Noncommutative Differential Calculus on the Kappa-Minkowski Space

TL;DR

This paper explores differential calculus on the noncommutative $ppa$-Minkowski space, revealing the existence of a unique 5D Lorentz covariant calculus and analyzing simpler 2D models.

Contribution

It demonstrates the non-existence of 4D bicovariant Lorentz covariant calculi and introduces a 5D calculus that satisfies these properties.

Findings

No 4D bicovariant Lorentz covariant calculus exists.

A 5D differential calculus is compatible with Lorentz covariance.

Analysis of 2D $ppa$-Minkowski space models.

Abstract

Following the construction of the $\kappa$-Minkowski space from the bicrossproduct structure of the $\kappa$-Poincare group, we investigate possible differential calculi on this noncommutative space. We discuss then the action of the Lorentz quantum algebra and prove that there are no 4D bicovariant differential calculi, which are Lorentz covariant. We show, however, that there exist a five-dimensional differential calculus, which satisfies both requirements. We study also a toy example of 2D $\kappa$-Minkowski space and and we briefly discuss the main properties of its differential calculi.