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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.

##### MSC:
 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.)
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