Mathematical Society of Japan Memoirs

Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras

Ivan Cherednik

Full-text: Open access

Abstract

This paper is the course of lectures delivered by the first author in Kyoto in 1996-97 and recorded by the others. We tried to follow closely the notes of the lectures not yielding to the temptation of giving more examples and names. The focus is on the relations of the Knizhnik-Zamolodchikov equations and Kac-Moody algebras to a new theory of spherical and hypergeometric functions based on affine and double affine Hecke algebras. Here mathematics and physics are closer than Siamese twins. We did not try to separate them, but the course turned out to be mainly about the mathematical issues. However we hope that the paper will be understandable for both physicists and mathematicians, for those who want to master the new Hecke algebra technique.

Chapter information

Source
Ivan Cherednik, Peter J. Forrester and Denis Uglov, Quantum Many-Body Problems and Representation Theory (Tokyo: The Mathematical Society of Japan, 1998), 1-96

Dates
First available in Project Euclid: 17 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.msjm/1389985793

Digital Object Identifier
doi:10.2969/msjmemoirs/00101C010

Mathematical Reviews number (MathSciNet)
MR1724948

Zentralblatt MATH identifier
1117.33300

Rights
Copyright © 1998, The Mathematical Society of Japan

Citation

Cherednik, Ivan. Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras. Quantum Many-Body Problems and Representation Theory, 1--96, The Mathematical Society of Japan, Tokyo, Japan, 1998. doi:10.2969/msjmemoirs/00101C010. https://projecteuclid.org/euclid.msjm/1389985793


