The Poincar\'e group as a Drinfel'd double

TL;DR

This paper explicitly constructs all Drinfel'd double structures for the Poincaré group in (2+1) dimensions, analyzes their associated r-matrices, and explores their implications for noncommutative spacetimes and Poisson homogeneous spaces.

Contribution

It provides a complete classification of Drinfel'd double structures for (2+1) Poincaré groups and links these to noncommutative geometries and quantum group structures.

Findings

Eight DD structures for (2+1) Poincaré group are constructed.

Two DD structures for (1+1) Poincaré with central extension are identified.

Some r-matrices lead to noncommutative Minkowski spacetimes.

Abstract

The eight nonisomorphic Drinfel'd double (DD) structures for the Poincar\'e Lie group in (2+1) dimensions are explicitly constructed in the kinematical basis. Also, the two existing DD structures for a non-trivial central extension of the (1+1) Poincar\'e group are also identified and constructed, while in (3+1) dimensions no Poincar\'e DD structure does exist. Each of the DD structures here presented has an associated canonical quasitriangular Poincar\'e $r$-matrix whose properties are analysed. Some of these $r$-matrices give rise to coisotropic Poisson homogeneous spaces with respect to the Lorentz subgroup, and their associated Poisson Minkowski spacetimes are constructed. Two of these (2+1) noncommutative DD Minkowski spacetimes turn out to be quotients by a Lorentz Poisson subgroup: the first one corresponds to the double of $\mathfrak{sl}(2)$ with trivial Lie bialgebra structure, and the second one gives rise to a quadratic noncommutative Poisson Minkowski spacetime. With these results, the explicit construction of DD structures for all Lorentzian kinematical groups in (1+1) and (2+1) dimensions is completed, and the connection between (anti-)de Sitter and Poincar\'e $r$-matrices through the vanishing cosmological constant limit is also analysed.