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

Symmetry Integrability and Geometry-Methods and Applications

article

Jean Avan¹  Baptiste Billaud²  Geneviève Rollet¹ 

[1] Laboratoire de Physique Théorique et Modélisation, Université de Cergy-Pontoise (CNRS UMR 8089);Laboratoire de Mathématiques ''Analyse, Université de Cergy-Pontoise (CNRS UMR 8088)

关键词: quantum integrable systems;    dynamical Yang–Baxter equation;    (weak) Hecke algebras;   

DOI  :  10.3842/SIGMA.2012.064

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown