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


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 


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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.

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