Hopf-algebra description of noncommutative-spacetime symmetries

TL;DR

This paper uses Hopf algebra to rigorously define and analyze the symmetries of noncommutative Minkowski spacetime, specifically kappaMinkowski, establishing a framework that preserves a relativistic invariant length scale.

Contribution

It introduces a Hopf-algebraic approach to characterize noncommutative spacetime symmetries, clarifying the role of the invariant length scale in kappaMinkowski.

Findings

Constructed a Poincare-like Hopf algebra with 10 generators for kappaMinkowski

Confirmed the necessity of Hopf algebra over Lie algebra for noncommutative symmetries

Clarified the role of the invariant length scale in noncommutative spacetime

Abstract

In the study of certain noncommutative versions of Minkowski spacetime there is still a large ambiguity concerning the characterization of their symmetries. Adopting as our case study the kappaMinkowski noncommutative space-time, on which a large literature is already available, we propose a line of analysis of noncommutative-spacetime symmetries that relies on the introduction of a Weyl map (connecting a given function in the noncommutative Minkowski with a corresponding function in commutative Minkowski) and of a compatible notion of integration in the noncommutative spacetime. We confirm (and we establish more robustly) previous suggestions that the commutative-spacetime notion of Lie-algebra symmetries must be replaced, in the noncommutative-spacetime context, by the one of Hopf-algebra symmetries. We prove that in kappaMinkowski it is possible to construct an action which is invariant under a Poincare-like Hopf algebra of symmetries with 10 generators, in which the noncommutativity length scale has the role of relativistic invariant. The approach here adopted does leave one residual ambiguity, which pertains to the description of the translation generators, but our results, independently of this ambiguity, are sufficient to clarify that some recent studies (gr-qc/0212128 and hep-th/0301061), which argued for an operational indistiguishability between theories with and without a length-scale relativistic invariant, implicitly assumed that the underlying spacetime would be classical.