Noncommutative Geometry and Cayley-smooth Orders: 1st Edition (Hardback) book cover

Noncommutative Geometry and Cayley-smooth Orders

1st Edition

By Lieven Le Bruyn

Chapman and Hall/CRC

592 pages | 75 B/W Illus.

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Hardback: 9781420064223
pub: 2007-08-24
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pub: 2007-08-24
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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.


"Summing up, this book is a nice introduction to the geometry of representation schemes … Along the way, the author develops a veritable arsenal of modern tools any prospective expert will need to know in this area … If the reader perseveres, the reviewer is positive the rewards will be great."

Mathematical Reviews

"The book is very well-ordered and is written in a nice and readable style. It contains a huge amount of interesting material related to many important topics in modern mathematics and mathematical physics. It could be greatly appreciated by talented students and young mathematicians as a well-written introduction to important fields of math."

EMS Newsletter, March 2009

"The final comment is then that this is a really entertaining book, covering most of the topics on noncommutative geometry. Also it is reasonable elementary, easy to read, easy to understand, and if the reader would like to go further into details, an extensive bibliography is given."

– Arvid Siqveland, in Zentralblatt Math, 2008/2009, Vol. 1131

Table of Contents



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


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


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



About the Series

Chapman & Hall/CRC Pure and Applied Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Algebra / General
MATHEMATICS / Number Theory
SCIENCE / Mathematical Physics