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

SIGMA 2 (2006), 084, 20 pages      nlin.SI/0612003      https://doi.org/10.3842/SIGMA.2006.084
Contribution to the Vadim Kuznetsov Memorial Issue

R-Matrix and Baxter Q-Operators for the Noncompact SL(N,C) Invariant Spin Chain

Sergey É. Derkachov a and Alexander N. Manashov b, c
a) St.-Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 27, 191023 St.-Petersburg, Russia
b) Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany
c) Department of Theoretical Physics, Sankt-Petersburg University, St.-Petersburg, Russia

Received October 30, 2006; Published online December 02, 2006

Abstract
The problem of constructing the SL(N,C) invariant solutions to the Yang-Baxter equation is considered. The solutions (R-operators) for arbitrarily principal series representations of SL(N,C) are obtained in an explicit form. We construct the commutative family of the operators Qk(u) which can be identified with the Baxter operators for the noncompact SL(N,C) spin magnet.

Key words: Yang-Baxter equation; Baxter operator.

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References

  1. Skljanin E.K., Tahtadzjan L.A., Faddeev L.D., The quantum inverse problem method. I, Teoret. Mat. Fiz., 1979, V.40, 194-220 (English transl.: Theor. Math. Phys., 1980, V.40, 688-706).
  2. Faddeev L.D., How algebraic Bethe ansatz works for integrable model (Les-Houches Lectures, 1995), hep-th/9605187.
  3. Drinfel'd V.G., Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR, 1985, V.283, 1060-1064 (English transl.: Soviet Math. Dokl., 1985, V.32, 254-258).
  4. Drinfel'd V.G., Quasi-Hopf algebras, Algebra i Analiz, 1989, V.1, N 6, 114-118 (English transl.: Leningrad Math. J., 1990, V.1, 1419-1457).
  5. Jimbo M., A q difference analog of U(g) and the Yang-Baxter equation, Lett. Math. Phys., 1985, V.10, 63-69.
  6. Rosso M., An analogue of P.B.W. theorem and the universal R-matrix for Uhsl(N+1), Comm. Math. Phys., 1989, V.124, 307-318.
  7. Kirillov A.N., Reshetikhin N., q Weyl group and a multiplicative formula for universal R matrices, Comm. Math. Phys., 1990, V.134, 421-431.
  8. Khoroshkin S.M., Tolstoy V.N., Universal R-matrix for quantized (super)algebras, Comm. Math. Phys., 1991, V.141, 599-617.
  9. Khoroshkin S.M., Tolstoy V.N., The uniqueness theorem for the universal R-matrix, Lett. Math. Phys., 1992, V.24, 231-244.
  10. Kulish P.P., Reshetikhin N.Y., Sklyanin E.K., Yang-Baxter equation and representation theory. I, Lett. Math. Phys., 1981, V.5, 393-403.
  11. Kulish P.P., Reshetikhin N.Y., On GL3-invariant solutions of the Yang-Baxter equation and associated quantum systems, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 1982, V.120, 92-121.
  12. Baxter R.J., Exactly solved models in statistical mechanics, London, Academic Press, 1982.
  13. Sklyanin E.K., The quantum Toda chain, in Nonlinear Equations in Classical and Quantum Field Theory (Meudon/Paris, 1983/1984), Lecture Notes in Phys., Vol. 226, Berlin, Springer, 1985, 196-233.
  14. Sklyanin E.K., Quantum inverse scattering method. Selected topics, in Quantum Group and Quantum Integrable Systems (Nankai Lectures in Mathematical Physics), Editor Mo-Lin Ge, Singapore, World Scientific, 1992, 63-97, hep-th/9211111.
  15. Derkachov S.E., Factorization of R-matrix. I, Zap. Nauch. Sem. POMI, 2006, V.335, 134-163, math.QA/0503396.
  16. Derkachov S.E., Korchemsky G.P., Manashov A.N., Noncompact Heisenberg spin magnets from high-energy QCD. I. Baxter Q-operator and separation of variables, Nuclear Phys. B, 2001, V.617, 375-440, hep-th/0107193.
