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

Abstract

For any affine Lie algebra ${\mathfrak g}$, we show that any finite dimensional representation of the universal dynamical $R$ matrix ${\cal R}(λ)$ of the elliptic quantum group ${\cal B}_{q,λ}({\mathfrak g})$ coincides with a corresponding connection matrix for the solutions of the $q$-KZ equation associated with $U_q({\mathfrak g})$. This provides a general connection between ${\cal B}_{q,λ}({\mathfrak g})$ and the elliptic face (IRF or SOS) models. In particular, we construct vector representations of ${\cal R}(λ)$ for ${\mathfrak g}=A_n^{(1)}$, $B_n^{(1)}$, $C_n^{(1)}$, $D_n^{(1)}$, and show that they coincide with the face weights derived by Jimbo, Miwa and Okado. We hence confirm the conjecture by Frenkel and Reshetikhin.