A Unified Approach to Poisson–Hopf Deformations of Lie–Hamilton Systems Based on $$\mathfrak {sl}$$(2)

Abstract

Based on a recently developed procedure to construct Poisson-Hopf deformations of Lie-Hamilton systems [4], a novel unified approach to nonequivalent deformations of Lie-Hamilton systems on the real plane with a Vessiot-Guldberg Lie algebra isomorphic to 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 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. 1

Geometric Models for Lie–Hamilton Systems on ℝ2

“…Although Lie-Hamilton systems on R 2 related to VG Lie algebras isomorphic to sl 2 and so 3 were very briefly studied in [4], our analysis here is much more detailed and it additionally shows, as a bonus, the existence of additional features of such Lie-Hamilton systems, which retrieves in a more natural and general manner results given in [13,16]. In the given basis, system (4) takes the form…”

“…Let us show how a Lie-Hamilton system on R 2 admitting a simple VG Lie algebra can be considered as the restriction of a certain type of Lie-Hamilton system on the dual to a Lie algebra to evendimensional symplectic leaves of the KKS bracket on such a dual [25]. Our approach extends the results and applications given in [4,Theorem 2]. Let g be a Lie algebra with a Lie bracket [·, ·] : g × g → g. If f ∈ C ∞ (g * ), where g * is the dual space to g, the canonical isomorphisms g ≃ g * * and T θ g * ≃ g * , for any θ ∈ g * , allow us to consider df θ : T θ g * → R as a vector in g ≃ g * * ≃ T * θ g * .…”

“…Our first model describes Lie-Hamilton systems on R 2 as the restriction of a Lie-Hamilton system on the dual, g * , of a Lie algebra g to an even-dimensional symplectic leaf on g * relative to the Poisson structure given by the Kirillov-Kostant-Souriau (KKS) bracket [25]. Our methods significantly extend and clarify the procedure briefly sketched in [4,Theorem 2], which was focused on studying only a particular simple example and it did not consider several difficulties for the application of the technique. Moreover, we now obtain as a byproduct that certain Lie-Hamilton systems on R 2 are endowed with a pseudo-Riemannian metric that is invariant relative to the Lie derivative with respect to the vector fields of the VG Lie algebra of the Lie-Hamilton system.…”

“…Although Lie-Hamilton systems on R 2 are the exception rather than the rule among general differential equations (cf. [6,17]), they admit a plethora of geometric properties and relevant applications [3,4,5,6,17], which motivates their analysis. For instance, Smorodinsky-Winternitz oscillators [11] or certain diffusion equations can be analysed via Lie-Hamilton systems on R 2 (see [5,6,17] and references therein).…”

“…Moreover, many works on Lie-Hamilton systems try to derive or to explain the existence of a Poisson bivector or symplectic form turning the elements of a VG Lie algebra into Hamiltonian vector fields [6,16]. This has been done by solving systems of partial differential equations (PDEs) [2] or using other algebraic and geometric techniques [6,4,16].…”