Poisson-Hopf algebra deformations of Lie-Hamilton systems

Abstract

Hopf algebra deformations are merged with a class of Lie systems of Hamiltonian type, the so-called Lie–Hamilton systems, to devise a novel formalism: the Poisson–Hopf algebra deformations of Lie–Hamilton systems. This approach applies to any Hopf algebra deformation of any Lie–Hamilton system. Remarkably, a Hopf algebra deformation transforms a Lie–Hamilton system, whose dynamic is governed by a finite-dimensional Lie algebra of functions, into a non-Lie–Hamilton system associated with a Poisson–Hopf algebra of functions that allows for the explicit description of its t-independent constants of the motion from deformed Casimir functions. We illustrate our approach by considering the Poisson–Hopf algebra analogue of the nonstandard quantum deformation of sl(2) and its applications to deform wellknown Lie–Hamilton systems describing oscillator systems, Milne–Pinney equations, and several types of Riccati equations. In particular, we obtain a new position-dependent mass oscillator system with a time-dependent frequency.