Poisson-Hopf algebra deformations of Lie-Hamilton systems

TL;DR

This paper introduces a new formalism combining Hopf algebra deformations with Lie-Hamilton systems, enabling explicit descriptions of constants of motion and applications to various classical systems, including a novel position-dependent mass oscillator.

Contribution

It develops the Poisson-Hopf algebra deformation framework for Lie-Hamilton systems, extending their analysis to non-Lie-Hamilton systems with explicit conserved quantities.

Findings

Deformation transforms Lie-Hamilton systems into non-Lie-Hamilton systems with explicit constants of motion.

Application to quantum deformations of $rak{sl}(2)$ yields new deformed oscillator and Riccati systems.

Discovery of a new position-dependent mass oscillator with a time-dependent frequency.

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 $\mathfrak{sl}(2)$ 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.