A geometric approach to Lie systems: formalism of Poisson-Hopf deformations
The notion of quantum algebras is merged with that of Lie systems in order to establish a new formalism called Poisson–Hopf algebra deformations of Lie systems. The procedure can be naturally applied to Lie systems endowed with a symplectic structure, the so-called Lie–Hamilton systems.This is quite...
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| Tipo de recurso: | tesis doctoral |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Universidad Complutense de Madrid (UCM) |
| Repositorio: | Docta Complutense |
| Idioma: | inglés |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/11559 |
| Acceso en línea: | https://hdl.handle.net/20.500.14352/11559 |
| Access Level: | acceso abierto |
| Palabra clave: | 512.554.3(043.2) Lie algebras Álgebras de Lie Álgebra 1201 Álgebra |
| Sumario: | The notion of quantum algebras is merged with that of Lie systems in order to establish a new formalism called Poisson–Hopf algebra deformations of Lie systems. The procedure can be naturally applied to Lie systems endowed with a symplectic structure, the so-called Lie–Hamilton systems.This is quite a general approach, as it can be applied to any quantum deformation and any underlying manifold. One of its main features is that, under quantum deformations, Lie systems are extended to generalized systems described by involutive distributions. As a consequence, a quantum deformed Lie system no longer has an underlying Vessiot–Guldberg Lie algebra or a quantum algebra one, but keeps a Poisson–Hopf algebra structure that enables us to obtain, in an explicit way, the t-independent constants of the motion from quantum deformed Casimir invariants, which are potentially useful in a further construction of the generalized notion of superposition rules. We illustrate this approach by considering the non-standard quantum deformation of sl(2) applied to well-known Lie systems, such as the oscillator problem or Milne–Pinney equation, as well as several types of Riccati equations. In this way, we obtain their new generalized (deformed) counterparts that cover, in particular, a new oscillator system with a time-dependent frequency and a position-dependent mass... |
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