Quaternionic and Poisson-Lie structures in 3d gravity: the cosmological constant as deformation parameter

TL;DR

This paper unifies the understanding of 3d gravity's local isometry groups using quaternionic and Poisson-Lie structures, revealing how the cosmological constant acts as a deformation parameter affecting the group's algebraic and geometric properties.

Contribution

It proves that local isometry groups in 3d gravity can be viewed as unit (split) quaternions over a ring depending on the cosmological constant, providing a unified framework for their Poisson structures.

Findings

Most local isometry groups are equipped with a Poisson-Lie structure of a classical double.

Derived a simple description of symplectic leaves of SU(2) and SL(2,R).

Computed Poisson structures on dual groups and Heisenberg doubles relevant for 3d gravity phase space.

Abstract

Each of the local isometry groups arising in 3d gravity can be viewed as the group of unit (split) quaternions over a ring which depends on the cosmological constant. In this paper we explain and prove this statement, and use it as a unifying framework for studying Poisson structures associated with the local isometry groups. We show that, in all cases except for Euclidean signature with positive cosmological constant, the local isometry groups are equipped with the Poisson-Lie structure of a classical double. We calculate the dressing action of the factor groups on each other and find, amongst others, a simple and unified description of the symplectic leaves of SU(2) and SL(2,R). We also compute the Poisson structure on the dual Poisson-Lie groups of the local isometry groups and on their Heisenberg doubles; together, they determine the Poisson structure of the phase space of 3d gravity in the so-called combinatorial description.