Derivatives, forms and vector fields on the kappa-deformed Euclidean space

TL;DR

This paper explores the structure of derivatives, forms, and vector fields on the kappa-deformed Euclidean space, revealing a differential calculus with unique properties that differ from conventional approaches, highlighting the emergence of derivative-valued quantities.

Contribution

It introduces new differential calculi on kappa-deformed Euclidean space with a consistent set of derivatives, forms, and vector fields, emphasizing their finite dimensionality and derivative-valued nature.

Findings

Developed a differential calculus with two sets of one-forms.

Constructed transformation laws for vector fields consistent with derivatives.

Showed that the number of derivatives and forms equals the spacetime dimension.

Abstract

The model of kappa-deformed space is an interesting example of a noncommutative space, since it allows a deformed symmetry. In this paper we present new results concerning different sets of derivatives on the coordinate algebra of kappa-deformed Euclidean space. We introduce a differential calculus with two interesting sets of one-forms and higher-order forms. The transformation law of vector fields is constructed in accordance with the transformation behaviour of derivatives. The crucial property of the different derivatives, forms and vector fields is that in an n-dimensional spacetime there are always n of them. This is the key difference with respect to conventional approaches, in which the differential calculus is (n+1)-dimensional. This work shows that derivative-valued quantities such as derivative-valued vector fields appear in a generic way on noncommutative spaces.