Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 064, 45 pages      arXiv:1204.2746

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

Jean Avan a, Baptiste Billaud b and Geneviève Rollet a
a) Laboratoire de Physique Théorique et Modélisation, Université de Cergy-Pontoise (CNRS UMR 8089), Saint-Martin 2, 2, av. Adolphe Chauvin, F-95302 Cergy-Pontoise Cedex, France
b) Laboratoire de Mathématiques ''Analyse, Géometrie Modélisation'', Université de Cergy-Pontoise (CNRS UMR 8088), Saint-Martin 2, 2, av. Adolphe Chauvin, F-95302 Cergy-Pontoise Cedex, France

Received April 24, 2012, in final form September 19, 2012; Published online September 28, 2012

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 $\{{\mathbb I}(i),\; i\in\{1,\dots,n\}\}$ 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, $\Lambda_{{\mathbb I}(i)}=\sum\limits_{j\in{\mathbb I}(i)}\lambda_j$, and $(\lambda_i)_{i\in\{1,\dots,n\}}$ denotes the family of dynamical coordinates.

Key words: quantum integrable systems; dynamical Yang-Baxter equation; (weak) Hecke algebras

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