Space-Time Symmetries of Noncommutative Spaces

TL;DR

This paper introduces a noncommutative Lorentz symmetry for canonical noncommutative spaces, demonstrating invariance of the star product and related actions under these transformations, while maintaining an undeformed metric.

Contribution

It defines a noncommutative Lorentz symmetry and proves invariance of key structures, extending the understanding of symmetries in noncommutative geometry.

Findings

Star product is invariant under noncommutative Lorentz transformations

Actions expanded via Seiberg-Witten maps are invariant under these transformations

The metric remains undeformed under general coordinate transformations

Abstract

We define a noncommutative Lorentz symmetry for canonical noncommutative spaces. The noncommutative vector fields and the derivatives transform under a deformed Lorentz transformation. We show that the star product is invariant under noncommutative Lorentz transformations. We then apply our idea to the case of actions obtained by expanding the star product and the fields taken in the enveloping algebra via the Seiberg-Witten maps and verify that these actions are invariant under these new noncommutative Lorentz transformations. We finally consider general coordinate transformations and show that the metric is undeformed.