Un enfoque geométrico a los sistemas de Lie : formalismo de las deformaciones de álgebras de Poisson–Hopf

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 22-01-2021The 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...La noción de álgebras cuánticas se fusiona con la de sistemas de Lie para establecer un nuevo formalismo, las deformaciones del álgebra de Poisson–Hopf de los sistemas de Lie. El procedimiento puede aplicarse a sistemas de Lie dotados de una estructura simpléctica, los denominados sistemas de Lie–Hamilton. Este es un enfoque bastante general, ya que se puede aplicar a cualquier deformación cuántica y a cualquier variedad subyacente. Una de sus principales características es que, bajo deformaciones cuánticas, los sistemas de Lie se extienden a distribuciones involutivas generalizadas. Como consecuencia, un sistema de Lie deformado cuánticamente ya no tiene un álgebra de Vessiot–Guldberg Lie subyacente o un álgebra cuántica, sino que mantiene una estructura de álgebra de Poisson–Hopf que permite obtener, de manera explícita, las constantes del movimientot-independientes a partir de los invariantes de Casimir deformados, que son potencialmente útiles en una construcción adicional de la noción generalizada de reglas de superposición. Ilustramos este enfoque considerando la deformación cuántica no estándar de sl(2) aplicada a sistemas de Lie conocidos, como el problema del oscilador o la ecuación de Milne–Pinney, así como varios tipos de ecuaciones de Riccati. De esta manera, se obtienen sus análogos generalizados (deformados) quedan lugar, en particular, a un nuevo sistema de tipo oscilatorio con una frecuencia dependiente del tiempo y una masa dependiente de la posición...Fac. de Ciencias MatemáticasTRUEunpu

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