In this new paper (arXiv:1303.3080) in collaboration with C. Meusburger, all possible Drinfel’d double structures for the anti-de Sitter Lie algebra so(2,2) and de Sitter Lie algebra so(3,1) in (2+1)-dimensions are explicitly constructed and analysed in terms of a kinematical basis adapted to (2+1)-gravity. Each of these structures provides in a canonical way a pairing among the (anti-)de Sitter generators, as well as a specific classical r-matrix, and the cosmological constant is included in them as a deformation parameter. It is shown that four of these structures give rise to a Drinfel’d double structure for the Poincaré algebra iso(2,1) in the limit where the cosmological constant tends to zero. We explain how these Drinfel’d double structures are adapted to (2+1)-gravity, and we show that the associated quantum groups are natural candidates for the quantum group symmetries of quantised (2+1)-gravity models and their associated non-commutative spacetimes.

In this paper (arXiv:1302.0684) the quantum deformations of (anti-)de Sitter  (A)dS algebras in (2+1) dimensions are revisited. In particular, the classification problem of (2+1) (A)dS Lie bialgebras is presented and the associated noncommutative quantum (A)dS spaces are also analysed. Moreover, the flat limit (or vanishing cosmological constant) of all these structures leading to (2+1) quantum Poincaré algebras is studied. Some results on the analogous (3+1) problem are sketched.

This new paper (arXiv:1212.3809) deals with a systematic computational approach for the explicit construction of any quantum Hopf algebra starting from the Lie bialgebra that gives the first-order deformation of the coproduct map. The procedure is based on the fact that any quantum algebra can be viewed as the quantization of a unique Poisson-Lie structure on the dual group.