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

TL;DR

This paper establishes a connection between the universal dynamical R matrix of elliptic quantum groups and solutions to the q-KZ equations, confirming a conjecture and constructing explicit vector representations for various affine Lie algebras.

Contribution

It demonstrates that finite dimensional representations of the dynamical R matrix coincide with connection matrices for q-KZ solutions, linking elliptic quantum groups to face models and confirming a conjecture.

Findings

Representation of R matrix matches q-KZ connection matrices

Constructed explicit vector representations for several affine Lie algebras

Confirmed the conjecture by Frenkel and Reshetikhin

Abstract

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.