• DOI:10.1088/1751-8121/aaa090
• Corpus ID: 56408303

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}
}

• Á. Ballesteros, R. Campoamor-Stursberg, +2 authors J. de Lucas
• Published 28 August 2017
• Mathematics
• Journal of Physics A: Mathematical and Theoretical

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 Poisson–Hopf algebra of functions that allows for the explicit description of its t-independent constants of the motion from deformed Casimir functions. We illustrate our approach by considering the Poisson–Hopf algebra analogue of the non-standard quantum deformation of sl(2) and its applications to deform well-known Lie–Hamilton systems describing oscillator systems, Milne–Pinney equations, and several types of Riccati equations. In particular, we obtain a new position-dependent mass oscillator system with a time-dependent frequency.