Lessons from (2+1)-dimensional quantum gravity

Proposals that quantum gravity gives rise to non-commutative spacetime geometry and deformations of Poincare symmetry are examined in the context of (2+1)-dimensional quantum gravity. The results are expressed in five lessons, which summarise how the gravitational constant, Planck’s constant and the cosmological constant enter the non-commutative and non-cocommutative structures arising in (2+1)-dimensional quantum gravity. It is emphasised that the much studied bicrossproduct kappa-Poincare algebra does not arise directly in (2+1)-dimensional quantum gravity.