Noncommutative Translations and $\star$-Product Formalism

TL;DR

This paper extends the $ abla$-product formalism to noncommutative space-times with Lie-algebraic structures, enabling representation of noncommutative translations via commutative ones and analyzing invariance properties.

Contribution

It introduces an extended $ abla$-product framework for noncommutative translations in Lie-algebraic noncommutative space-times, preserving translational invariance.

Findings

Translational invariance of noncommutative bilinear actions established.

Extension of $ abla$-product to Lie-algebraic noncommutative spaces.

Brief discussion on quadratic noncommutativity effects.

Abstract

We consider the noncommutative space-times with Lie-algebraic noncommutativity (e.g. $\kappa$-deformed Minkowski space). In the framework with classical fields we extend the $\star$-product in order to represent the noncommutative translations in terms of commutative ones. We show the translational invariance of noncommutative bilinear action with local product of noncommutative fields. The quadratic noncommutativity is also briefly discussed.