Dynamical R matrices of elliptic quantum groups and connection matrices for the q-KZ equations

Abstract

For any affine Lie algebra g, we show that any finite dimensional representation of the universal dynamical R matrix R(λ) of the elliptic quantum group Bq,λ(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,λ(g) and the elliptic face (IRF or SOS) models. In particular, we construct vector representations of R(λ) for g = A(1)n, B(1)n, C(1)n, D(1)n, and show that they coincide with the face weights derived by Jimbo, Miwa and Okado. We hence confirm the conjecture by Frenkel and Reshetikhin.