Principal fiber bundles in non-commutative geometry

TL;DR

This paper introduces non-commutative principal fiber bundles within non-commutative geometry, utilizing Hopf algebras, and illustrates the theory with quantum groups like $U_q\mathfrak{sl}(2)$, expanding the mathematical framework for quantum field theory.

Contribution

It develops a theory of non-commutative principal fiber bundles and applies it to quantum groups, bridging geometry and algebra in quantum field theory contexts.

Findings

Constructed a framework for non-commutative principal fiber bundles.

Applied the theory to quantum enveloping algebra $U_q\mathfrak{sl}(2)$.

Explored various aspects of non-commutative bundles in quantum algebra.

Abstract

These are the expanded notes of a course given at the Summer school "Geometric, topological and algebraic methods for quantum field theory" held at Villa de Leyva, Colombia in July 2015. We first give an introduction to non-commutative geometry and to the language of Hopf algebras. We next build up a theory of non-commutative principal fiber bundles and consider various aspects of such objects. Finally, we illustrate the theory using the quantum enveloping algebra $U_q\mathfrak{sl}(2)$ and related Hopf algebras.