Drinfel'd doubles for (2+1)-gravity

TL;DR

This paper explicitly constructs and analyzes all Drinfel'd double structures for the (2+1)-dimensional anti-de Sitter and de Sitter Lie algebras, linking them to quantum symmetries in (2+1)-gravity.

Contribution

It provides a complete classification of Drinfel'd double structures for these algebras and connects them to quantum group symmetries relevant for (2+1)-gravity models.

Findings

Four structures lead to Drinfel'd doubles for the Poincaré algebra as the cosmological constant vanishes.

Each structure yields a classical r-matrix incorporating the cosmological constant as a deformation parameter.

The associated quantum groups are potential symmetries for quantized (2+1)-gravity and non-commutative spacetimes.

Abstract

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 Poincar\'e 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.