Classification of Non-Affine Non-Hecke Dynamical R-Matrices

TL;DR

This paper classifies all non-affine dynamical quantum R-matrices satisfying the Gervais-Neveu-Felder equation, revealing their structure based on partitions and signs, without relying on Hecke conditions.

Contribution

It provides a complete classification of non-affine dynamical R-matrices without assuming Hecke conditions, introducing a new parametrization based on partitions and signs.

Findings

Solutions built from elementary blocks satisfying weak Hecke condition

Characterization by partitions and sign families

Analytical form involving trigonometric or rational functions

Abstract

A complete classification of non-affine dynamical quantum $R$-matrices obeying the ${\mathcal G}l_n({\mathbb C})$-Gervais-Neveu-Felder equation is obtained without assuming either Hecke or weak Hecke conditions. More general dynamical dependences are observed. It is shown that any solution is built upon elementary blocks, which individually satisfy the weak Hecke condition. Each solution is in particular characterized by an arbitrary partition ${{\mathbb I}(i), i\in{1,...,n}}$ of the set of indices ${1,...,n}$ into classes, ${\mathbb I}(i)$ being the class of the index $i$, and an arbitrary family of signs $(\epsilon_{\mathbb I})_{{\mathbb I}\in{{\mathbb I}(i), i\in{1,...,n}}}$ on this partition. The weak Hecke-type $R$-matrices exhibit the analytical behaviour $R_{ij,ji}=f(\epsilon_{{\mathbb I}(i)}\Lambda_{{\mathbb I}(i)}-\epsilon_{{\mathbb I}(j)}\Lambda_{{\mathbb I}(j)})$, where $f$ is a particular trigonometric or rational function, $\Lambda_{{\mathbb I}(i)}=\sum\limits_{j\in{\mathbb I}(i)}\lambda_j$, and $(\lambda_i)_{i\in{1,...,n}}$ denotes the family of dynamical coordinates.