  17. Derkachov S.E., Korchemsky G.P., Kotanski J., Manashov A.N., Noncompact Heisenberg spin magnets from high-energy QCD. II. Quantization conditions and energy spectrum, Nuclear Phys. B, 2002, V.645, 237-297, hep-th/0204124.
  18. Bazhanov V.V., Lukyanov S.L., Zamolodchikov A.B., Integrable structure of conformal field theory. III. The Yang-Baxter relation, Comm. Math. Phys., 1999, V.200, 297-324, hep-th/9805008.
  19. Bazhanov V.V., Hibberd A.N., Khoroshkin S.M., Integrable structure of W(3) conformal field theory, quantum Boussinesq theory and boundary affine Toda theory, Nuclear Phys. B, 2002, V.622, 475-547, hep-th/0105177.
  20. Kuznetsov V.B., Nijhoff F.W., Sklyanin E.K., Separation of variables for the Ruijsenaars system, Comm. Math. Phys., 1997, V.189, 855-877, solv-int/9701004.
  21. Kharchev S., Lebedev D., Integral representation for the eigenfunctions of quantum periodic Toda chain, Lett. Math. Phys., 1999, V.50, 53-77, hep-th/9910265.
  22. Kharchev S., Lebedev D., Semenov-Tian-Shansky M., Unitary representations of Uq(sl(2,R)), the modular double, and the multiparticle q-deformed Toda chains, Comm. Math. Phys., 2002, V.225, 573-609, hep-th/0102180.
  23. Derkachov S.E., Korchemsky G.P., Manashov A.N., Separation of variables for the quantum SL(2,R) spin chain, JHEP, 2003, N 07, Paper 047, 30 pages, hep-th/0210216.
  24. Derkachov S.E., Korchemsky G.P., Manashov A.N., Baxter Q-operator and separation of variables for the open SL(2,R) spin chain, JHEP, 2003, N 10, Paper 053, 31 pages, hep-th/0309144.
  25. Kuznetsov V.B., Mangazeev V.V., Sklyanin E.K., Q-operator and factorised separation chain for Jack's symmetric polynomials, Indag. Mathem., 2003, V.14, 451-482, math.CA/0306242.
  26. Silantyev A., Transition function for the Toda chain model, nlin.SI/0603017.
  27. Iorgov N., Eigenvectors of open Bazhanov-Stroganov quantum chain, SIGMA, 2006, V.2, Paper 019, 10 pages, nlin.SI/0602010.
  28. Kuznetsov V.B., Sklyanin E.K., Few remarks on Bäcklund transformations for many-body systems, J. Phys. A: Math. Gen., 1998, V.31, 2241-2251, solv-int/9711010.
  29. Sklyanin E.K., Bäcklund transformations and Baxter's Q-operator, nlin.SI/0009009.
  30. Gel'fand I.M., Naimark M.A., Unitary representations of the classical groups, Trudy Mat. Inst. Steklov., Vol. 36, Moscow - Leningrad, Izdat. Nauk SSSR, 1950 (German transl.: Berlin, Academie - Verlag, 1957).
  31. Zhelobenko D.P., Shtern A.I., Representations of Lie groups, Moscow, Nauka, 1983 (in Russian).
  32. Knapp A.W., Stein E.M., Intertwining operators for semisimple groups, Ann. of Math., 1971, V.93, 489-578.
  33. Knapp A.W., Representation theory of semisimple groups, Princeton, New Jersey, Princeton University Press, 1986.
  34. Derkachov S.E., Factorization of R-matrix and Baxter's Q-operator, math.QA/0507252.
  35. Perk J.H.H., Au-Yang H., Yang-Baxter equations, in Encyclopedia of Mathematical Physics, Editors J.-P. Françoise, G.L. Naber and S.T. Tsou, Oxford, Elsevier, 2006, Vol. 5, 465-473.
  36. Isaev A.P., Multi-loop Feynman integrals and conformal quantum mechanics, Nuclear Phys. B, 2003, V.662, 461-475, hep-th/0303056.
  37. Derkachov S.E., Manashov A.N., Factorization of the transfer matrices for the quantum sl(2) spin chains and Baxter equation, J. Phys. A: Math. Gen., 2006, V.39, 4147-4159, nlin.SI/0512047.
  38. Derkachov S.E., Manashov A.N., Baxter operators for the quantum sl(3) invariant spin chain, J. Phys. A: Math. Gen., 2006, V.39, 13171-13190, nlin.SI/0604018.
  39. Derkachov S.E., Baxter's Q-operator for the homogeneous XXX spin chain, J. Phys. A: Math. Gen., 1999, V.32, 5299-5316, solv-int/9902015.

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