Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 091, 25 pages      math.QA/0612558      https://doi.org/10.3842/SIGMA.2006.091
Contribution to the Proceedings of the O'Raifeartaigh Symposium

Dynamical R Matrices of Elliptic Quantum Groups and Connection Matrices for the q-KZ Equations

Hitoshi Konno
Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8521, Japan

Received October 02, 2006, in final form November 28, 2006; Published online December 19, 2006

Abstract
For any affine Lie algebra g, we show that any finite dimensional representation of the universal dynamical R matrix R(l) of the elliptic quantum group Bq,l(g) coincides with a corresponding connection matrix for the solutions of the q-KZ equation associated with Uq(g). This provides a general connection between Bq,l(g) and the elliptic face (IRF or SOS) models. In particular, we construct vector representations of R(l) for g = An(1), Bn(1), Cn(1), Dn(1), and show that they coincide with the face weights derived by Jimbo, Miwa and Okado. We hence confirm the conjecture by Frenkel and Reshetikhin.

Key words: elliptic quantum group; quasi-Hopf algebra.

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References

  1. Jimbo M., A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys., 1985, V.10, 63-69.
  2. Jimbo M., Quantum R matrix for the generalized Toda system, Comm. Math. Phys., 1986, V.102, 537-547.
  3. Jimbo M., Miwa T., Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in Mathematics, Vol. 85, Amer. Math. Soc., 1995.
  4. Konno H., An elliptic algebra Uq,p(sl2) and the fusion RSOS models, Comm. Math. Phys., 1998, V.195, 373-403, q-alg/9709013.
  5. Jimbo M., Konno H., Odake S., Shiraishi J., Elliptic algebra Uq,p(sl2): Drinfel'd currents and vertex operators, Comm. Math. Phys., 1999, V.199, 605-647, math.QA/9802002.
  6. Kojima T., Konno H., The elliptic algebra Uq,p(slN) and the Drinfel'd realization of the elliptic quantum group Bq,l(slN), Comm. Math. Phys., 2003, V.239, 405-447, math.QA/0210383.
  7. Kojima T., Konno H., The Drinfel'd realization of the elliptic quantum group Bq,l(A2(2)), J. Math. Phys., 2004, V.45, 3146-3179, math.QA/0401055.
  8. Kojima T., Konno H., Weston R., The vertex-face correspondence and correlation functions of the fusion eight-vertex models. I. The general formalism, Nuclear Phys. B, 2005, V.720, 348-398, math.QA/0504433.
  9. Baxter R.J., Partition function of the eight-vertex lattice model, Ann. Phys., 1972, V.70, 193-228.
  10. Belavin A., Dynamical symmetry of integrable quantum systems, Nuclear Phys. B, 1981, V.180, 189-200.
  11. Andrews G.E., Baxter R.J., Forrester P.J., Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Stat. Phys., 1984, V.35, 193-266.
  12. Jimbo M., Miwa T., Okado M., Solvable lattice models whose states are dominant integral weights of A(1)n-1, Lett. Math. Phys., 1987, V.14, 123-131.
  13. Jimbo M., Miwa T., Okado M., Solvable lattice models related to the vector representation of classical simple Lie algebras, Comm. Math. Phys., 1988, V.116, 507-525.
  14. Kuniba A., Exact solution of solid-on-solid models for twisted affine Lie algebras A(2)2n and A(2)2n-1, Nuclear Phys. B, 1991, V.355, 801-821.
  15. Kuniba A., Suzuki J., Exactly solvable G(1)2 solid-on-solid models, Phys. Lett. A, 1991, V.160, 216-222.
