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SIGMA, 2012, Volume 8, 064, 45 pages (Mi sigma741)  

This article is cited in 1 scientific paper (total in 1 paper)

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

Jean Avana, Baptiste Billaudb, Geneviéve Rolleta

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

Abstract: 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 $\{\mathbb I(i),i\in\{1,…,n\}\}$ 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, $\Lambda_{\mathbb I(i)}=\sum_{j\in\mathbb I(i)}\lambda_j$, and $(\lambda_i)_{i\in\{1,…,n\}}$ denotes the family of dynamical coordinates.

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

DOI: https://doi.org/10.3842/SIGMA.2012.064

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Full text: http://emis.mi.ras.ru/.../064
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ArXiv: 1204.2746
MSC: 16T25; 17B37; 81R12; 81R50
Received: April 24, 2012; in final form September 19, 2012; Published online September 28, 2012
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Citation: Jean Avan, Baptiste Billaud, Geneviéve Rollet, “Classification of non-affine non-Hecke dynamical $R$-matrices”, SIGMA, 8 (2012), 064, 45 pp.

Citation in format AMSBIB
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\by Jean Avan, Baptiste Billaud, Genevi\'eve Rollet
\paper Classification of non-affine non-Hecke dynamical $R$-matrices
\jour SIGMA
\yr 2012
\vol 8
\papernumber 064
\totalpages 45
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\crossref{https://doi.org/10.3842/SIGMA.2012.064}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Jean Avan, Eric Ragoucy, “Rational Calogero–Moser model: explicit form and $r$-matrix of the second Poisson structure”, SIGMA, 8 (2012), 079, 13 pp.  mathnet  crossref  mathscinet
  • Symmetry, Integrability and Geometry: Methods and Applications
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