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 non-standard quantum deformation of and its applications to deform well-known 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.