Quantum groups appeared during the eighties as the underlying algebraic symmetries of several two-dimensional integrable models. They are noncommutative generalizations of Lie groups endowed with a Hopf algebra structure, and the possibility of defining noncommutative spaces that are covariant under quantum group (co)actions soon provided a fruitful link with noncommutative geometry. At the same time, when quantum group analogues of the Lie groups of spacetime symmetries (Galilei, Poincare’ and (anti-) de Sitter) were constructed, they attracted the attention of quantum gravity researchers. In fact, they provided a possible mathematical framework to model the “quantum” geometry of space-time and the quantum deformations of its kinematical symmetries at the Planck scale, where nontrivial features are expected to arise because of the interplay between gravity and quantum theory.
This Special Issue is open to contributions dealing with any of the many facets of quantum group symmetry and their generalizations. On the more formal side, possible topics include the theory of Poisson-Lie groups and Poisson homogeneous spaces as the associated semiclassical objects; Hopf algebras; the classification of quantum groups and spaces, their representation theory and its connections with q-special functions; the construction of noncommutative differential calculi; and the theory of quantum bundles. On application side, possible topics are: classical and quantum integrable models with quantum group invariance; the applications of quantum groups in different (2+1) quantum gravity contexts (like combinatorial quantisation, state sum models or spin foams); and quantum kinematical groups and their noncommutative spacetimes in connection with deformed special relativity and quantum gravity phenomenology.
Prof. Angel Ballesteros
Dr. Giulia Gubitosi
Prof. Francisco J. Herranz