Quantum D = 3 Euclidean and Poincar\xe9 symmetries from contraction limits

Jerzy Kowalski-Glikman,Jerzy Lukierski,Tomasz Trześniewski

doi:10.1007/jhep09(2020)096

Abstract

Following the recently obtained complete classification of quantum-deformed mathfrak{o} (4), mathfrak{o} (1, 3) and mathfrak{o} (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 Poincaré 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 with- out 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.

Highlights

Symmetries, described in the language of Hopf algebras, especially the quantum Poincare algebras and quantum Poincare groups, as well as quantum versions of thede Sitter and conformal symmetries
Following the recently obtained complete classification of quantum-deformed o(4), o(3, 1) and o(2, 2) algebras, characterized by classical r-matrices, we study their inhomogeneous D = 3 quantum IW contractions, with Euclidean or Lorentzian signature
A classical r-matrix r is linear in the deformation parameters qi and determines the coboundary local isometry (Lie) bialgebra structure of the algebra g, with the coproduct of algebra elements g ∈ g given by the perturbative formula

Summary

Summary of the contraction results

As we already explained for complex r-matrices (2.15), (2.27) and (2.34), the most general quantum IW contraction limits for rII, rIII and rV are combinations of r-matrices of the type rand r. The relevant (Hermitian) r-matrices for positive cosmological constant Λ = R−2 > 0 are o(3, 1) r-matrices listed in subsection 4.3 but with all parameters set to be real (let us remind that the combinations γ − γand γ + γare actual parameters in the case of rIII/rIaII). They satisfy the following Yang-Baxter equations (the results are the same for rI and rIa, rIII and rIaII, etc.). O (2, 2) r-matrices satisfy the Yang-Baxter equations (again, the results are the same for rI and rIa, rIII and rIaII, etc.).

Drinfeld double r-matrices

Conclusions and outlook