Differential calculi on quantum principal bundles in the Durdevic approach

TL;DR

This thesis explores the Durdevic approach to differential calculi on quantum principal bundles using Hopf algebra methods, extending the theory and providing explicit examples on noncommutative spaces.

Contribution

It extends Durdevic's theory of differential calculi on quantum principal bundles and offers explicit realizations on noncommutative geometries.

Findings

Extended Durdevic theory of quantum principal bundles.

Constructed explicit differential calculi on noncommutative spaces.

Compared new results with existing literature.

Abstract

In this thesis we study the Durdevic theory of differential calculi on quantum principal bundles within the domain of noncommutative geometry. Throughout the exposition, an algebraic approach based on Hopf algebras is employed. We begin by briefly recalling the foundational concepts of Hopf algebras and comodule algebras. Hopf-Galois extensions are introduced, along with an important correspondence result which relates crossed product algebras and cleft extensions. Differential calculus over algebras is reviewed, with a particular focus on the covariant case. The Durdevic theory is presented in a modern language. We compare this theory to existing literature on the topic. We extend the Durdevic theory and provide explicit realisations of quantum principal bundles and complete differential calculi on the noncommutative algebraic 2-torus, the quantum Hopf fibration and on crossed product algebras, resulting in an original contribution within this thesis.