Classification of non-affine non-Hecke dynamical $R$-matrices

Jean Avan ; Baptiste Billaud ; Geneviéve Rollet

Symmetry, integrability and geometry: methods and applications, Tome 8 (2012)

Résumé

A complete classification of non-affine dynamical quantum $R$-matrices obeying the $\mathcal Gl_n(\mathbb C)$-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 $\{\mathbb I(i),i\in\{1,\dots,n\}\}$ of the set of indices $\{1,\dots,n\}$ into classes, $\mathbb I(i)$ being the class of the index $i$, and an arbitrary family of signs $(\epsilon_\mathbb I)_{\mathbb I\in\{\mathbb I(i),\,i\in\{1,\dots,n\}\}}$ 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, $\Lambda_{\mathbb I(i)}=\sum_{j\in\mathbb I(i)}\lambda_j$, and $(\lambda_i)_{i\in\{1,\dots,n\}}$ denotes the family of dynamical coordinates.

Détail

Keywords: quantum integrable systems; dynamical Yang–Baxter equation; (weak) Hecke algebras.

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