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Quantum Toda chains intertwined


Authors: A. Gerasimov, D. Lebedev and S. Oblezin
Original publication: Algebra i Analiz, tom 22 (2010), nomer 3.
Journal: St. Petersburg Math. J. 22 (2011), 411-435
MSC (2010): Primary 81Q12
DOI: https://doi.org/10.1090/S1061-0022-2011-01149-5
Published electronically: March 17, 2011
MathSciNet review: 2729942
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Abstract: 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.


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  • [F] L. D. Faddeev, How the algebraic Bethe ansatz works for integrable models, Symétries Quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, pp. 149-219. MR 1616371 (2000b:82010)
  • [GKLO] A. Gerasimov, S. Kharchev, D. Lebedev, and S. Oblezin, On a Gauss-Givental representation for quantum Toda chain wave function, Int. Math. Res. Not. 2006, Art. ID 96489; arXiv:math.RT/0505310. MR 2219213 (2007f:17041)
  • [GLO1] A. Gerasimov, D. Lebedev, and S. Oblezin, Givental integral representation for classical groups, arXiv:math. RT/ 0608152.
  • [GLO2] -, New integral representations of Whittaker functions for classical groups, math.RT/ 0705.2886.
  • [GLO3] -, Baxter operator and Archimedean Hecke algebra, Comm. Math. Phys. 284 (2008), 867-896; DOI 10.1007/s00220-008-0547-9, arXiv:math.RT/0706.3476. MR 2452597 (2009k:17049)
  • [Gi] A. 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; arXiv:alg-geom/9612001. MR 1767115 (2001d:14063)
  • [Ha] M. Hashizume, Whittaker functions on semi-simple Lie groups, Hiroshima Math. J. 12 (1982), 259-293. MR 0665496 (84d:22018)
  • [He] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure Appl. Math., vol. 80, Acad. Press, Inc., New York-London, 1978. MR 0514561 (80k:53081)
  • [I] V. Inozemtsev, The finite Toda lattices, Comm. Math. Phys. 121 (1989), 629-638. MR 0990995 (90f:58086)
  • [K] V. Kac, Infinite-dimensional Lie algebras, Cambridge University Press, Cambridge, 1990. MR 1104219 (92k:17038)
  • [KS] V. Kuznetsov and E. K. Sklyanin, Bäcklund transformation for the BC-type Toda lattice, Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007), 080, 17 pp.; arXiv:0707.1950. MR 2366942 (2009j:70024)
  • [PG] V. Pasquier and M. Gaudin, The periodic Toda chain and a matrix generalization of the Bessel function recursion relations, J. Phys. A 25 (1992), 5243-5252. MR 1192958 (94a:82023)
  • [RSTS] A. G. Reĭman and M. A. Semenov-Tyan-Shanskiĭ, Integrable systems. Group-theoretical approach, Inst. Kompyuter. Issled., Moscow-Izhevsk, 2003. (Russian)
  • [S] E. K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A 21 (1988), 2375-2389. MR 0953215 (89h:81258)
  • [STS] Dynamical Systems-7, Itogi Nauki i Tekhniki. Sovrem. Probl. Mat. Fundam. Naprav., vol. 16, VINITI, Moscow, 1987; English transl., Encyclopaedia Math. Sci., vol. 16, Springer-Verlag, Berlin, 1994, pp. 226-259. MR 0922069 (88g:58004)
  • [T] M. Toda, Theory of nonlinear lattices, Springer Ser. in Solid-State Sci., vol. 20, Springer-Verlag, Berlin-New York, 1981. MR 0618652 (82k:58052b)

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Additional Information

A. Gerasimov
Affiliation: Institute for Theoretical and Experimental Physics, Moscow 117259, Russia; School of Mathematics, Trinity College, Dublin 2, Ireland; and Hamilton Mathematics Institute, Trinity College, Dublin 2, Ireland
Email: anton@maths.tcd.ie

D. Lebedev
Affiliation: Institute for Theoretical and Experimental Physics, Moscow 117259, Russia
Email: lebedev@itep.ru

S. Oblezin
Affiliation: Institute for Theoretical and Experimental Physics, Moscow 117259, Russia
Email: Sergey.Oblezin@itep.ru

DOI: https://doi.org/10.1090/S1061-0022-2011-01149-5
Keywords: Quantum Toda Hamiltonians, elementary intertwining operator, recursive operator, quantization Pasquier–Gaudin integral $Q$-operator
Received by editor(s): January 11, 2010
Published electronically: March 17, 2011
Additional Notes: Supported by RFBR (grant nos. 08-01-00931-a and 09-01-93108-NCNIL-a). A. Gerasimov was also partly supported by a grant from Science Foundation Ireland. S. Oblezin gratefully acknowledges the support from Deligne’s 2004 Balzan prize in mathematics
Dedicated: To Ludwig Dmitrievich Faddeev on the occasion of his 75th birthday \flushright The most convenient model for exploring such relationships is the Toda chain. L. D. Faddeev (Preprint LOMI, P-2-79) \endflushright
Article copyright: © Copyright 2011 American Mathematical Society

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