Quantum double and $\kappa$-Poincar\'e symmetries in (2+1)-gravity and Chern-Simons theory

Abstract

We review the role of Drinfeld doubles and kappa-Poincare symmetries in quantised (2+1)-gravity and Chern-Simons theory. We discuss the conditions under which a given Hopf algebra symmetry is compatible with a Chern-Simons theory and determine this compatibility explicitly for the Drinfeld doubles and kappa-Poincare symmetries associated with the isometry groups of (2+1)-gravity. In particular, we explain that the usual kappa-Poincare symmetries with a timelike deformation are not directly associated with (2+1)-gravity. These kappa-Poincare symmetries are linked to Chern-Simons theory only in the de Sitter case, and the relevant Chern-Simons theory is physically inequivalent to (2+1)-gravity.