Quantum $D = 3$ Euclidean and Poincar\'{e} symmetries from contraction limits

TL;DR

This paper classifies quantum deformations of 3D Euclidean and Poincaré symmetries via contraction limits, connecting them to known models in 3D quantum gravity and expanding the understanding of their algebraic structures.

Contribution

It provides a comprehensive analysis of inhomogeneous D=3 quantum contractions, linking them to existing Euclidean and Poincaré quantum deformations and exploring their relevance in quantum gravity models.

Findings

IW contractions recover all Stachura deformations.

The results apply to 3D quantum gravity models with and without cosmological constant.

Some deformations are associated with Drinfeld double structures.

Abstract

Following the recently obtained complete classification of quantum-deformed $\mathfrak{o}(4)$, $\mathfrak{o}(3,1)$ and $\mathfrak{o}(2,2)$ algebras, characterized by classical $r$-matrices, we study their inhomogeneous $D = 3$ quantum IW contractions (i.e. the limit of vanishing cosmological constant), with Euclidean or Lorentzian signature. Subsequently, we compare our results with the complete list of $D = 3$ inhomogeneous Euclidean and $D = 3$ Poincar\'{e} quantum deformations obtained by P.~Stachura. It turns out that the IW contractions allow us to recover all Stachura deformations. We further discuss the applicability of our results in the models of 3D quantum gravity in the Chern-Simons formulation (both with and without the cosmological constant), where it is known that the relevant quantum deformations should satisfy the Fock-Rosly conditions. The latter deformations in part of the cases are associated with the Drinfeld double structures, which also have been recently investigated in detail.