Quantum Toda chains intertwined. (English. Russian original) Zbl 1219.81128

St. Petersbg. Math. J. 22, No. 3, 411-435 (2011); translation from Algebra Anal. 2010, No. 3, 107-141 (2010).
Summary: An explicit construction of integral operators intertwining various quantum Toda chains is conjectured. Compositions of the intertwining operators provide recursive and \( \mathcal{Q}\)-operators for quantum Toda chains. In particular the authors’ earlier results on Toda chains corresponding to classical Lie algebras are extended to the generic \( BC_n\)- and Inozemtsev-Toda chains. Also, an explicit form of \( \mathcal{Q}\)-operators is conjectured for the closed Toda chains corresponding to the Lie algebras \( B_{\infty}, C_{\infty}, D_{\infty}\), the affine Lie algebras \( B^{(1)}_n, C^{(1)}_n, D^{(1)}_n, D^{(2)}_n, A^{(2)}_{2n-1}, A^{(2)}_{2n}\), and the affine analogs of \( BC_n\)- and Inozemtsev-Toda chains.


81Q12 Nonselfadjoint operator theory in quantum theory including creation and destruction operators
81S05 Commutation relations and statistics as related to quantum mechanics (general)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
Full Text: DOI arXiv


[1] L. D. Faddeev, How the algebraic Bethe ansatz works for integrable models, Symétries quantiques (Les Houches, 1995) North-Holland, Amsterdam, 1998, pp. 149 – 219. · Zbl 0934.35170
[2] A. Gerasimov, S. Kharchev, D. Lebedev, and S. Oblezin, On a Gauss-Givental representation of quantum Toda chain wave function, Int. Math. Res. Not. , posted on (2006), Art. ID 96489, 23. · Zbl 1142.17019
[3] A. Gerasimov, D. Lebedev, and S. Oblezin, Givental integral representation for classical groups, arXiv:math. RT/ 0608152. · Zbl 1267.17007
[4] -, New integral representations of Whittaker functions for classical groups, math.RT/ 0705.2886.
[5] A. Gerasimov, D. Lebedev, and S. Oblezin, Baxter operator and Archimedean Hecke algebra, Comm. Math. Phys. 284 (2008), no. 3, 867 – 896. · Zbl 1163.17010
[6] Alexander Givental, Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture, Topics in singularity theory, Amer. Math. Soc. Transl. Ser. 2, vol. 180, Amer. Math. Soc., Providence, RI, 1997, pp. 103 – 115. · Zbl 0895.32006
[7] Michihiko Hashizume, Whittaker functions on semisimple Lie groups, Hiroshima Math. J. 12 (1982), no. 2, 259 – 293. · Zbl 0524.43005
[8] Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. · Zbl 0451.53038
[9] V. I. Inozemtsev, The finite Toda lattices, Comm. Math. Phys. 121 (1989), no. 4, 629 – 638. · Zbl 0677.35086
[10] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. · Zbl 0716.17022
[11] Vadim Kuznetsov and Evgeny Sklyanin, Bäcklund transformation for the BC-type Toda lattice, SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007), Paper 080, 17. · Zbl 1157.70012
[12] V. Pasquier and M. Gaudin, The periodic Toda chain and a matrix generalization of the Bessel function recursion relations, J. Phys. A 25 (1992), no. 20, 5243 – 5252. · Zbl 0768.58023
[13] A. G. Reĭman and M. A. Semenov-Tyan-Shanskiĭ, Integrable systems. Group-theoretical approach, Inst. Kompyuter. Issled., Moscow-Izhevsk, 2003. (Russian)
[14] E. K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A 21 (1988), no. 10, 2375 – 2389. · Zbl 0685.58058
[15] Современные проблемы математики. Фундаментал\(^{\приме}\)ные направления. Том 16, Итоги Науки и Техники. [Прогресс ин Сциенце анд Течнологы], Акад. Наук СССР, Всесоюз. Инст. Научн. и Техн. Информ., Мосцощ, 1987 (Руссиан). Динамические системы. 7. [Дынамицал сыстемс. 7].
[16] Morikazu Toda, Nonlinear lattice dynamics, Iwanami Shoten, Tokyo, 1978 (Japanese). Morikazu Toda, Theory of nonlinear lattices, Springer Series in Solid-State Sciences, vol. 20, Springer-Verlag, Berlin-New York, 1981. Translated from the Japanese by the author.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.