Noncommutative Differential Forms on the kappa-deformed Space

TL;DR

This paper develops a differential calculus framework on kappa-deformed space, introducing explicit forms, derivatives, and star-products, with applications to noncommutative geometry.

Contribution

It constructs a family of differential forms and derivatives on kappa-deformed space, providing explicit formulas and analyzing their algebraic properties.

Findings

Explicit expressions for exterior derivatives and forms in various realizations

Star-product of differential forms is well-defined under certain conditions

Exterior derivative satisfies undeformed Leibniz rule in specific realizations

Abstract

We construct a differential algebra of forms on the kappa-deformed space. For a given realization of the noncommutative coordinates as formal power series in the Weyl algebra we find an infinite family of one-forms and nilpotent exterior derivatives. We derive explicit expressions for the exterior derivative and one-forms in covariant and noncovariant realizations. We also introduce higher-order forms and show that the exterior derivative satisfies the graded Leibniz rule. The differential forms are generally not graded-commutative, but they satisfy the graded Jacobi identity. We also consider the star-product of classical differential forms. The star-product is well-defined if the commutator between the noncommutative coordinates and one-forms is closed in the space of one-forms alone. In addition, we show that in certain realizations the exterior derivative acting on the star-product satisfies the undeformed Leibniz rule.