Drinfel’d doubles for (2+1)-gravity

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. PACS: 02.20.Uw 04.60.-m