Quantum Integrable Toda-Like Systems in Deformation Quantization

Abstract

We define and discuss the notion of quantum integrability of a classically integrable system within the framework of deformation quantization, i.e. the question whether the classical conserved quantities (which are already in involution with respect to the Poisson bracket) commute with respect to some star product on the phase space after possible quantum corrections. As an example of this method, we show by means of suitable 2 by 2 quantum R-matrices that a list of Toda-like classical integrable systems given by Y. B. Suris is quantum integrable with respect to the usual star product of the Weyl type in flat 2n-dimensional space.