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 $\mathcal{R}(\lambda )$ of the elliptic quantum group ${\mathcal{B}}_{q,\lambda }(\mathfrak{g})$ 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 ${\mathcal{B}}_{q,\lambda }(\mathfrak{g})$ and the elliptic face (IRF or SOS) models. In particular, we construct vector representations of $\mathcal{R}(\lambda )$ 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.

This is a contribution to the Proc. of the O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 2006, Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/