Space-time from Symmetry: The Moyal Plane from the Poincare-Hopf Algebra

TL;DR

This paper derives the non-commutative Moyal plane structure from the deformation of the Poincare' group coproduct using Drinfel'd twist, revealing a deep connection between spacetime algebra and group symmetries.

Contribution

It introduces a method to obtain the Moyal plane from the twisted coproduct of the Poincare' group, highlighting the reciprocal influence between spacetime algebra and group symmetries.

Findings

Non-commutative product derived from Poincare' group deformation

Identification of spacetime algebra with invariant functions under Lorentz action

Extension involves cohomological features and reciprocal influence

Abstract

We show how to get a non-commutative product for functions on space-time starting from the deformation of the coproduct of the Poincare' group using the Drinfel'd twist. Thus it is easy to see that the commutative algebra of functions on space-time (R^4) can be identified as the set of functions on the Poincare' group invariant under the right action of the Lorentz group provided we use the standard coproduct for the Poincare' group. We obtain our results for the noncommutative Moyal plane by generalizing this result to the case of the twisted coproduct. This extension is not trivial and involves cohomological features. As is known, spacetime algebra fixes the coproduct on the dffeomorphism group of the manifold. We now see that the influence is reciprocal: they are strongly tied.