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arXiv · 1803.07404

A unified approach to Poisson-Hopf deformations of Lie-Hamilton systems based on sl(2)

Angel Ballesteros·Rutwig Campoamor-Stursberg·Eduardo Fernandez-Saiz·Francisco J. Herranz·Javier de Lucas
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Abstract

Based on a recently developed procedure to construct Poisson-Hopf deformations of Lie-Hamilton systems, a novel unified approach to nonequivalent deformations of Lie-Hamilton systems on the real plane with a Vessiot-Guldberg Lie algebra isomorphic to $\mathfrak{sl}(2)$ is proposed. This, in particular, allows us to define a notion of Poisson-Hopf systems in dependence of a parameterized family of Poisson algebra representations. Such an approach is explicitly illustrated by applying it to the three non-diffeomorphic classes of $\mathfrak{sl}(2)$ Lie-Hamilton systems. Our results cover deformations of the Ermakov system, Milne-Pinney, Kummer-Schwarz and several Riccati equations as well as of the harmonic oscillator (all of them with $t$-dependent coefficients). Furthermore $t$-independent constants of motion are given as well. Our methods can be employed to generate other Lie-Hamilton systems and their deformations for other Vessiot-Guldberg Lie algebras and their deformations.

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Angel Ballesteros, Rutwig Campoamor-Stursberg, Eduardo Fernandez-Saiz, Francisco J. Herranz, Javier de Lucas. 2018-03-20. A unified approach to Poisson-Hopf deformations of Lie-Hamilton systems based on sl(2). https://doi.org/10.1007/978-981-13-2715-5_23

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