## Abstract

We revisit the representation theory of the quantum double of the universal cover of the Lorentz group in 2 + 1 dimensions, motivated by its role as a deformed Poincaré symmetry and symmetry algebra in (2 + 1)-dimensional quantum gravity. We express the unitary irreducible representations in terms of covariant, infinite-component fields on curved momentum space satisfying algebraic spin and mass constraints. Adapting and applying the method of group Fourier transforms, we obtain covariant fields on (2 + 1)-dimensional Minkowski space which necessarily depend on an additional internal and circular dimension. The momentum space constraints turn into differential or exponentiated differential operators, and the group Fourier transform induces a star product on Minkowski space and the internal space which is essentially a version of Rieffel’s deformation quantisation via convolution.

Original language | English |
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Pages (from-to) | 1-39 |

Number of pages | 39 |

Journal | Letters in Mathematical Physics |

Early online date | 27 Dec 2018 |

DOIs | |

Publication status | E-pub ahead of print - 27 Dec 2018 |

## Keywords

- 3d quantum gravity
- Anyons
- Group Fourier transform
- Non-commutative spacetime
- Quantum double

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics