TY - BOOK

T1 - Non-commutative covering spaces and their symmetries

AU - Canlubo, Clarisson

PY - 2016

Y1 - 2016

N2 - The main goal of this thesis is to propose a notion analogous to covering spaces in classicalgeometry. This is motivated by the author's long term goal of dening the (etale) fundamentalgroup of a non-commutative space and put forth a good notion of monodromy.We will present a notion of a non-commutative covering space using Galois theory of Hopfalgebroids. We will look at basic properties of classical covering spaces that generalize to thenon-commutative framework. Afterwards, we will explore a series of examples. We will startwith coverings of a point and central coverings of commutative spaces and see how these areclosely tied up. Coupled Hopf algebras will be presented to give a general description of coveringsof a point. We will give a complete description of the geometry of the central coverings ofcommutative spaces using the coverings of a point. A topologized version of Hopf categories willbe dened and its corresponding Galois theory. Using this and basic concepts from algebraic geometryand spectral theory, we will give a full description of the general structure of non-centralcoverings. Examples of coverings of the rational and irrational non-commutative tori will alsobe studied. Using the non-commutative analogue of the hyperelliptic involution, we will showthat unlike the classical case, the non-commutative sphere is a covering of the non-commutativetorus. There is a purely non-commutative phenomenon happening to non-commutative coverings,namely, their symmetry is two-sided. We will explain this and relate it to bi-Galois theory.Using the OZ-transform, we will show that non-commutative covering spaces come in pairs.Several categories of covering spaces will be dened and studied. Appealing to Tannaka duality,we will explain how this lead to a notion of an etale fundamental group. Finally, in the lastchapter we will discuss possible future projects.

AB - The main goal of this thesis is to propose a notion analogous to covering spaces in classicalgeometry. This is motivated by the author's long term goal of dening the (etale) fundamentalgroup of a non-commutative space and put forth a good notion of monodromy.We will present a notion of a non-commutative covering space using Galois theory of Hopfalgebroids. We will look at basic properties of classical covering spaces that generalize to thenon-commutative framework. Afterwards, we will explore a series of examples. We will startwith coverings of a point and central coverings of commutative spaces and see how these areclosely tied up. Coupled Hopf algebras will be presented to give a general description of coveringsof a point. We will give a complete description of the geometry of the central coverings ofcommutative spaces using the coverings of a point. A topologized version of Hopf categories willbe dened and its corresponding Galois theory. Using this and basic concepts from algebraic geometryand spectral theory, we will give a full description of the general structure of non-centralcoverings. Examples of coverings of the rational and irrational non-commutative tori will alsobe studied. Using the non-commutative analogue of the hyperelliptic involution, we will showthat unlike the classical case, the non-commutative sphere is a covering of the non-commutativetorus. There is a purely non-commutative phenomenon happening to non-commutative coverings,namely, their symmetry is two-sided. We will explain this and relate it to bi-Galois theory.Using the OZ-transform, we will show that non-commutative covering spaces come in pairs.Several categories of covering spaces will be dened and studied. Appealing to Tannaka duality,we will explain how this lead to a notion of an etale fundamental group. Finally, in the lastchapter we will discuss possible future projects.

UR - https://soeg.kb.dk/permalink/45KBDK_KGL/fbp0ps/alma99122937902505763

M3 - Ph.D. thesis

BT - Non-commutative covering spaces and their symmetries

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -