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 z 1,...,z N , 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.
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References
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.
Bernard, D.: On the Wess-Zumino-Witten models on the torus, Nuclear Phys. B 303 (1988), 77-93.
Chari, V. and Pressley, A.: A Guide to Quantum Groups, Cambridge Univ. Press, 1994.
Drinfeld, V. G.: On almost cocommutative Hopf algebras, Leningrad Math. J. 1(2) (1990), 321-342.
Etingof, P.: Representations of affine Lie algebras, elliptic r-matrix systems, and special functions, Comm. Math. Phys. 159(3) (1994), 471-502.
Etingof, P.: Difference equations with elliptic coefficients and quantum affine algebras, Preprint hep-th/9312057 (1993).
Etingof, P.: Central elements for quantum affine algebras and affine Macdonald's operators, Math. Res. Lett. 2(5) (1995), 611-628.
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.
Etingof, P. and Kirillov, A. Jr.: Macdonald's polynomials and representations of quantum groups, Math. Res. Lett. 1(3) (1994), 279-296.
Etingof, P. and Kirillov, A. Jr.: On the affine analogue of Jack and Macdonald polynomials, Duke Math. J. 78(2) (1995), 229-256.
Etingof, P. and de Moura, A.: On the quantum Kazhdan-Lusztig functor, math.QA 0203003.
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.
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.
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.
Etingof, P. and Varchenko, A.: Traces of intertwiners for quantum groups and difference equations, I, Duke Math. J. 104(3) (2000), 391-432.
Felder, G., Tarasov, V. and Varchenko, A.: Monodromy of solutions of the elliptic quantum Knizhnik-Zamolodchikov-Bernard difference equations, q-alg/9705017.
Frenkel, I. and Reshetikhin, N.: Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146(1) (1992), 1-60.
Felder, G. and Varchenko, A.: The q-deformed Knizhnik-Zamolodchikov-Bernard heat equation, Comm. Math. Phys. 221(3) (2001), 549-571.
Felder, G. and Varchenko, A.: q-deformed KZB heat equation: completeness, modular properties and SL(3,Z), math.QA/0110081.
Jimbo, M., Odake, S., Konno, H. and Shiraishi, J.: Quasi-Hopf twistors for elliptic quantum groups, Transform. Groups 4(4) (1999), 303-327.
Jimbo, M. and Miwa, T.: Algebraic Analysis of Solvable Lattice Models, CBMS Regional Conf. Ser. Math. 85, Amer. Math. Soc., Providence, RI, 1995.
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.
Kazhdan, D. and Soibelman, Y.: Representations of quantum affine algebras, Selecta Math. (NS) 1(3) (1995), 537-595.
Konno, H.: Modern Phys. Lett. A 9 (1994), 1253-1266.
Konno, H.: Nuclear Phys. B 432 (1994), 457-486.
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.
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.
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Etingof, P., Schiffmann, O. & Varchenko, A. Traces of Intertwiners for Quantum Groups and Difference Equations. Letters in Mathematical Physics 62, 143–158 (2002). https://doi.org/10.1023/A:1021619920915
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DOI: https://doi.org/10.1023/A:1021619920915