Hopf images of coactions and effective symmetry of quantum principal bundles

TL;DR

This paper introduces Hopf images of coactions to analyze the effective quantum symmetry of principal bundles, providing a classification and rigidity results for quantum principal bundles with applications to quantum groups.

Contribution

It develops the concept of Hopf images of coactions and applies it to classify quantum principal bundles by their effective symmetry, including a rigidity theorem.

Findings

Every quantum principal bundle admits a reduction to an inner-faithful quantum symmetry.

The classification of quantum principal bundles up to effective symmetry is established.

Examples from quantum groups illustrate the theory.

Abstract

We introduce Hopf images of coactions of Hopf algebras and develop their role in the geometry of quantum principal bundles. Assuming cosemisimplicity of the structure Hopf algebra, we show that every quantum principal bundle equipped with a right-covariant first-order differential calculus admits a canonical and functorial reduction to one with inner-faithful quantum symmetry. This yields a classification of quantum principal bundles up to effective quantum symmetry and a rigidity result identifying the minimal effective symmetry acting on the reduced total space. Examples from quantum groups are discussed.