The main research subjects of this Project are quantum groups, Poisson-Lie groups, homogeneous spaces and their applications in the fields of quantum gravity, quantum communication theory and classical and quantum integrable systems. The three main scientific objectives of the Project are the following:

A. To obtain new results in the theory of Poisson homogeneous spaces, kinematical quantum groups and non-commutative space-times, both Lorentzian and Galilean ones, and to develope applications of the latter as models for space-times in quantum gravity. Non- commutative space-times of geodesics will be also constructed, and their covariance properties under quantum kinematical groups will be studied.

B. To study applications of Hopf algebras and quantum kinematical groups as groups of inertial transformations for quantum reference frames in a context of quantum communication theory, as well as to apply Hopf algebras constructed from quantum doubles in order to construct new Kitaev-type models for topological quantum computation.

C. To develope new applications of Poisson-Lie groups and homogenepus spaces in the field of classical and quantum integrabe systems, including the theory of nonlinear integrable equations, Lie-Hamilton systems and quantum integrable systems on curved surfaces, among them those ones which are suitable to model bidimensional nanostructures.