Topology Change for Fuzzy Physics: Fuzzy Spaces as Hopf Algebras

TL;DR

This paper demonstrates that fuzzy spaces, constructed via quantization of Lie group orbits, possess rich Hopf algebra structures, enabling modeling of topology-changing processes like splitting fuzzy spheres into multiple components.

Contribution

It reveals that fuzzy spaces are Hopf algebras with enhanced structures, providing a framework for quantum symmetries and topology change in non-commutative geometry.

Findings

Fuzzy spaces are shown to be Hopf algebras.

Fuzzy spaces have structures beyond standard Hopf algebras.

Modeling of fuzzy space splitting processes like sphere decomposition.

Abstract

Fuzzy spaces are obtained by quantizing adjoint orbits of compact semi-simple Lie groups. Fuzzy spheres emerge from quantizing S^2 and are associated with the group SU(2) in this manner. They are useful for regularizing quantum field theories and modeling spacetimes by non-commutative manifolds. We show that fuzzy spaces are Hopf algebras and in fact have more structure than the latter. They are thus candidates for quantum symmetries. Using their generalized Hopf algebraic structures, we can also model processes where one fuzzy space splits into several fuzzy spaces. For example we can discuss the quantum transition where the fuzzy sphere for angular momentum J splits into fuzzy spheres for angular momenta K and L.