Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries

TL;DR

This paper explores the mathematical structures of noncommutative Minkowski spacetimes through Lie bialgebras, revealing new hybrid gauge symmetries and noncommutative geometries that extend traditional symmetry frameworks.

Contribution

It classifies all Lie bialgebra structures on centrally-extended Poincaré and (A)dS algebras compatible with gauge symmetries, introducing novel hybrid noncommutative spacetime models.

Findings

Classified all coisotropic Lie bialgebra structures on extended Poincaré and (A)dS algebras.

Identified multiple types of hybrid noncommutative geometries involving spacetime and gauge coordinates.

Connected noncommutative spacetime structures to quantum homogeneous spaces of quantum Poincaré algebra.

Abstract

We study the Lie bialgebra structures that can be built on the one-dimensional central extension of the Poincar\'e and (A)dS algebras in (1+1) dimensions. These central extensions admit more than one interpretation, but the simplest one is that they describe the symmetries of (the noncommutative deformation of) an Abelian gauge theory, $U(1)$ or $SO(2)$ on the (1+1) dimensional Minkowski or (A)dS spacetime. We show that this highlights the possibility that the algebra of functions on the gauge bundle becomes noncommutative. This is a new way in which the Coleman-Mandula theorem could be circumvented by noncommutative structures, and it is related to a mixing of spacetime and gauge symmetry generators when they act on tensor-product states. We obtain all Lie bialgebra structures on centrally-extended Poincar\'e and (A)dS which are coisotropic w.r.t. the Lorentz algebra, and therefore all of them admit the construction of a noncommutative principal gauge bundle on a quantum homogeneous Minkowski spacetime. It is shown that several different types of hybrid noncommutativity between the spacetime and gauge coordinates are allowed by introducing quantum extended Poincar\'e symmetries. In one of these cases, an alternative interpretation of the central extension leads to a new description of the well-known canonical noncommutative spacetime as the quantum homogeneous space of a quantum Poincar\'e algebra of symmetries.