Star Product and Invariant Integration for Lie type Noncommutative Spacetimes

TL;DR

This paper develops a star product and invariant integration framework for Lie type noncommutative spacetimes, including canonical cases, with explicit examples like kappa-Minkowski and the Heisenberg algebra, enhancing mathematical tools for quantum spacetime models.

Contribution

Introduces a star product and invariant integral for Lie type noncommutative spaces, including a central generator and quasi-cyclicity property, with explicit examples.

Findings

Defined a star product compatible with translations.

Established an invariant integral with quasi-cyclicity.

Provided explicit examples for kappa-Minkowski and Heisenberg algebra.

Abstract

We present a star product for noncommutative spaces of Lie type, including the so called ``canonical'' case by introducing a central generator, which is compatible with translations and admits a simple, manageable definition of an invariant integral. A quasi-cyclicity property for the latter is shown to hold, which reduces to exact cyclicity when the adjoint representation of the underlying Lie algebra is traceless. Several explicit examples illuminate the formalism, dealing with kappa-Minkowski spacetime and the Heisenberg algebra (``canonical'' noncommutative 2-plane).