Letters in Mathematical Physics

, Volume 62, Issue 2, pp 143–158 | Cite as

Traces of Intertwiners for Quantum Groups and Difference Equations

  • P. Etingof
  • O. Schiffmann
  • A. Varchenko

Abstract

In this Letter we study twisted traces of products of intertwining operators for quantum affine algebras. They are interesting special functions, depending on two weights λ,μ, three scalar parameters q,ω,k, and spectral parameters z1,...,zN, which may be regarded as q-analogs of conformal blocks of the Wess–Zumino–Witten model on an elliptic curve. It is expected that in the rank 1 case they essentially coincide with the elliptic hypergeometric functions defined by Felder and Varchenko. Our main result is that after a suitable renormalization the traces satisfy four systems of difference equations – the Macdonald–Ruijsenaars equation, the q-Knizhnik–Zamolodchikov–Bernard equation, and their dual versions. We also show that in the case when the twisting automorphism is trivial, the trace functions are symmetric under the permutation λ ↔ μ, k ↔ ω. Thus, our results generalize those of Etingof and Schiffmann, dealing with the case q=1, and Etingof, Varchenko, and Schiffmann, dealing with the finite-dimensional case.

difference equation intertwining operators quantum group 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [ABRR]
    Arnaudon, D., Buffenoir, E., Ragoucy, E. and Roche, Ph.: Universal solutions of quantum dynamical Yang-Baxter equations, Lett. Math. Phys. 44(3) (1998), 201-214.Google Scholar
  2. [Be]
    Bernard, D.: On the Wess-Zumino-Witten models on the torus, Nuclear Phys. B 303 (1988), 77-93.Google Scholar
  3. [CP]
    Chari, V. and Pressley, A.: A Guide to Quantum Groups, Cambridge Univ. Press, 1994.Google Scholar
  4. [D]
    Drinfeld, V. G.: On almost cocommutative Hopf algebras, Leningrad Math. J. 1(2) (1990), 321-342.Google Scholar
  5. [E1]
    Etingof, P.: Representations of affine Lie algebras, elliptic r-matrix systems, and special functions, Comm. Math. Phys. 159(3) (1994), 471-502.Google Scholar
  6. [E2]
    Etingof, P.: Difference equations with elliptic coefficients and quantum affine algebras, Preprint hep-th/9312057 (1993).Google Scholar
  7. [E3]
    Etingof, P.: Central elements for quantum affine algebras and affine Macdonald's operators, Math. Res. Lett. 2(5) (1995), 611-628.Google Scholar
  8. [EFK]
    Etingof, P., Frenkel, I. and Kirillov, A. Jr.: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Math. Surveys Monogr. 58, Amer. Math. Soc., Providence, RI, 1998.Google Scholar
  9. [EK]
    Etingof, P. and Kirillov, A. Jr.: Macdonald's polynomials and representations of quantum groups, Math. Res. Lett. 1(3) (1994), 279-296.Google Scholar
  10. [EK1]
    Etingof, P. and Kirillov, A. Jr.: On the affine analogue of Jack and Macdonald polynomials, Duke Math. J. 78(2) (1995), 229-256.Google Scholar
  11. [EM]
    Etingof, P. and de Moura, A.: On the quantum Kazhdan-Lusztig functor, math.QA 0203003.Google Scholar
  12. [ESS]
    Etingof, P., Schedler, T. and Schiffmann, O.: Explicit quantization of dynamical r-matrices for finite-dimensional simple Lie algebras, J. Amer. Math. Soc. 13 (2000), 595-609.Google Scholar
  13. [ES1]
    Etingof, P. and Schiffmann, O.: Twisted traces of intertwiners for Kac-Moody algebras and classical dynamical r-matrices corresponding to generalized Belavin-Drinfeld triples, Math. Res. Lett 6 (1999), 593-612.Google Scholar
  14. [ES2]
    Etingof, P. and Schiffmann, O.: Twisted traces of quantum intertwiners and quantum dynamical R-matrices corresponding to generalized Belavin-Drinfeld triples, to appear in Comm. Math. Phys. Google Scholar
  15. [EV]
    Etingof, P. and Varchenko, A.: Traces of intertwiners for quantum groups and difference equations, I, Duke Math. J. 104(3) (2000), 391-432.Google Scholar
  16. [FTV]
    Felder, G., Tarasov, V. and Varchenko, A.: Monodromy of solutions of the elliptic quantum Knizhnik-Zamolodchikov-Bernard difference equations, q-alg/9705017.Google Scholar
  17. [FR]
    Frenkel, I. and Reshetikhin, N.: Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146(1) (1992), 1-60.Google Scholar
  18. [FV1]
    Felder, G. and Varchenko, A.: The q-deformed Knizhnik-Zamolodchikov-Bernard heat equation, Comm. Math. Phys. 221(3) (2001), 549-571.Google Scholar
  19. [FV2]
    Felder, G. and Varchenko, A.: q-deformed KZB heat equation: completeness, modular properties and SL(3,Z), math.QA/0110081.Google Scholar
  20. [JKOS]
    Jimbo, M., Odake, S., Konno, H. and Shiraishi, J.: Quasi-Hopf twistors for elliptic quantum groups, Transform. Groups 4(4) (1999), 303-327.Google Scholar
  21. [JM]
    Jimbo, M. and Miwa, T.: Algebraic Analysis of Solvable Lattice Models, CBMS Regional Conf. Ser. Math. 85, Amer. Math. Soc., Providence, RI, 1995.Google Scholar
  22. [JMN]
    Jimbo, M., Miwa, T. and Nakayashiki, A.: Difference equations for the correlation functions of the eight-vertex model, J. Phys. A 26(9) (1993), 2199-2209.Google Scholar
  23. [KS]
    Kazhdan, D. and Soibelman, Y.: Representations of quantum affine algebras, Selecta Math. (NS) 1(3) (1995), 537-595.Google Scholar
  24. [Ko1]
    Konno, H.: Modern Phys. Lett. A 9 (1994), 1253-1266.Google Scholar
  25. [Ko2]
    Konno, H.: Nuclear Phys. B 432 (1994), 457-486.Google Scholar
  26. [Mo]
    Moura, A.: Elliptic dynamical R-matrices from the monodromy of the q-Knizhnik-Zamolodchikov equations for the standard representation of Uq(s1(n+1)), math.RT/ 0112145.Google Scholar
  27. [T]
    Takhtajan, L. A.: Solutions of the triangle equations with Zn x Zn-symmetry and matrix analogues of the Weierstrass zeta and sigma functions. Differential geometry, Lie groups and mechanics, VI, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 133 (1984), 258-276.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • P. Etingof
    • 1
  • O. Schiffmann
    • 2
  • A. Varchenko
    • 3
  1. 1.MIT Mathematics DepartmentCambridgeU.S.A.
  2. 2.Yale Mathematics DepartmentNew HavenU.S.A.
  3. 3.Department of MathematicsUniversity of North CarolinaChapel HillU.S.A.

Personalised recommendations