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.
