**SUMMARY**:

This research project is focused on the applications of the theory of quantum groups, Poisson-Lie groups and Lie symmetries in the fields of mathematical quantum gravity and of classical and quantum integrable systems.

The three main scientific objectives of the project are the following:

- To perform a systematic study of the role of quantum groups and Poisson-Lie symmetries in (2+1) and (3+1) mathematical quantum gravity. In particular, to classify, construct and analyse the applications in this context of the quantum Lorentzian isometry groups and their associated noncommutative spacetimes.
- To apply Hopf algebra and Poisson-Lie techniques in order to obtain and solve new classical and quantum integrable systems on curved spaces, including pseudoRiemannian spaces with non constant curvature, and to explore their applications in nanophysics.
- To apply Poisson-Hopf algebra and Lie symmetry techniques to the integrability problem for certain first order ODEs, including non-autonomous and coupled systems, and to contribute with them to new developments in the theory of superposition rules for Lie-Hamilton systems.

**RESEARCH GROUP**:

Ángel Ballesteros, Alfonso Blasco, Francisco J. Herranz, Pedro Naranjo (U. of Burgos)

Decio Levi, Fabio Musso, Orlando Ragnisco (Roma Tre University)

Rutwig Campoamor-Stursberg (U. Complutense)

Rafael Hernández Heredero (U. Politécnica of Madrid)

Catherine Meusburger (FAU Erlangen-Nürnberg)

Danilo Riglioni (CRM, U. Montréal)

**SUPPORTED BY**:

Ministerio de Economía y Competitividad (Spain), Project MTM2013-43820-P

The Project is running from 01/01/2014 until 31/12/2016