Export citation

References

  • 1. K. Aomoto, A note on holonomic $q$ -difference systems, Algebraic Analysis,1, Eds. M. Kashiwara, T. Kawai, Academic Press, San Diego (1988), 22-28.
  • 2. R. Askey and M.E.H. Ismail, A generalization of ultraspherical polynomials, in Studies in Pure Mathematics (ed. P. Erdös), Birkhäuser (1983), 55-78.
  • 3. H.M. Babujan and R. Flume, Off shell Bethe ansatz equation for Gaudin magnets and solutions of Knizhnik-Zamolodchikov equations, Preprint Bonn-HE-93-30.
  • 4. A.A. Belavin and V.G. Drinfeld, Solutions of the classical Yang-Baxter equations for simple Lie algebras, Funct. Anal. Appl. 16 (1982), 159-180.
  • 5. D. Bernard, On the Wess-Zumino-Witten model on the torus, Nucl. Phys. B303 (1988), 77-93.
  • 6. J. Birman, On Braid groups, Commun. Pure Appl. Math., 22 (1969), 41-72.
  • 7. I. Cherednik, A definition of $\tau$ -functions for generalized affine Lie algebras, Funct. Anal. Appl. 17:3 (1983), 93-95.
  • 8. I. Cherednik, Functional realizations of basic representations of factorizable groups and Lie algebras Funct. Anal. Appl. 19:3 (1985), 36-52.
  • 9. I. Cherednik, Factorized particles on a half-line and root systems, Theor. Math. Phys. 61:1 (1984), 35-43
  • 10. I. Cherednik, Generalized braid groups and local $r$ -matrix systems, Doklady Akad. Nauk SSSR 307:1 (1989), 27-34.
  • 11. I. Cherednik, Monodromy representations for generalized Knizhnik-Zamolodchikov equations and Hecke algebras, Publ. RIMS 27 (1991), 711-726.
  • 12. I. Cherednik, Affine extensions of Knizhnik-Zamolodchikov equations and Lusztig's isomorphisms, in: `Special Functions', ICM-90 Satellite Conference Proceedings, Eds. M. Kashiwara and T. Miwa, Springer (1991), 63-77.
  • 13. I. Cherednik, Integral solutions of trigonometric Knizhnik-Zamolodchikov equations and Kac-Moody algebras, Publ. RIMS 27 (1991), 727-744.
  • 14. I. Cherednik, A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Invent. Math. 106 (1991), 411-431.
  • 15. I. Cherednik, Quantum Knizhnik-Zamolodchikov equations and affine root systems, Commun. Math. Phys. 150 (1992), 109-136.
  • 16. I. Cherednik, Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald's operators, Int. Math. Res. Notices 6 (1992), 171-179.
  • 17. I. Cherednik, The Macdonald constant-term conjecture, Int. Math. Res. Notices 6 (1993), 165-177.
  • 18. I. Cherednik, Induced representations of double affine Hecke algebras and applications, Math. Res. Lett. 1 (1994), 319-337.
  • 19. I. Cherednik, Integration of quantum many-body problems by an affine Knizhnik-Zamolodchikov equations, Preprint RIMS-776 (1991), Adv. Math. 106 (1994), 65-95.
  • 20. I. Cherednik, Macdonald's evaluation conjectures and difference Fourier transform, Invent. Math. 122 (1995), 119-145.
  • 21. I. Cherednik, Nonsymmetric Macdonald polynomials, Int. Math. Res. Notices 10 (1995), 483-515.
  • 22. I. Cherednik, Elliptic quantum many-body problem and double affine Knizhnik - Zamolodchikov equation, Commun. Math. Phys. 169:2 (1995), 441-461.
  • 23. I. Cherednik, Difference-elliptic operators and root systems, IMRN 1 (1995), 43-59.
  • 24. I. Cherednik, Intertwining operators for double affine Hecke algebras, Preprint RIMS-1079 (1996), Selecta Math.
  • 25. E. Date and M. Jimbo and M. Kashiwara and T. Miwa, Operator approach to the Kadomtsev-Petviashvili equation. Transformation groups for soliton equations III, J. Phys. Soc. Japan 50 (1981), 3806-3812.
  • 26. E. Date and M. Jimbo and A. Matsuo and T. Miwa, Hypergeometric-type integrals and the $SL_{2}(\mathbb{C})$ Knozhnik-Zamolodchikov equation, Preprint RIMS 667 (1989), Kyoto.
  • 27. V. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl. 21:1 (1986), 69-70.
  • 28. V. Drinfeld, Quantum groups, Proc.ICM-86 (Berkeley) 1. Amer. Math. Soc. (1987), 798-820.
  • 29. C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. AMS. 311 (1989) 167-183.
  • 30. P. Etingof and A. Kirillov Jr., Representations of affine Lie algebras, parabolic equations, and Lamé functions, Duke Math. J., (1993).
  • 31. P. Etingof and A. Kirillov Jr., Representation-theoretic proof of the inner product and symmetry identities for Macdonald's polynomials, Compositio Mathematica (1995).
  • 32. B. Feigin and E. Frenkel and N. Reshetikhin, Gaudin model, Bethe ansatz and correlation functions at critical level, Comm. Math. Phys 166 (1994), 27-62.
  • 33. G. Felder, Conformal field theory and integrable systems associated to elliptic curves, Proceed. of the International Congress of Mathematicians, Zürich (1994).
