Noncommutative Geometry and Quantum Group Symmetries

TL;DR

This paper explores the application of Connes' noncommutative geometry to quantum groups and quantum homogeneous spaces, aiming to interpret these algebraic structures as noncommutative spaces.

Contribution

It provides a framework for understanding quantum groups within the context of noncommutative geometry, bridging algebraic and geometric perspectives.

Findings

Quantum groups can be interpreted as noncommutative spaces.

A geometric framework for quantum homogeneous spaces is developed.

Connections between algebraic structures and geometric intuition are established.

Abstract

Quantum groups and quantum homogeneous spaces - developed by several authors since the 80's - provide a large class of examples of algebras which for many reasons we interpret as `coordinate algebras' over noncommutative spaces. This dissertation is an attempt to understand them from the point of view of Connes' noncommutative geometry.