# Poisson–Hopf algebra deformations of Lie–Hamilton systems

@article{Ballesteros2017PoissonHopfAD, title={Poisson–Hopf algebra deformations of Lie–Hamilton systems}, author={{\'A}ngel Ballesteros and Rutwig Campoamor-Stursberg and Eduardo Fern{\'a}ndez-Saiz and Francisco J. Herranz and Javier de Lucas}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2017}, volume={51} }

Hopf algebra deformations are merged with a class of Lie systems of Hamiltonian type, the so-called Lie–Hamilton systems, to devise a novel formalism: the Poisson–Hopf algebra deformations of Lie–Hamilton systems. This approach applies to any Hopf algebra deformation of any Lie–Hamilton system. Remarkably, a Hopf algebra deformation transforms a Lie–Hamilton system, whose dynamic is governed by a finite-dimensional Lie algebra of functions, into a non-Lie–Hamilton system associated with a…

## 9 Citations

### Poisson–Hopf deformations of Lie–Hamilton systems revisited: deformed superposition rules and applications to the oscillator algebra

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2021

The formalism for Poisson–Hopf (PH) deformations of Lie–Hamilton (LH) systems, recently proposed in Ballesteros Á et al (2018 J. Phys. A: Math. Theor. 51 065202), is refined in one of its crucial…

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

- Mathematics
- 2017

Based on a recently developed procedure to construct Poisson–Hopf deformations of Lie–Hamilton systems [9], a novel unified approach to nonequivalent deformations of Lie–Hamilton systems on the real…

### Geometric Models for Lie–Hamilton Systems on ℝ2

- MathematicsMathematics
- 2019

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…

### Multisymplectic structures and invariant tensors for Lie systems

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2019

A Lie system is a non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a…

### Quasi-Lie schemes for PDEs

- MathematicsInternational Journal of Geometric Methods in Modern Physics
- 2019

The theory of quasi-Lie systems, i.e. systems of first-order ordinary differential equations that can be related via a generalized flow to Lie systems, is extended to systems of partial differential…

### Reduction and reconstruction of multisymplectic Lie systems

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2022

A Lie system is a non-autonomous system of first-order ordinary differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional real Lie…

### Exact solutions and superposition rules for Hamiltonian systems generalizing stochastic SIS epidemic models with variable infection rates

- Mathematics
- 2023

Using the theory of Lie-Hamilton systems, formal generalized stochastic Hamiltonian systems that enlarge a recently proposed stochastic SIS epidemic model with a variable infection rate are…

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

- Mathematics
- 2020

A general solution is obtained 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 that will be crucial to display the expected behavior of the curve of infections during the epidemic.

### A generalization of a SIS epidemic model with fluctuations

- MathematicsMathematical Methods in the Applied Sciences
- 2021

In a recent paper (Nakamura and Martinez, 2019), the classical epidemic compartmental susceptible‐infectious‐susceptible (SIS) model has been upgraded to a form which permits fluctuations in terms of…

## 93 References

### Lie–Hamilton systems on curved spaces: a geometrical approach

- Mathematics
- 2016

A Lie–Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional Lie…

### Poisson–Lie groups, bi-Hamiltonian systems and integrable deformations

- Mathematics
- 2016

Given a Lie–Poisson completely integrable bi-Hamiltonian system on Rn, we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation…

### From constants of motion to superposition rules for Lie–Hamilton systems

- Mathematics
- 2013

A Lie system is a non-autonomous system of first-order differential equations possessing a superposition rule, i.e. a map expressing its general solution in terms of a generic finite family of…

### Lie symmetries for Lie systems: Applications to systems of ODEs and PDEs

- MathematicsAppl. Math. Comput.
- 2016

### Quantum algebras as quantizations of dual Poisson–Lie groups

- Mathematics
- 2013

A systematic computational approach for the explicit construction of any quantum Hopf algebra (Uz(g), Δz) starting from the Lie bialgebra (g, δ) that gives the first-order deformation of the…

### Non-coboundary Poisson–Lie structures on the book group

- Mathematics
- 2012

All possible Poisson–Lie (PL) structures on the 3D real Lie group generated by a dilation and two commuting translations are obtained. Their classification is fully performed by relating these PL…