  • 34. G. Felder and A. Varchenko, Integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equation, IMRN 5 (1995), 221-233.
  • 35. G. Felder, C. Wieczerkowski, The Knizhnik - Zamolodchikov - Bernard equation on the torus, Proceedings of the Vancower Summer School on Mathematical Quantum Field Theory (August 1993).
  • 36. G. Felder, C. Wieczerkowski, Topological representations of $U_{q}(sl_{2})$ , Commun. Math. Phys. 138 (1991), 583-605.
  • 37. I.B. Frenkel and N.Yu. Reshetikhin, Quantum affine algebras and holonomic difference equations, Commun. Math. Phys. (1991).
  • 38. G.J. Heckman and E.M. Opdam, Root systems and hypergeometric functions I, Comp. Math. 64 (1987), 329-352.
  • 39. G.J. Heckman, Root systems and hypergeometric functions II, Comp. Math. 64 (1987), 353-373.
  • 40. G.J. Heckman, An elementary approach to the hypergeometric shift operator of Opdam. Invent. math. 103 (1991), 341-350.
  • 41. S. Helgason, Differential geometry, Lie groups and symmetric spaces, Acad. Press (1978).
  • 42. V.G. Kac, Infinite dimensional Lie algebras, Cambridge University Press, Cambridge (1990).
  • 43. V.G. Kac and D. Peterson, Spin and wedge representations of infinite dimensional Lie algebras and groups, Proc. Nat. Acad. Sci. USA 78 (1981), 3308-3312.
  • 44. S. Kato, Irreducibility of principal series representations for Hecke algebras of affine type, J. Fac. Sci. Univ. Tokyo, IA, 28:3 (1983), 929-943.
  • 45. D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), 153-215.
  • 46. D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. III, J. of AMS 7(1994), 335-381.
  • 47. A. Kirillov, Jr., Inner product on conformal blocks and Macdonald's polynomials at roots of unity, Preprint (1995).
  • 48. V.G. Knizhnik and A.B. Zamolodchikov, Current algebra and Wess-Zumino models in two-dimensions, Nuclear Physics. B247, (1984), 83-103.
  • 49. T. Kohno, Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier Grenouble 37 (1987) 139-160.
  • 50. P. Kulish and E. Sklyanin, On solutions of the Yang-Baxeter equation, J. Sov. Math. 19:5 (1982), 1596-1620.
  • 51. H. van der Lek, Extended Artin groups, Proc. Symp. Pure Math. 40:2(1983), 117-122.
  • 52. G. Lusztig, Affine Hecke algebras and their graded version, Jour. Amer. Math. Soc. 2 (1989), 599-635.
  • 53. I.G. Macdonald, Orthogonal polynomials associated with root systems, preprint 1987.
  • 54. I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Séminaire BOURBAKI, 47ème année, 1994-95, n$^{\textrm{o}}$ 797.
  • 55. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Second Edition, Clarendon Press, Oxford, 1995.
  • 56. A. Matsuo, Integrable connections related to zonal spherical functions, Invent. Math. 110 (1992), 96-121.
  • 57. M. Noumi, Macdomald's symmetric functions on some quantum homogeneous spaces, Preprint (1992).
  • 58. H. Ochiai and T. Oshima and H. Sekiguchi, Commuting families of symmetric differential operators, Proc. of the Japan Acad., A70:2 (1994), 62-68.
  • 59. M.A. Olshanetsky and A.M. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404.
  • 60. E.M. Opdam, Some applications of hypergeometric shift operators, Invent. Math. 98 (1989), 1-18.
  • 61. E.M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175(1995), 75-121.
  • 62. J. Rogawski, On modules over the Hecke algebra of a p-adic group, Invent. Math. 79 (1985), 443-465.
  • 63. K. Saito, Extended affine root systems I., Publ. RIMS, Kyoto Univ., 21:1 (1985), 75-179.
  • 64. M. Semenov-Tjan-Shanskii, What is a classical $r$-matrix?, Funct. Anal. Appl., 17:4 (1983), 259-272.
  • 65. V.V. Schechtman and A.A. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Inventiones Math. 106 (1991) 139-194.
  • 66. F. Smirnov, General formula for soliton formfactors in Sine-Gordon Model. J. Phys. A : Math. Gen. 19 (1986), L575-L580.
  • 67. B. Sutherland, Exact results for a quantum many-body problem in one-dimension, Phys. Rev. A4 (1971), 2019-2021, Phys. Rev. A5 (1971), 1372-1376.
  • 68. A. Tsuchiya and Y. Kanie, Vertex operators in conformal field theory on $P^{1}$ and monodromy representations of braid groups, Adv. Stud. Pure. Math. 16 (1988), 297-372.
  • 69. A. Varchenko, Multidimensional hypergeometnc functions and representation theory of quantum groups, Adv. Ser. Math. Phys. 21, World Sci, River Edge, NJ (1995).
  • 70. E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B300(FS22) (1987), 360-376.