Our research is aimed to apply different algebraic and geometric structures such as Lie algebras and groups, representation theory, homogeneous spaces, spacetime symmetries, Hopf algebras, quantum groups, Lie bialgebras, Poisson-Lie groups, Poisson algebras and Poisson geometry, integrability theory, quantization techniques and noncommutative geometry, for the study of classical and quantum Hamiltonian systems, with special emphasis on their symmetry and integrability properties.
Nowadays, our principal research subjects are the following:
- Classical and quantum (super)integrable Hamiltonian systems on curved spaces.
- Bertrand´s theorem for spaces with nonconstant curvature.
- Hamiltonian structure and integrability properties of Lotka-Volterra systems.
- Integrable Hamiltonian systems with quantum group symmetry.
- Geometry and representation theory of the symmetry groups of spacetimes.
- Contractions of Lie algebras, groups and quantum groups.
- Classification of quantum deformations of spacetime symmetries.
- Lie bialgebra quantization techniques.
- Non-commutative spacetimes with cosmological constant.
- Quantum groups and Poisson-Lie groups in (2+1) gravity.