Quantum $D=3$ Euclidean and Poincaré symmetries from contraction limits

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 $D=3$ 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 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.

arXiv comment28 pages, v2 analysis of applicability of r-matrices in 3D gravity with $\Lambda \ne 0$ in Subsec. VI.C, generalization of procedure of quantum IW contractions in Subsec. IV.E, some conventions changed, redundant formulae removed, references added

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• Spectral dimensions and dimension spectra of quantum spacetimes
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• Quantum symmetries in 2+1 dimensions: Carroll, (a)dS-Carroll, Galilei and (a)dS-Galilei
• BMS algebras in 4 and 3 dimensions, their quantum deformations and duals
• Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups
• On Carrollian and Galilean contractions of BMS algebra in 3 and 4 dimensions
• 3-dimensional $\Lambda$-BMS Symmetry and its Deformations