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

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

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