Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 079, 23 pages      arXiv:1403.3038      https://doi.org/10.3842/SIGMA.2014.079
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

Group Momentum Space and Hopf Algebra Symmetries of Point Particles Coupled to 2+1 Gravity

Michele Arzano a, Danilo Latini b and Matteo Lotito c
a) Dipartimento di Fisica and INFN, ''Sapienza'' University of Rome, P.le A. Moro 2, 00185 Roma, Italy
b) Dipartimento di Fisica and INFN, Università  di Roma Tre, Via Vasca Navale 84, I-00146 Roma, Italy
c) Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221-0011, USA

Received March 13, 2014, in final form July 15, 2014; Published online July 24, 2014

Abstract
We present an in-depth investigation of the ${\rm SL}(2,\mathbb{R})$ momentum space describing point particles coupled to Einstein gravity in three space-time dimensions. We introduce different sets of coordinates on the group manifold and discuss their properties under Lorentz transformations. In particular we show how a certain set of coordinates exhibits an upper bound on the energy under deformed Lorentz boosts which saturate at the Planck energy. We discuss how this deformed symmetry framework is generally described by a quantum deformation of the Poincaré group: the quantum double of ${\rm SL}(2,\mathbb{R})$. We then illustrate how the space of functions on the group manifold momentum space has a dual representation on a non-commutative space of coordinates via a (quantum) group Fourier transform. In this context we explore the connection between Weyl maps and different notions of (quantum) group Fourier transform appeared in the literature in the past years and establish relations between them.

Key words: $2+1$ gravity; Lie group momentum space; deformed symmetries; Hopf algebra.

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