Star product realizations of kappa-Minkowski space

TL;DR

This paper introduces new star products for $$-Minkowski space, demonstrating their algebraic properties, extensions to smooth functions, and explicit realization of the $$-Poincare9 algebra acting on this space.

Contribution

It defines a family of star products and involutions for $$-Minkowski space, showing their algebraic structures and explicit realizations of the $$-Poincare9 algebra.

Findings

Star products have isomorphic Banach algebra completions.

Certain star products extend to polynomially bounded smooth functions.

Explicit realization of $$-Poincare9 algebra action on the space.

Abstract

We define a family of star products and involutions associated with $\kappa$-Minkowski space. Applying corresponding quantization maps we show that these star products restricted to a certain space of Schwartz functions have isomorphic Banach algebra completions. For two particular star products it is demonstrated that they can be extended to a class of polynomially bounded smooth functions allowing a realization of the full Hopf algebra structure on $\kappa$-Minkowski space. Furthermore, we give an explicit realization of the action of the $\kappa$-Poincar\'e algebra as an involutive Hopf algebra on this representation of $\kappa$-Minkowski space and initiate a study of its properties.