The $q$-linked complex Minkowski space, its real forms and deformed isometry groups

TL;DR

This paper explores the duality between real forms of quantum deformations of 4D orthogonal groups and their classical isometry groups, introducing a new $q$-linked deformation with associated isometry coactions.

Contribution

It introduces the $q$-linked deformation of classical 4D isometry groups and establishes their duality with quantum group real forms, extending prior classification work.

Findings

Established duality between quantum deformations and classical isometry groups.

Constructed the $q$-linked deformation for Euclidean, Kleinian, and Lorentzian spaces.

Defined coactions of the deformed isometry groups on the quantum spaces.

Abstract

We establish duality between real forms of the quantum deformation of the 4-dimensional orthogonal group studied by Fioresi et al. and the classification work made by Borowiec et al.. Classically these real forms are the isometry groups of $\mathbb{R}^4$ equipped with Euclidean, Kleinian or Lorentzian metric. A general deformation, named $q$-linked, of each of these spaces is then constructed, together with the coaction of the corresponding isometry group.