Le Bruyn Noncommutative Geometry and Cayley-smooth Orders

1. Auflage 2007
ISBN: 978-1-4200-6423-0
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 592 Seiten

Reihe: Chapman & Hall/CRC Pure and Applied Mathematics

Noncommutative Geometry and Cayley-smooth Orders explains the theory of Cayley-smooth orders in central simple algebras over function fields of varieties. In particular, the book describes the étale local structure of such orders as well as their central singularities and finite dimensional representations.

After an introduction to partial desingularizations of commutative singularities from noncommutative algebras, the book presents the invariant theoretic description of orders and their centers. It proceeds to introduce étale topology and its use in noncommutative algebra as well as to collect the necessary material on representations of quivers. The subsequent chapters explain the étale local structure of a Cayley-smooth order in a semisimple representation, classify the associated central singularity to smooth equivalence, describe the nullcone of these marked quiver representations, and relate them to the study of all isomorphism classes of n-dimensional representations of a Cayley-smooth order. The final chapters study Quillen-smooth algebras via their finite dimensional representations.

Noncommutative Geometry and Cayley-smooth Orders provides a gentle introduction to one of mathematics' and physics' hottest topics.

Zielgruppe

Mathematicians and graduate students in algebra, algebraic geometry, and number theory.

Autoren/Hrsg.

Le Bruyn, Lieven

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik

Inhaltsverzeichnis

Preface

Introduction
Noncommutative algebra
Noncommutative geometry
Noncommutative desingularizations

Cayley-Hamilton Algebras
Conjugacy classes of matrices
Simultaneous conjugacy classes
Matrix invariants and necklaces
The trace algebra
The symmetric group
Necklace relations
Trace relations
Cayley-Hamilton algebras

Reconstructing Algebras
Representation schemes
Some algebraic geometry
The Hilbert criterium
Semisimple modules
Some invariant theory
Geometric reconstruction
The Gerstenhaber-Hesselink theorem
The real moment map

Étale Technology
Étale topology
Central simple algebras
Spectral sequences
Tsen and Tate fields
Coniveau spectral sequence
The Artin-Mumford exact sequence
Normal spaces
Knop-Luna slices

Quiver Technology
Smoothness
Local structure
Quiver orders
Simple roots
Indecomposable roots
Canonical decomposition
General subrepresentations
Semistable representations

Semisimple Representations
Representation types
Cayley-smooth locus
Reduction steps
Curves and surfaces
Complex moment map
Preprojective algebras
Central smooth locus
Central singularities

Nilpotent Representations
Cornering matrices
Optimal corners
Hesselink stratification
Cornering quiver representations
Simultaneous conjugacy classes
Representation fibers
Brauer-Severi varieties
Brauer-Severi fibers

Noncommutative Manifolds
Formal structure
Semi-invariants
Universal localization
Compact manifolds
Differential forms
deRham cohomology
Symplectic structure
Necklace Lie algebras

Moduli Spaces
Moment maps
Dynamical systems
Deformed preprojective algebras
Hilbert schemes
Hyper Kähler structure
Calogero particles
Coadjoint orbits
Adelic Grassmannian

References

Index