Ballesteros, Ángel; Herranz, Francisco J.; Meusburger, Catherine Drinfel’d doubles for \((2+1)\)-gravity. (English) Zbl 1273.83127 Classical Quantum Gravity 30, No. 15, Article ID 155012, 20 p. (2013). Summary: 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 when 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 quantized \((2+1)\)-gravity models and their associated non-commutative spacetimes. Cited in 9 Documents MSC: 83C80 Analogues of general relativity in lower dimensions 17B45 Lie algebras of linear algebraic groups 83C45 Quantization of the gravitational field 83C65 Methods of noncommutative geometry in general relativity PDFBibTeX XMLCite \textit{Á. Ballesteros} et al., Classical Quantum Gravity 30, No. 15, Article ID 155012, 20 p. (2013; Zbl 1273.83127) Full Text: DOI arXiv