This article is cited in 7 papers

Unitary Representations of the Quantum Lorentz Group and Quantum Relativistic Toda Chain

M. A. Olshanetsky^a, V.-B. K. Rogov<sup>b

a</sup> Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)
^b Moscow State University of Railway Communications

Abstract: We give a group theory interpretation of the three types of $q$-Bessel functions. We consider a family of quantum Lorentz groups and a family of quantum Lobachevsky spaces. For three values of the parameter of the quantum Lobachevsky space, the Casimir operators correspond to the two-body relativistic open Toda-chain Hamiltonians whose eigenfunctions are the modified $q$-Bessel functions of the three types. We construct the principal series of unitary irreducible representations of the quantum Lorentz groups. Special matrix elements in the irreducible spaces given by the $q$-Macdonald functions are the wave functions of the two-body relativistic open Toda chain. We obtain integral representations for these functions.

Received: 08.10.2001

DOI: 10.4213/tmf307

 English version: Theoretical and Mathematical Physics, 2002, 130:3, 299–322

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