Poisson–Hopf algebra deformations of Lie–Hamilton systems

Ballesteros, Ángel; Campoamor-Stursberg, Rutwig; Fernández‐Saiz, Eduardo; Herranz, Francisco J.; Lucas, J. de

doi:10.1088/1751-8121/aaa090

Cited by 14 publications

(55 citation statements)

References 70 publications

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“…However, the success of quantum groups [18,24] and the coalgebra formalism within the analysis of superintegrable systems [3,7,9], and the fact that quantum algebras appear as deformations of Lie algebras suggested the possibility of extending the notion and techniques of Lie-Hamilton systems beyond the range of application of the Lie theory. An approach in this direction was recently proposed in [4], where a method to construct quantum deformed Lie-Hamilton systems (LH systems in short) by means of the coalgebra formalism and quantum algebras was given.…”

Section: Introductionmentioning

confidence: 99%

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

Ballesteros

Campoamor-Stursberg

Fernández‐Saiz

et al. 2018

Springer Proceedings in Mathematics &Amp; Statistics

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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

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Section: Introductionmentioning

confidence: 99%

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

Ballesteros

Campoamor-Stursberg

Fernández‐Saiz

et al. 2018

Springer Proceedings in Mathematics &Amp; Statistics

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“…Now, we show the graphics for < ρ >= q(t) and σ 2 = 1/p 2 using the two particular solutions in Figure 2 and Figure 3 provided in the introduction. Notice that we have renamed c A Lie-Hamilton system admits a Poisson-Hopf deformation, see [11]. The interest of Poisson-Hopf deformations of these models resides in the fact that many of the outcome deformed systems happen to be other systems that are already identified in the physics and mathematics literature.…”

Section: The Sisf Model Is Lie-hamiltonianmentioning

confidence: 99%

“…To obtain a deformation of the Lie-Hamilton realization of the SISf model we make use of deformed Poisson-Hopf algebras. Following [11], we summarize the procedure for the planar systems as follows:…”

Section: The Poisson-hopf Algebra Deformations Of Lie-hamilton Systemsmentioning

confidence: 99%

Geometry and solutions of an epidemic SIS model permitting fluctuations and quantization

Esen¹,

Fernández‐Saiz²,

Sardón³

et al. 2020

Preprint

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Some recent works reveal that there are models of differential equations for the mean and variance of infected individuals that reproduce the SIS epidemic model at some point. This stochastic SIS epidemic model can be interpreted as a Hamiltonian system, therefore we wondered if it could be geometrically handled through the theory of Lie-Hamilton systems, and this happened to be the case. The primordial result is that we are able to obtain a general solution for the stochastic/ SIS-epidemic model (with fluctuations) in form of a nonlinear superposition rule that includes particular stochastic solutions and certain constants to be related to initial conditions of the contagion process. The choice of these initial conditions will be crucial to display the expected behavior of the curve of infections during the epidemic. We shall limit these constants to nonsingular regimes and display graphics of the behavior of the solutions. As one could expect, the increase of infected individuals follows a sigmoid-like curve.Lie-Hamiltonian systems admit a quantum deformation, so does the stochastic SIS-epidemic model. We present this generalization as well. If one wants to study the evolution of an SIS epidemic under the influence of a constant heat source (like centrally heated buildings), one can make use of quantum stochastic differential equations coming from the so-called quantum deformation.

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“…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).…”

Section: Introductionmentioning

confidence: 99%

“…, X r of the VG Lie algebra on R 2 under inspection. Let us also define J := X L 1 ∧ X L 2 , which is not necessarily a Poisson bivector as 3…”

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confidence: 99%

Geometric Models for Lie–Hamilton Systems on ℝ2

Lange

Lucas

2019

Mathematics

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This paper provides a geometric description for Lie-Hamilton systems on R 2 with locally transitive Vessiot-Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie-Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie-Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give rise to a natural framework for the analysis of Lie-Hamilton systems on R 2 while retrieving known results in a natural manner. Our methods may be extended to study Lie-Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures.

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Poisson–Hopf algebra deformations of Lie–Hamilton systems

Cited by 14 publications

References 70 publications

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

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

Geometry and solutions of an epidemic SIS model permitting fluctuations and quantization

Geometric Models for Lie–Hamilton Systems on ℝ2

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