Star product realizations of kappa-Minkowski 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 $\kappa$-Poincaré algebra as an involutive Hopf algebra on this representation of $\kappa$-Minkowski space and initiate a study of its properties.

35 Pages

Cited by in corpus (25)

• Localization and Reference Frames in $\kappa$-Minkowski Spacetime
• The $\kappa$-(A)dS noncommutative spacetime
• $\kappa$-Poincaré invariant quantum field theories with KMS weight
• Toward the classification of differential calculi on $\kappa$-Minkowski space and related field theories
• Gauge theories on quantum spaces
• $\kappa$-Poincaré invariant orientable field theories at 1-loop
• Gauge theories on $\kappa$-Minkowski spaces: Twist and modular operators
• $k$-Minkowski-deformation of $U(1)$ gauge theory
• Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries
• $\kappa$-Minkowski star product in any dimension from symplectic realization
• Spectral dimensions and dimension spectra of quantum spacetimes
• Single Extra Dimension from $\kappa$-Poincaré and Gauge Invariance
• Involutive representations of coordinate algebras and quantum spaces
• Vacuum energy and the cosmological constant problem in $\kappa$-Poincaré invariant field theories
• Twisted BRST symmetry in gauge theories on $\kappa$-Minkowski
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• A short introduction to $\kappa$-deformation
• Lie-Poisson gauge theories and $\kappa$-Minkowski electrodynamics
• Quantum instability of gauge theories on $\kappa$-Minkowski space
• Field theories on $\rho$-deformed Minkowski space-time
• Fuzzy de Sitter Space from kappa-Minkowski Space in Matrix Basis
• The Weyl-Mellin quantization map for $\kappa$-Minkowski Noncommutative Spacetime
• Quantum causality constraints on kappa-Minkowski space-time
• Quantum causality in $\kappa$-Minkowski and related constraints
• Star-products for Lie-algebraic noncommutative Minkowski space-times