Drinfel'd double structures for Poincaré and Euclidean groups

Abstract

All non-isomorphic three-dimensional Poisson homogeneous Euclidean spaces are constructed and analyzed, based on the classification of coboundary Lie bialgebra structures of the Euclidean group in 3-dimensions, and the only Drinfel'd double structure for this group is explicitly given. The similar construction for the Poincaré case is reviewed and the striking differences between the Lorentzian and Euclidean cases are underlined. Finally, the contraction scheme starting from Drinfel'd double structures of the $\mathfrak{so}(3,1)$ Lie algebra is presented.

arXiv comment12 pages. Based on the contribution presented at "The 32nd International Colloquium on Group Theoretical Methods in Physics" (Group32), July 9-13, 2018

References in corpus (16)

• Combinatorial quantization of the Hamiltonian Chern-Simons theory I
• Quantum Mechanics of a Point Particle in 2+1 Dimensional Gravity
• Symplectic structure of the moduli space of flat connections on a Riemann surface
• Two dimensional lattice gauge theory based on a quantum group
• Noncommutative geometry of angular momentum space U(su(2))
• Three Dimensional Quantum Geometry and Deformed Poincare Symmetry
• Poisson structure and symmetry in the Chern-Simons formulation of (2+1)-dimensional gravity
• Topological field theory and the quantum double of SU(2)
• Noncommutative Model with Spontaneous Time Generation and Planckian bound
• Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations
• Quantum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o^*(4) symmetries in unified o(4;C) setting
• Drinfel'd doubles for (2+1)-gravity
• A (2+1) non-commutative Drinfel'd double spacetime with cosmological constant
• Quantum deformations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak{o}}^{\star }(4)$ symmetries in unified $\mathfrak{o}(4;\mathbb{C})$ setting -- Addendum
• The Poincaré group as a Drinfel'd double
• Quantum groups and noncommutative spacetimes with cosmological constant

Cited by in corpus (3)

• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• Quantum $D=3$ Euclidean and Poincaré symmetries from contraction limits
• The Quantum Gravity Disk: Discrete Current Algebra