Representation theory and quantum integrability

Progress in Mathematics

Volumn 237, Issue , 2005, Pages 133-156

  Gerasimov, A ^a,b   Kharchev, S ^a   Lebedev, D ^a  

^a INSTITUTE FOR THEORETICAL AND EXPERIMENTAL PHYSICS   (Russian Federation)

^b TRINITY COLLEGE DUBLIN   (Ireland)

Author keywords

Gelfand Zetlin representation; Infinite dimensional representations of Uq(sl(N)); Modular double; Toda systems; Whittaker modules

Indexed keywords

EID: 33646426744     PISSN: 07431643     EISSN: 2296505X     Source Type: Book Series    
DOI: 10.1007/3-7643-7341-5_4     Document Type: Chapter

References (33)

1. V.G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32, 254–258, (1985).
2. M. Jimbo, A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10, (1985), 63–69.
3. L.D. Faddeev, Quantum completely integrable models in field theory, Sov. Sci. Rev., Sect. C (Math. Phys. Rev.) 1, (1980), 107–155.
4. P.P. Kulish, E.K. Sklyanin, Quantum spectral transform method. Recent developments, Lecture Notes in Phys. 151, pp. 61–119, Springer, Berlin-New York, 1982.
5. E.K. Sklyanin, Separation of variables - new trends, Quantum field theory, integrable models and beyond, (Kyoto, 1994). Progr. Theor. Phys. Suppl. 118, (1995), 35–60.
6. A. Gerasimov, S. Kharchev, D. Lebedev, Representation Theory and Quantum Inverse Scattering Method: The Open Toda Chain and the Hyperbolic Sutherland Model, Int. Math. Res. Notices 2004, no.17, (2004), 823–854.
7. A. Gerasimov, S. Kharchev, D. Lebedev, On a class of integrable systems connected with GL(N, R), arXiv:math.QA/0301025.
8. I.M. Gelfand, M.L. Tsetlin, Finite-dimensional representations of the group of uni-modular matrices, Dokl. Akad. Nauk SSSR 71, (1950), 825–828.
9. I.M. Gelfand, M.I. Graev, Finite-dimensional irreducible representations of the unitary and the full linear groups, and related special functions, Izv. Akad. Nauk SSSR, Ser. Mat. 29, (1965), 1329–1356; [Transl., II Ser., Am. Math. Soc. 64, (1965), 116– 146].
10. F. Lemire, J. Patera, Formal analytic continuation of Gelfand’s finite-dimensional representations of gl(n,C), J. Math. Phys. 20, (1979), 820–829.
11. V.G. Drinfeld, Quantum groups, Proc. Int. Congr. Math. Berkeley, California, (1986), Providence (1987), 718–820.
12. V.G. Drinfeld, A new realization of Yangians and of quantum affine algebras, (Russian) Dokl. Akad. Nauk SSSR 296, (1987), no. 1, 13–17; translation in Soviet Math. Dokl. 36, (1988), 212–216.
13. L.D. Faddeev, Discrete Heisenberg-Weyl group and modular group, Lett. Math. Phys. 34, (1995), 249–254.
14. L.D. Faddeev, Modular double of a quantum group, in: Conférence Moshé Flato 1999, Quantization, Deformations, and Symmetries. Vol. I, 149–156, Kluwer Acad. Publ., Dordrecht, 2000.
15. q(sl(2, R)), the modular double and the multiparticle q-deformed Toda chains, Comm. Math. Phys. 225, (2002), 573–609.
16. G. Lusztig, Introduction to quantum groups, Progress in Mathematics, 110, Birkhäuser Boston, Inc., Boston, MA, 1993
17. V. Chari, A. Pressley, A guide to quantum groups, Cambridge Univ. Press, Cambridge, 1994.
18. A. Connes, Non commutative geometry, Academic Press, 1994.
19. ∗ -algebras associated with irrational rotations, Pacific J. Math. 93, (1981), 415–430.
20. J. Adams, D. Barbasch and D. Vogan Jr. The Langlands Classification and Irreducible Characters for Real Reductive Groups, Progress in Mathematics 104, Birkhäuser 1992.
21. S. Kharchev, D. Lebedev, Integral representation for the eigenfunctions of a quantum periodic Toda chain, Lett. Math. Phys. 50, (1999), 53–77.
22. S. Kharchev, D. Lebedev, Eigenfunctions of GL(N, R) Toda chain: The Mellin-Barnes representation, JETP Lett. 71, (2000), 235–238.
23. S. Kharchev, D. Lebedev, Integral representations for the eigenfunctions of quantum open and periodic Toda chains from QISM formalism, J.Phys. A34, (2001), 2247– 2258.
24. B. Kostant, On Whittaker vectors and representation theory, Inventiones Math. 48, (1978), 101–184.
25. R. Howe, T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290, (1991), 569–619.
26. I. Gelfand, A. Kirillov, Sur les corps liés aux algèbres enveloppantes des algèbres de Lie, Publ. Math. de l’IHÉS 31, (1966), 509–523.
27. A. Gerasimov, S. Kharchev, D. Lebedev, in preparation.
28. A. Sevostyanov, Regular nilpotent elements and quantum groups, Comm. Math. Phys. 204, (1999), 1–16.
29. A. Molev, M. Nazarov, G. Olshanski, Yangians and classical Lie algebras, Russian Math. Surveys 51, (1996), 205–282.
30. M. Nazarov, V. Tarasov, Yangians and Gelfand-Zetlin bases, Publ. Res. Inst. Math. Sci. 30, (1994), 459–478.
31. I.V. Cherednik, A new interpretation of Gelfand-Tzetlin bases, Duke Math. J. 54, (1987), 563–577.
32. E.K. Sklyanin, The quantum Toda chain, Lect. Notes in Phys. 226, (1985), 196–233.
33. H. Flaschka, D. McLaughlin, Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions, Progr. Theor. Phys. 55, (1976), 438–456.