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

SIGMA 5 (2009), 110, 22 pages      arXiv:0908.4064      https://doi.org/10.3842/SIGMA.2009.110
Contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions”

Manin Matrices, Quantum Elliptic Commutative Families and Characteristic Polynomial of Elliptic Gaudin Model

Vladimir Rubtsov a, c, Alexey Silantyev b and Dmitri Talalaev c
a) LAREMA, Université d'Angers, 2 Boulevard Lavoisier, 49045 Angers, France
b) Department of Mathematics, University Gardens, University of Glasgow, G12 8QW, UK
c) ITEP, B. Cheremushkinskaja 25, 117218 Moscow, Russia

Received March 30, 2009, in final form December 12, 2009; Published online December 24, 2009

Abstract
In this paper we construct the quantum spectral curve for the quantum dynamical elliptic gln Gaudin model. We realize it considering a commutative family corresponding to the Felder's elliptic quantum group Eτ,h(gln) and taking the appropriate limit. The approach of Manin matrices here suits well to the problem of constructing the generation function of commuting elements which plays an important role in SoV and Langlands concept.

Key words: Manin matrices; L-operators; elliptic Felder R-matrix; Gaudin models.

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References

  1. Kulish P., Sklyanin E., Quantum spectral transform method. Recent developments, Lecture Notes in Phys., Vol. 151, Springer, Berlin - New York, 1982, 61-119.
  2. Kirillov A., Reshetikhin N., The Yangians, Bethe ansatz and combinatorics, Lett. Math. Phys. 12 (1986), 199-208.
  3. 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.
  4. Felder G., Elliptic quantum groups, in Proceedings of the XIth International Congress of Mathematical Physics (Paris, 1994), Int. Press, Cambridge, MA, 1995, 211-218, hep-th/9412207.
  5. Chervov A., Falqui G., Manin matrices and Talalaev's formula, J. Phys. A: Math. Theor. 41 (2008), 194006, 28 pages, arXiv:0711.2236.
  6. Chervov A., Falqui G., Rubtsov V., Algebraic properties of Manin matrices. I, Adv. in Appl. Math. 43 (2009), 239-315, arXiv:0901.0235.
  7. Felder G., Varchenko A., Elliptic quantum groups and Ruijsenaars models, J. Statist. Phys. 89 (1997), 963-980, q-alg/9704005.
  8. Zhao S.-Y., Shi K.-J., Yue R.-H., The center for the elliptic quantum group Eτ,η(sln), Commun. Theor. Phys. 39 (2003), 67-72, math.QA/0412100.
  9. Nazarov M., Olshanski G., Bethe subalgebras in twisted Yangians, Comm. Math. Phys. 178 (1996), 483-506, q-alg/9507003.
  10. Molev A.I., Yangians and their applications, in Handbook of Algebra, Vol. 3, Editor M. Hazewinkel, North-Holland, Amsterdam, 2003, 907-959, math.QA/0211288.
  11. 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.
  12. Talalaev D., Universal R-matrix formalism for the spin Calogero-Moser system and its difference counterpart, Internat. Math. Res. Notices 2000 (2000), no. 11, 597-606, math.QA/9909014.
  13. Talalaev D., Quantization of the Gaudin system, Funct. Anal. Appl. 40 (2006), no. 1, 73-77, hep-th/0404153.
  14. Tarasov V., Varchenko A., Small elliptic quantum group eτ,γ(slN), Mosc. Math. J. 1 (2001), 243-286, math.QA/0011145.
  15. Chervov A., Falqui G., Rubtsov V., Silantyev A., Algebraic properties of Manin matrices II. q-analogues and integrable systems, to appear.
  16. Kojima T., Konno H., The elliptic algebra Uq,p(^slN) and the Drinfel'd realization of the elliptic quantum group Bq,λ(^slN), Comm. Math. Phys. 239 (2003), 405-447, math.QA/0210383.
  17. Konno H., Elliptic quantum group Uq,p(^sl2), Hopf algebroid structure and elliptic hypergeometric series, J. Geom. Phys. 59 (2009), 1485-1511, arXiv:0803.2292.
  18. Enriquez B., Feigin B., Rubtsov V., Separation of variables for Gaudin-Calogero systems, Compositio Math. 110 (1998), 1-16, q-alg/9605030.
  19. Felder G., Schorr A., Separation of variables for quantum integrable systems on elliptic curves, J. Phys. A: Math. Gen. 32 (1999), 8001-8022, math.QA/9905072.
  20. Pakuliak S., Rubtsov V., Silantyev A., Classical elliptic current algebras. I, J. Gen. Lie Theory Appl. 2 (2008), 65-78.
    Pakuliak S., Rubtsov V., Silantyev A., Classical elliptic current algebras. II, J. Gen. Lie Theory Appl. 2 (2008), 79-93, arXiv:0709.3592.
  21. Enriquez B., Felder G., Elliptic quantum groups Eτ,η(sl2) and quasi-Hopf algebras, Comm. Math. Phys. 195 (1998), 651-689, q-alg/9703018.
  22. Chervov A., Talalaev D., Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence, hep-th/0604128.
  23. Chervov A., Talalaev D., KZ equation, G-opers, quantum Drinfeld-Sokolov reduction and quantum Cayley-Hamilton identity, J. Math. Sci. 158 (2009), 904-911, hep-th/0607250.
  24. Gould M.D., Zhang Y.-Z., Zhao S.-Y., Elliptic Gaudin models and elliptic KZ equations, Nuclear Phys. B 630 (2002), 492-508, nlin.SI/0110038.

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