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

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

2006-12-19

arxiv.org/abs/math/0612558v1

SIGMA 2 (2006), 091, 25 pages

Mathematics

Quantum Algebra

This is a contribution to the Proc. of the O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (

Scientific paper

10.3842/SIGMA.2006.091

For any affine Lie algebra ${\mathfrak g}$, we show that any finite dimensional representation of the universal dynamical $R$ matrix ${\cal R}(\lambda)$ of the elliptic quantum group ${\cal B}_{q,\lambda}({\mathfrak g})$ 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 ${\cal B}_{q,\lambda}({\mathfrak g})$ and the elliptic face (IRF or SOS) models. In particular, we construct vector representations of ${\cal R}(\lambda)$ 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.

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