Speaker:** Iván Gutiérrez-Sagredo (University of Burgos)**

Date and time: October 23, 14:15

Place: Mathematics Department, FAU Erlangen-Nürnberg

# Author: angballesteros

# Seminar “Some new aspects about symmetries for central forces”

Speaker:** Stephen Anco (Brock University, Canada)**

Date and time: October 11, 17:30

Place: Seminario del Departamento de Física, Facultad de Ciencias

# Poisson-Hopf algebra deformations of Lie-Hamilton systems

In this paper (**arXiv:1708.08185**) 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.

# Curved momentum spaces from quantum groups with cosmological constant

In this paper (**arXiv:1707.09600**) we bring the concept that quantum symmetries describe theories with nontrivial momentum space properties one step further, looking at quantum symmetries of spacetime in presence of a nonvanishing cosmological constant Λ. In particular, the momentum space associated to the κ-deformation of the de Sitter algebra in (1+1) and (2+1) dimensions is explicitly constructed as a dual Poisson-Lie group manifold parametrized by Λ. Such momentum space includes both the momenta associated to spacetime translations and the `hyperbolic’ momenta associated to boost transformations, and has the geometry of (half of) a de Sitter manifold. Known results for the momentum space of the κ-Poincaré algebra are smoothly recovered in the limit Λ→0, where hyperbolic momenta decouple from translational momenta. The approach here presented is general and can be applied to other quantum deformations of kinematical symmetries, including (3+1)-dimensional ones.

# Seminar “Superintegrable Classical and Quantum Lissajous Systems on the Sphere”

Speaker:** Sengul Kuru (University of Ankara)**

Date and time: May 29, 12:30

Place: Aula 14, Facultad de Ciencias

# Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti-)de Sitter groups

The κ-deformation of the (2+1)D anti-de Sitter, Poincaré and de Sitter groups is presented through a unified approach in which the curvature of the spacetime (or the cosmological constant) is considered as an explicit parameter. The Drinfel’d-double and the Poisson-Lie structure underlying the κ-deformation are explicitly given, and the three quantum kinematical groups are obtained as quantizations of such Poisson-Lie algebras. As a consequence, the non-commutative (2+1)D spacetimes that generalize the κ-Minkowski space to the (anti-)de Sitter ones are obtained. Moreover, noncommutative 4D spaces of (time-like) geodesics can be defined, and they can be interpreted as a novel possibility to introduce non-commutative worldlines. Furthermore, quantum (anti-)de Sitter algebras are presented both in the known basis related with 2+1 quantum gravity and in a new one which generalizes the bicrossproduct one. In this framework, the quantum deformation parameter is related with the Planck length, and the existence of a kind of “duality” between the cosmological constant and the Planck scale is also envisaged. This paper (**arXiv:hep-th/0401244**) is an updated review version of a 2004 manuscript with the same title and authors.

# Quantum groups and noncommutative spacetimes with cosmological constant

Noncommutative spacetimes are widely believed to model some properties of the quantum structure of spacetime at the Planck regime. In this contribution (**arXiv:1702.04704**) the construction of (anti-)de Sitter noncommutative spacetimes obtained through quantum groups is reviewed. In this approach the quantum deformation parameter z is related to a Planck scale, and the cosmological constant Λ plays the role of a second deformation parameter of geometric nature, whose limit Λ→0 provides the corresponding noncommutative Minkowski spacetimes.