Based on a recently developed procedure to construct Poisson-Hopf deformations of Lie-Hamilton systems, a novel unified approach to nonequivalent deformations of Lie-Hamilton systems on the real plane with a Vessiot-Guldberg Lie algebra isomorphic to 𝔰𝔩(2) is proposed in the paper **arXiv:1803.07404**. This, in particular, allows us to define a notion of Poisson-Hopf systems in dependence of a parameterized family of Poisson algebra representations. Such an approach is explicitly illustrated by applying it to the three non-diffeomorphic classes of 𝔰𝔩(2) Lie-Hamilton systems. Our results cover deformations of the Ermakov system, Milne-Pinney, Kummer-Schwarz and several Riccati equations as well as of the harmonic oscillator (all of them with t-dependent coefficients). Furthermore t-independent constants of motion are given as well. Our methods can be employed to generate other Lie-Hamilton systems and their deformations for other Vessiot-Guldberg Lie algebras and their deformations.

# Seminar “On symmetry techniques for differential equations”

Speaker:** María Luz Gandarias (Universidad de Cádiz)**

Date and time: December 12, 17:00

Place: Seminario del Departamento de Física, Facultad de Ciencias

# Seminar “Quantum field theory on kappa-Minkowski spacetime”

Speaker:** Flavio Mercati (La Sapienza, Rome)**

Date and time: November 29, 17:00

Place: Seminario del Departamento de Física, Facultad de Ciencias

# Curved momentum spaces from quantum (Anti-)de Sitter groups in (3+1) dimensions

In this new paper (**arXiv:1711.05050**), curved momentum spaces associated to the κ-deformation of the (3+1) de Sitter and Anti-de Sitter algebras are constructed as orbits of suitable actions of the dual Poisson-Lie group associated to the κ-deformation with non-vanishing cosmological constant. The κ-de Sitter and κ-Anti-de Sitter curved momentum spaces are separately analysed, and they turn out to be, respectively, half of the (6+1)-dimensional de Sitter space and half of a space with SO(4,4) invariance. Such spaces are made of the momenta associated to spacetime translations and the “hyperbolic” momenta associated to boost transformations. The known κ-Poincaré curved momentum space is smoothly recovered as the vanishing cosmological constant limit from both of the constructions.

# Seminar “From Poisson geometry to non-commutative spacetime”

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

Date and time: October 23, 14:15

Place: Mathematics Department, FAU Erlangen-Nürnberg

# 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.