St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



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
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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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

D. Lebedev
Affiliation: Institute for Theoretical and Experimental Physics, Moscow 117259, Russia

S. Oblezin
Affiliation: Institute for Theoretical and Experimental Physics, Moscow 117259, Russia

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