On the semiduals of local isometry groups in 3d gravity

TL;DR

This paper explores the semiduals of local isometry groups in 3d gravity, constructing associated quantum groups and clarifying their physical interpretations related to cosmological and Planck scales.

Contribution

It introduces a method to derive quantum groups from 3d gravity symmetries using semidualisation, linking them to cosmological and Planck scale exchanges.

Findings

Constructs bicrossproduct quantum groups from 3d gravity isometry groups.

Identifies quantum doubles and kappa-Poincare algebras with various deformation parameters.

Provides interpretation of semiduality as exchange of cosmological length and Planck mass.

Abstract

We use factorisations of the local isometry groups arising in 3d gravity for Lorentzian and Euclidean signatures and any value of the cosmological constant to construct associated bicrossproduct quantum groups via semidualisation. In this way we obtain quantum doubles of the Lorentz and rotation groups in 3d, as well as kappa-Poincare algebras whose associated r-matrices have spacelike, timelike and lightlike deformation parameters. We confirm and elaborate the interpretation of semiduality proposed in [13] as the exchange of the cosmological length scale and the Planck mass in the context of 3d quantum gravity. In particular, semiduality gives a simple understanding of why the quantum double of the Lorentz group and the kappa-Poincare algebra with spacelike deformation parameter are both associated to 3d gravity with vanishing cosmological constant, while the kappa-Poincare algebra with a timelike deformation parameter can only be associated to 3d gravity if one takes the Planck mass to be imaginary.