A geometric approach to Lie systems: formalism of Poisson-Hopf algebra deformations

Abstract

The notion of quantum algebras is merged with that of Lie systems in order to establish a new formalism called Poisson-Hopf algebra deformations of Lie systems. The procedure can be naturally applied to Lie systems endowed with a symplectic structure, the so-called Lie-Hamilton systems. This is quite a general approach, as it can be applied to any quantum deformation and any underlying manifold. One of its main features is that, Lie systems are extended to generalized systems described by involutive distributions. In this way, we obtain their new generalized (deformed) counterparts that cover, in particular, a new oscillator system with a time-dependent frequency and a position-dependent mass. Based on a recently developed procedure to construct Poisson-Hopf deformations of Lie-Hamilton systems \cite{Ballesteros5}, a novel unified approach to deformations of Lie-Hamilton systems on the real plane with a Vessiot-Guldberg Lie algebra isomorphic to $\mathfrak{sl}(2)$ is proposed. In addition, we study the deformed systems obtained from Lie-Hamilton systems associated to the oscillator algebra $\mathfrak{h}_4$, seen as a subalgebra of the 2-photon algebra $\mathfrak{h}_6$. As a particular application, we propose an epidemiological model of SISf type that uses the solvable Lie algebra $\mathfrak{b}_2$ as subalgebra of $\mathfrak{sl}(2)$, by restriction of the corresponding quantum deformed systems.