. P. (1994) Right ideals in rings of differential operators, J. Algebra, 167, pp. 116–141.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Berest, Y. and Wilson, G. (2000) Automorphisms and ideals of the Weyl algebra, Math. Ann. 318(1), pp. 127–147.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Wilson, G. (1993) Bispectral commutative ordinary differential operators, J. Reine Angew. Math. 442, pp. 177–204.MATHMathSciNetGoogle Scholar
  7. 7.
    Bakalov, B., Horozov, E., and Yakimov, M. (1998) Highest weight modules over W1+∞, and the bispectral problem, Duke Math. J. 93, pp. 41–72.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Günbaum, F. A. (1982) The limited angle reconstruction problem in computer tomography, in: Processings of Symposium on Applied Mathematics, Vol. 27, AMS, ed. L. Shepp, pp. 43–61.Google Scholar
  9. 9.
    Duistermaat, J. J. and Grünbaum, F. A. (1986) Differential equations in the spectral parameter, Commun. Math. Phys. 103, pp. 177–240.MATHCrossRefADSGoogle Scholar
  10. 10.
    Bakalov, B., Horozov, E., and Yakimov, M. (1996) Tau-functions as highest weight vectors for W1+∞ algebra, J. Phys. A: Math. Gen. 29, pp. 5565–5573, hep-th/9510211.MATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Berest, Y. and Wilson, G. (2001) Ideal classes of the Weyl algebra and noncommutative projective geometry, arXiv.math.AG/0104240.Google Scholar
  12. 12.
    Sato, M. (1981) Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds, RIMS Kokyuroku 439, pp. 30–40.Google Scholar
  13. 13.
    Date, E., Jimbo, M., Kashiwara, M., and Miwa, T. (1983) Transformation groups for solution equations, in: Proceedings of Research Institute for Mathematical Sciences Symposium on Nonlinear Integrable systems—Classical and Quantum Theory (Kyoto 1981), World Scientific, Singapore, eds. M. Jimbo and T. Miwa, pp. 39–111.Google Scholar
  14. 14.
    Segal, G. and Wilson, G. (1985) Loop groups and equations of KdV type, Publ. Math. IHES 61, pp. 5–65.MATHMathSciNetGoogle Scholar
  15. 15.
    Kac, V. G. and Raina, A. (1987) Bombay lectures on highest weight representations of infinite dimensional Lie algebras, Advanced Series in Mathematical Physics 2, World Scientific, Singapore.Google Scholar
  16. 16.
    van Moerbeke, P. (1994) Integrable foundations of string theory (CIMPA-Summer school at Sophia-Antipolis 1991), in: Lectures on Integrable Systems, eds. O. Babelon et al., World Scientific, Singapore.Google Scholar
  17. 17.
    Bakalov, B., Horozov, E., and Yakimov, M. (1996) Bäcklund-Darboux transformations in Sato’s Grassmannian, Serdica Math. J. 4, q-alg/9602010.Google Scholar
  18. 18.
    Bakalov, B., Horozov, E., and Yakimov, M. (1997) Bispectral algebras of commuting ordinary differential operators, Comm. Mat. Phys. 190, pp. 331–373, q-alg/9602011.MATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Horozov, E. (2002) Dual algebras of differential operators, in: Kowalevski property (Montréal), CRM Proceedings Lecture Notes 32, Surveys from Kowalevski Workshop on Mathematical methods of Regular Dynamics, Leeds, April 2000, American Mathematical Society, Providence, pp. 121–148.Google Scholar
  20. 20.
    Kac, V. G. and Radul, A. (1993) Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Commun. Math. Phys. 157, pp. 429–457, hep-th/9308153.MATHCrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Fuchsteiner, B. (1983) Master-symmetries, higher order time-dependent symmetries and conserved densities of nonlinear evolution equations, Progr. Theor. Phys. 70(6), pp. 1508–1522.CrossRefADSGoogle Scholar
  22. 22.
    Orlov, A. Y. and Schulman, E. I. (1989) Additional symmetries for integrable and conformal algebra representation, Lett. Math. Phys. 12, pp. 171–179.CrossRefMathSciNetADSGoogle Scholar
  23. 23.
    Adler, M., Shiota, T., and van Moerbeke, P. (1995) A Lax representation for the vertex operator and the central extension, Commun. Math. Phys. 171, pp. 547–588.MATHCrossRefADSGoogle Scholar
  24. 24.
    Dickey, L. (1991) Soliton Equations and Integrable Systems. World Scientific, Singapore.Google Scholar
  25. 25.
    Kac, V. G. and Peterson, D. H. (1981) Spin and wedge representations of infinitedimensional Lie algebras and groups, Proc. Natl. Acad. Sci. USA. 78, pp. 3308–3312.MATHCrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Kasman, A. (1995) Bispectral KP solutions and linearization of Calogero-Moser particle systems, Commun. Math. Phys. 172, pp. 427–448.MATHCrossRefADSMathSciNetGoogle Scholar

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© Springer 2006

Authors and Affiliations

  • E. Horozov
    • 1
  1. 1.Institute of Mathematics and InformaticsSofiaBulgaria

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