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

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.

21 pages, 1 figure. Based on the contribution presented at the "X International Symposium on Quantum Theory and Symmetries" (QTS-10), June 19-25, 2017, Varna, Bulgaria

References in corpus (3)

• Superposition rules, Lie theorem and partial differential equations
• Lie-Hamilton systems on the plane: Applications and superposition rules
• Poisson-Hopf algebra deformations of Lie-Hamilton systems

Cited by in corpus (3)

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• Lie-Hamilton systems on Riemannian and Lorentzian spaces from conformal transformations and some of their applications
• Geometric models for Lie--Hamilton systems on ${\mathbb{R}}^2$