  16. Frenkel I.B., Reshetikhin N.Yu., Quantum affine algebras and holonomic difference equations, Comm. Math. Phys., 1992, V.146, 1-60.
  17. Date E., Jimbo M., Okado M., Crystal base and q-vertex operators, Comm. Math. Phys., 1993, V.155, 47-69.
  18. Sklyanin E.K., Some algebraic structures connected with the Yang-Baxter equation, Funct. Anal. Appl., 1982, V.16, 27-34.
  19. Foda O., Iohara K., Jimbo M., Kedem R., Miwa T., Yan H., An elliptic quantum algebra for sl2, Lett. Math. Phys., 1994, V.32, 259-268, hep-th/9403094.
  20. Felder G., Elliptic quantum groups, in Proceedings XIth International Congress of Mathematical Physics (1994, Paris), Cambridge, Int. Press, 1995, 211-218, hep-th/9412207.
  21. Frønsdal C., Quasi-Hopf deformations of quantum groups, Lett. Math. Phys., 1997, V.40, 117-134, q-alg/9611028.
  22. Enriquez B., Felder G., Elliptic quantum groups Et,h(sl2) and quasi-Hopf algebras, Comm. Math. Phys., 1998, V.195, 651-689, q-alg/9703018.
  23. Jimbo M., Konno H., Odake S., Shiraishi J., Quasi-Hopf twistors for elliptic quantum groups, Transformation Groups, 1999, V.4, 303-327, q-alg/9712029.
  24. Drinfel'd V.G., Quasi-Hopf algebras, Leningrad Math. J., 1990, V.1, 1419-1457.
  25. Babelon O., Bernard D., Billey E., A quasi-Hopf algebra interpretation of quantum 3j- and 6j-symbols and difference equations, Phys. Lett. B, 1996, V.375, 89-97, q-alg/9511019.
  26. Felder G., Varchenko A., On representations of the elliptic quantum groups Et,h(sl2), Comm. Math. Phys., 1996, V.181, 741-761, q-alg/9601003.
  27. Etingof P., Varchenko A., Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups, Comm. Math. Phys., 1998, V.196, 591-640, q-alg/9708015.
  28. Etingof P., Varchenko A., Exchange dynamical quantum groups, Comm. Math. Phys., 1999, V.205, 19-52, math.QA/9801135.
  29. Koelink E., van Norden Y., Rosengren H., Elliptic U(2) quantum group and elliptic hypergeometric series, Comm. Math. Phys., 2004, V.245, 519-537, math.QA/0304189.
  30. Kac V.G., Infinite dimensional Lie algebras, 3rd ed., Cambridge University Press, 1990.
  31. Tanisaki T., Killing forms, Harish-Chandra isomorphysims, and universal R matrices for quantum algebras, Internat. J. Modern Phys. A, 1991, V.7, 941-961.
  32. Arnaudon D., Buffenoir E., Ragoucy E., Roche P., Universal solutions of quantum dynamical Yang-Baxter equations, Lett. Math. Phys., 1998, V.44, 201-214, q-alg/9712037.
  33. Idzumi M., Iohara K., Jimbo M., Miwa T., Nakashima T., Tokihiro T., Quantum affine symmetry in vertex models, Internat. J. Modern Phys. A, 1993, V.8, 1479-1511, hep-th/9208066.
  34. Bourbaki N., Groupes et Algebres de Lie, Chaps. 4-6, Paris, Hermann, 1968.
  35. Date E., Okado M., Calculation of excited spectra of the spin model related with the vector representation of the quantized affine algebra of type An(1), Internat. J. Modern Phys. A, 1994, V.9, 399-417.
  36. Davies B., Okado M., Excitation spectra of spin models constructed from quantized affine algebras of types Bn(1) and Dn(1), Internat. J. Modern Phys. A, 1996, V.11, 1975-2018, hep-th/9506201.
  37. Jing N., Misra K.C., Okado M., q-wedge modules for quantized enveloping algebras of classical type, J. Algebra, 2000, V.230, 518-539, math.QA/9811013.
  38. Drinfel'd V.G., On almost co-commutative Hopf algebras, Leningrad Math. J., 1990, V.1, 231-431.

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