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


SIGMA 2 (2006), 091, 25 pages      math.QA/0612558      http://dx.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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