Differential calculus on q-Minkowski space

TL;DR

This paper presents a new approach to non-commutative calculus on q-Minkowski space using reflection equations, clarifying algebraic structures and invariance properties under the q-Lorentz group.

Contribution

It introduces a reflection equation-based framework for differential calculus on q-Minkowski space, addressing ambiguities and establishing algebraic relations among generators.

Findings

Derived commutation relations for coordinates, derivatives, and forms.

Compared new algebraic structures with previous results.

Discussed invariance under q-Lorentz group actions.

Abstract

We wish to report here on a recent approach to the non-commutative calculus on $q$-Minkowski space which is based on the reflection equations with no spectral parameter. These are considered as the expression of the invariance (under the coaction of the $q$-Lorentz group) of the commutation properties which define the different $q$-Minkowski algebras. This approach also allows us to discuss the possible ambiguities in the definition of $q$-Minkowski space ${\cal M}_q$ and its differential calculus. The commutation relations among the generators of ${\cal M}_q$ (coordinates), ${\cal D}_q$ (derivatives), $\Lambda_q$ (one-forms) and a few invariant (scalar) operators are established and compared with earlier results.