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Unitary representations of $$U_{q} (\mathfrak s\mathfrak l(2,\mathbb R))$$, the modular double and the multiparticle $$q$$-deformed Toda chain. (English) Zbl 1001.37067
The paper deals with analytic theory of the quantum two-particle $$q$$-deformed Toda chains and the corresponding theory of joint eigenfunctions of the quantum Toda Hamiltonians that are called the generalized Whittaker functions. This is the simplest nontrivial example clarifying the role of the modular duality concept in the representation theory of noncompact semisimple quantum groups.
In Section 1 the elementary representation of the algebra $$U_q(sl(2,\mathbb R)$$ is considered. Then the theory of Whittaker vectors and Whittaker functions for the corresponding modular double and with the 2-particle $$q$$-deformed open Toda chain is given. The explicit formulae for the Whittaker vectors in terms of the double sine functions of Barnes are obtained. Moreover, the integral representations for solutions to the one-parameter family of two-particle relativistic Toda chains in the framework of representation theory are derived; all these solutions enjoy the dual symmetry. The article has an Appendix devoted to the analytical properties of the double sine functions.

##### MSC:
 37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B80 Applications of Lie algebras and superalgebras to integrable systems 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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