Covariant realizations of kappa-deformed space

TL;DR

This paper explores covariant realizations of a kappa-deformed space with a focus on algebraic structures, coproducts, and star products, providing new insights into their physical equivalence and applications to scalar fields.

Contribution

It introduces a comprehensive framework for covariant realizations of kappa-deformed space, including new relationships between coproducts and star products, and discusses their physical implications.

Findings

Multiple covariant realizations are constructed and shown to be physically equivalent.

New relationships between coproducts and star products are established.

Invariant integration and scalar fields are analyzed within the natural realization.

Abstract

We study a Lie algebra type $\kappa$-deformed space with undeformed rotation algebra and commutative vector-like Dirac derivatives in a covariant way. Space deformation depends on an arbitrary vector. Infinitely many covariant realizations in terms of commuting coordinates of undeformed space and their derivatives are constructed. The corresponding coproducts and star products are found and related in a new way. All covariant realizations are physically equivalent. Specially, a few simple realizations are found and discussed. The scalar fields, invariants and the notion of invariant integration is discussed in the natural realization.