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Drinfel’d doubles for \((2+1)\)-gravity. (English) Zbl 1273.83127

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.

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
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