Kappa-Minkowski spacetime, Kappa-Poincar\'{e} Hopf algebra and realizations

TL;DR

This paper unifies kappa-Minkowski spacetime with Lorentz algebra using Lie algebra, constructs related deformed algebras and realizations, and explores their properties including star products, coproducts, and dualities.

Contribution

It introduces a unified framework for kappa-Minkowski spacetime and Lorentz algebra, providing explicit realizations, star products, and analyzing their algebraic and geometric properties.

Findings

Unified Lie algebra for kappa-Minkowski and Lorentz algebra.

Constructed all matrices specifying deformed algebras.

Analyzed properties of star product, coproduct, and dual realizations.

Abstract

We unify kappa-Minkowki spacetime and Lorentz algebra in unique Lie algebra. Introducing commutative momenta, a family of kappa-deformed Heisenberg algebras and kappa-deformed Poincare algebras are defined. They are specified by the matrix depending on momenta. We construct all such matrices. Realizations and star product are defined and analyzed in general and specially, their relation to coproduct of momenta is pointed out. Hopf algebra of the Poincare algebra, related to the covariant realization, is presented in unified covariant form. Left-right dual realizations and dual algebra are introduced and considered. The generalized involution and the star inner product are analyzed and their properties are discussed. Partial integration and deformed trace property are obtained in general. The translation invariance of the star product is pointed out. Finally, perturbative approach up to the first order in $a$ is presented in Appendix.