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Raising Operators for the Whittaker Wave Functions of the Toda Chain and Intertwining Operators

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REFERENCES

  1. E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, “Transformation groups for soliton equations,” In: Proc. RIMS Symp. Nonlinear Integrable Systems-Classical Theory and Quantum Theory, M. Jimbo and T. Miwa (Eds.), World Scientific, Singapore (1983), p. 39.

    Google Scholar 

  2. M. A. Olshanetsky and A. M. Perelomov, “Classical integrable finite-dimensional systems related to Lie algebras,” Phys. Rept., 71 313–400 (1981).

    Google Scholar 

  3. M. Olshanetsky and A. Perelomov, “Quantum integrable systems related to Lie algebras,” Phys. Rept., 94 313–404 (1983).

    Google Scholar 

  4. J.-L. Verdier, “Algebras de Lie, systems Hamiltonienes, courbes algebriques,” In: Seminaire Bourbaki, 34-e annee, No. 566 (1980/81), pp. 1–10.

  5. J.-L. Verdier, “Les representations des algebras de Lie affines: applications a quelques problemes de physique,” In: Seminaire Bourbaki, 34-e annee, No. 596 (1981/82), pp. 1–13.

    Google Scholar 

  6. A. Gorsky, “Integrable many body systems in the field theories,” Theor. Mat. Phys., 103 No.3 (1995).

    Google Scholar 

  7. M. Jimbo and T. Miwa, “Algebraic analysis of solvable lattice models,” In: Conference Board of the Math. Sci., Regional Conference Series in Mathematics, 85 (1995).

  8. E. Frenkel, Five lectures on soliton equations, q-alg/9712005.

  9. I. Cherednik, Lectures on Affine Knizhnik-Zamoldchikov Equations, Quantum Many-Body Problems, Hecke Algebras, and Macdonald Theory, Research Inst. Math. Studies (1998).

  10. S. Kharchev, “ Kadomtsev-Petviashvili hierachy and generalized Kontsevich model,” hep-th/9810091.

  11. V. Drinfeld and V. V. Sokolov, “Lie algebras and evolutions of KdV type,” J. Sov. Math., 30 1975–2036 (1985).

    Google Scholar 

  12. A. Pressley and G. Segal, Loop Groups, Clarendon Press, Oxford (1986).

    Google Scholar 

  13. V. Kac, Infinite-Dimensional Lie Algebras, Cambridge University Press, Cambridge (1990).

    Google Scholar 

  14. A. N. Leznov and M. V. Saveliev, Theory group methods for integrating non-linear dynamical systems [in Russian], Nauka, Moscow (1985).

    Google Scholar 

  15. P. Etingof, I. Frenkel, and A. Kirillov Representation Theory and the Kniznik-Zamolodchikov Equation, AMS, Providence, Rhode Island (1998).

    Google Scholar 

  16. A. Mironov, “Tau-function within group theory approach and its quantization,” hep-th/9711006.

  17. A. Gerasimov, S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and M. Olshanetsky, “Liouville type models in group theory framework. I. Finite-dimensional algebras,” preprint FIAN/TD-18/95, ITEP M4/TH-7/95; hep-th/9601161.

  18. A. Gerasimov, S. Khoroshkin, D. Lebedev, A. Mironov, and A. Morozov, “Generalized Hirota equations and representation theory. I. The case of SL(2) and SLq(2),” Int. J. Mod. Phys., A10 2589–2614 (1995).

    Google Scholar 

  19. M. Golenisheva-Kutozova and D. Lebedev, Intertwining operators and soliton equations, hep-th/9805186.

  20. A. Antonov and B. Feigin, “Quantum group representations and Baxter equation,” Phys. Lett., B392 115–122 (1997).

    Google Scholar 

  21. A. Tsuchiya and Y. Kanie, “Vertex operators in conformal field theory on P 1 and monodromy representations of braid group,” Adv. Stud. Pure Math., 16 297–372 (1988).

    Google Scholar 

  22. I. Frenkel and N. Reshetikhin, “Quantum affine algebras and holonomic difference equations,” Commun. Math. Phys., 146 No.1 (1992).

    Google Scholar 

  23. N. Vilenkin and A. Klimyk, Representations of Lie Groups and Special Functions, Kluwer Academic Publishers (1995).

  24. J. Van der Jeugt and R. Jagannathan, “Realizations of su(1, 1) and U q (su(1, 1)) and generating functions for orthogonal polynomials,” math-ph/9807019.

  25. O. I. Bogoyavlensky, “On perturbations of the periodic Toda lattice,” Comm. Math. Phys., 51 201–209 (1976).

    Google Scholar 

  26. M. Semenov-Tian-Shansky, “Quantization of the open Toda chains,” In: Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Vol. 16 [in Russian], All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1987), pp. 194–226.

    Google Scholar 

  27. B. Kostant, “Whittaker vectors and representation theory,” Invent. Math., 48 101–184 (1978).

    Google Scholar 

  28. A. N. Kirillov and Masatoshi Noumi, “Affine Hecke algebras and raising operators for Macdonald polynomials,” q-alg/9605004.

  29. A. N. Kirillov and Masatoshi Noumi, “q-Difference raising operators for Macdonald polynomials and the integrality of transition coefficients,” q-alg/9605005.

  30. Luc Lapointe and Luc Vinet, “Creation operators for the Macdonald and Jack polynomials,” q-alg/9607024.

  31. Luc Lapointe and Luc Vinet, “Rodrigues formulas for the Macdonald polynomials,” q-alg/9607025.

  32. Yasushi Kajihara and Masatoshi Noumi, “Raising operators of row type for Macdonald polynomials,” math/9803151.

  33. I. Gradshtein and I. Ryzhik, Tables of Integrals, Sums, Series and Products [in Russian], Fizmatgiz, Moscow (1963).

    Google Scholar 

  34. P. Etingof, “Whittaker functions on quantum groups and q-deformed Toda operators,” math. QA/9901053.

  35. A. Sevostyanov, “The Whittaker model of the center of the quantum group and Hecke algebras,” math. QA/9904075.

  36. V. Bazhanov, S. Lukyanov, and A. Zamolodchikov, “Integrable structure of conformal field theory I, II, III,” hep-th/9412229, hep-th/9604044, hep-th/9805008.

  37. P. Etingof, “Integral formulas for wave functions of quantum many-body problems and representations of gl n ,” hep-th/9405038.

  38. B. L. Feigin and E. V. Frenkel, “Representations of affine Kac-Moody algebras and bosonization,” In: Physics and Mathematics of Strings. Knizhnik Memorial Volume, World Scientific, Singapore (1990), pp. 271–316.

  39. G. Heckmann, “An elementary approach to the hypergeometric shift operators of Opdam,” Invent. Math., 98 341–350 (1991).

    Google Scholar 

  40. C. Dunkl, “Differential-difference operators associated with reflection groups,” Trans. Amer. Math. Soc., 311 No.1, 167–183 (1989).

    Google Scholar 

  41. E. Opdam, “Some applications of hypergeometric shift operators,” Invent. Math., 98 No.1, 1–18 (1989).

    Google Scholar 

  42. J. Denef and F. Loeser, “Character sums associated to. nite Coxeter groups,” math. AG/9803033.

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya.Tematicheskie Obzory. Vol. 117, Geometry, 2004.

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Chervov, A. Raising Operators for the Whittaker Wave Functions of the Toda Chain and Intertwining Operators. J Math Sci 128, 3121–3141 (2005). https://doi.org/10.1007/s10958-005-0259-z

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