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

SIGMA 3 (2007), 004, 10 pages      math.QA/0612086
Contribution to the Proceedings of the Coimbra Workshop on Geometric Aspects of Integrable Systems

Construction of the Bethe State for the Eτ,η(so3) Elliptic Quantum Group

Nenad Manojlović and Zoltán Nagy
Departamento de Matemática, FCT, Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugal

Received October 31, 2006, in final form December 28, 2006; Published online January 05, 2007

Elliptic quantum groups can be associated to solutions of the star-triangle relation of statistical mechanics. In this paper, we consider the particular case of the Eτ,η(so3) elliptic quantum group. In the context of algebraic Bethe ansatz, we construct the corresponding Bethe creation operator for the transfer matrix defined in an arbitrary representation of Eτ,η(so3).

Key words: elliptic quantum group; algebraic Bethe ansatz.

pdf (215 kb)   ps (169 kb)   tex (13 kb)


  1. Avan J., Babelon O., Billey E., The Gervais-Neveu-Felder equation and the quantum Calogero-Moser system, Comm. Math. Phys. 178 (1996), 281-300.
  2. Billey E., Algebraic nested Bethe ansatz for the elliptic Ruijsenaars model, math.QA/9806068.
  3. Faddeev L.D., How algebraic Bethe ansatz works for integrable models, in Quantum Symmetries / Symétries quantiques, Proceedings of the "Les Houches Summer School, Session LXIV" (1 August - 8 September, 1995, Les Houches, France), Editors A. Connes, K. Gawedzki and J. Zinn-Justin, North-Holland, Amsterdam, 1998, 149-219, hep-th/9605187.
  4. Felder G., Conformal field theory and integrable systems associated to elliptic curves, hep-th/9407154.
  5. Felder G., Varchenko A., Algebraic Bethe ansatz for the elliptic quantum group Eτ,η(sl2), Nuclear Phys. B 480 (1996), 485-503, q-alg/9605024.
  6. Felder G., Varchenko A., Elliptic quantum groups and Ruijsenaars models, J. Statist. Phys. 89 (1997), 963-980, q-alg/9704005.
  7. Felder G., Varchenko A., On representations of the elliptic quantum group Eτ,η(sl2), Comm. Math. Phys. 181 (1996), 741-761, q-alg/9601003.
  8. Hou B.Y., Sasaki R., Yang W.-L., Algebraic Bethe ansatz for the elliptic quantum group Eτ,η(sln) and its applications, Nuclear Phys. B 663 (2003), 467-486, hep-th/0303077.
  9. Jimbo M., Miwa T., Okado M., Solvable lattice models related to the vector representation of classical simple Lie algebras, Comm. Math. Phys. 116 (1988), 507-525.
  10. Korepin V.E., Boguliubov N.M., Izergin A.G., Quantum inverse scattering method and correlation functions, Cambridge Monograph on Mathematical Physics, Cambridge University Press, 1993.
  11. Kulish P.P., Sklyanin E.K., Quantum spectral transform method. Recent developments, Lecture Notes in Phys. 151, Editors J. Hietarinta and C. Montonen, Springer, New York, 1982, 61-119.
  12. Manojlović N., Nagy Z., Algebraic Bethe ansatz for the elliptic quantum group Eτ,η(so3), in preparation.
  13. Tarasov V.O., Algebraic Bethe ansatz for the Izergin-Korepin R-matrix, Theoret. and Math. Phys. 76 (1988), 793-803.

Previous article   Next article   Contents of Volume 3 (2007)