SIGMA 8 (2012), 064, 45 pages      arXiv:1204.2746      https://doi.org/10.3842/SIGMA.2012.064

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

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 $\{{\mathbb I}(i),\; i\in\{1,\dots,n\}\}$ 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

pdf (780 kb)   tex (50 kb)

References

1. Avan J., Babelon O., Billey E., The Gervais-Neveu-Felder equation and the quantum Calogero-Moser systems, Comm. Math. Phys. 178 (1996), 281-299, hep-th/9505091.
2. Avan J., Kulish P.P., Rollet G., Reflection $K$-matrices related to Temperley-Lieb $R$-matrices, Theoret. and Math. Phys. 169 (2011), 1530-1538, arXiv:1012.3012.
3. Avan J., Ragoucy E., Rational Calogero-Moser model: explicit forms and $r$-matrix structure of the second Poisson structure, arXiv:1207.5368.
4. Avan J., Talon M., Classical $R$-matrix structure for the Calogero model, Phys. Lett. B 303 (1993), 33-37, hep-th/9210128.
5. Balog J., Dabrowski L., Fehér L., Classical $r$-matrix and exchange algebra in WZNW and Toda theories, Phys. Lett. B 244 (1990), 227-234.
6. Balog J., Fehér L., Palla L., Chiral extensions of the WZNW phase space, Poisson-Lie symmetries and groupoids, Nuclear Phys. B 568 (2000), 503-542, hep-th/9910046.
7. Bernard D., On the Wess-Zumino-Witten models on Riemann surfaces, Nuclear Phys. B 309 (1988), 145-174.
8. Bernard D., On the Wess-Zumino-Witten models on the torus, Nuclear Phys. B 303 (1988), 77-93.
9. Calogero F., Solution of a three-body problem in one dimension, J. Math. Phys. 10 (1969), 2191-2196.
10. Donin J., Mudrov A., Dynamical Yang-Baxter equation and quantum vector bundles, Comm. Math. Phys. 254 (2005), 719-760, math.QA/0306028.
11. Enriquez B., Etingof P., Quantization of classical dynamical $r$-matrices with nonabelian base, math.QA/0311224.
12. Etingof P., On the dynamical Yang-Baxter equation, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 555-570, math.QA/0207008.
13. Etingof P., Schiffmann O., Lectures on the dynamical Yang-Baxter equations, in Quantum Groups and Lie Theory (Durham, 1999), London Math. Soc. Lecture Note Ser., Vol. 290, Cambridge Univ. Press, Cambridge, 2001, 89-129, math.QA/9908064.
14. Etingof P., Varchenko A., Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups, Comm. Math. Phys. 196 (1998), 591-640, math.QA/9801135.
15. Felder G., Conformal field theory and integrable systems associated to elliptic curves, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 1247-1255, hep-th/9407154.
16. Felder G., Elliptic quantum groups, in XIth International Congress of Mathematical Physics (Paris, 1994), Int. Press, Cambridge, MA, 1995, 211-218, hep-th/9412207.
17. Furlan P., Hadjiivanov L.K., Isaev A.P., Ogievetsky O.V., Pyatov P.N., Todorov I.T., Quantum matrix algebra for the ${\rm SU}(n)$ WZNW model, J. Phys. A: Math. Gen. 36 (2003), 5497-5530, hep-th/0003210.
18. Gervais J.L., Neveu A., Novel triangle relation and absence of tachyons in Liouville string field theory, Nuclear Phys. B 238 (1984), 125-141.
19. Hadjiivanov L.K., Stanev Y.S., Todorov I.T., Regular basis and $R$-matrices for the $\widehat{\rm su}(n)_k$ Knizhnik-Zamolodchikov equation, Lett. Math. Phys. 54 (2000), 137-155, hep-th/0007187.
20. Isaev A.P., Twisted Yang-Baxter equations for linear quantum (super)groups, J. Phys. A: Math. Gen. 29 (1996), 6903-6910, q-alg/95110006.
21. Ju G., Luo X., Wang S., Wu K., A free-fermion type solution of quantum dynamical Yang-Baxter equation, Commun. Theor. Phys. 32 (1999), 557-562.
22. Knizhnik V.G., Zamolodchikov A.B., Current algebra and Wess-Zumino model in two dimensions, Nuclear Phys. B 247 (1984), 83-103.
23. Moser J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975), 197-220.
24. Ruijsenaars S.N.M., Schneider H., A new class of integrable systems and its relation to solitons, Ann. Physics 170 (1986), 370-405.
25. Xu P., Quantum dynamical Yang-Baxter equation over a nonabelian base, Comm. Math. Phys. 226 (2002), 475-495, math.QA/0104071.