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AMS Member Price:  $79.20 
Hardcover ISBN:  9781470451745 
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Hardcover ISBN:  9781470451745 
Product Code:  COLL/66 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470456955 
Product Code:  COLL/66.E 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
Hardcover ISBN:  9781470451745 
eBook ISBN:  9781470456955 
Product Code:  COLL/66.B 
List Price:  $198.00 $148.50 
MAA Member Price:  $178.20 $133.65 
AMS Member Price:  $158.40 $118.80 

Book DetailsColloquium PublicationsVolume: 66; 2020; 630 ppMSC: Primary 16; 15; 14
A polynomial identity for an algebra (or a ring) \(A\) is a polynomial in noncommutative variables that vanishes under any evaluation in \(A\). An algebra satisfying a nontrivial polynomial identity is called a PI algebra, and this is the main object of study in this book, which can be used by graduate students and researchers alike.
The book is divided into four parts. Part 1 contains foundational material on representation theory and noncommutative algebra. In addition to setting the stage for the rest of the book, this part can be used for an introductory course in noncommutative algebra. An expert reader may use Part 1 as reference and start with the main topics in the remaining parts. Part 2 discusses the combinatorial aspects of the theory, the growth theorem, and Shirshov's bases. Here methods of representation theory of the symmetric group play a major role. Part 3 contains the main body of structure theorems for PI algebras, theorems of Kaplansky and Posner, the theory of central polynomials, M. Artin's theorem on Azumaya algebras, and the geometric part on the variety of semisimple representations, including the foundations of the theory of Cayley–Hamilton algebras. Part 4 is devoted first to the proof of the theorem of Razmyslov, Kemer, and Braun on the nilpotency of the nil radical for finitely generated PI algebras over Noetherian rings, then to the theory of Kemer and the Specht problem. Finally, the authors discuss PI exponent and codimension growth. This part uses some nontrivial analytic tools coming from probability theory. The appendix presents the counterexamples of Golod and Shafarevich to the Burnside problem.
ReadershipGraduate students and researchers interested in the theory of (non)commutative rings and their application.

Table of Contents

Chapters

Introduction

Foundations

Noncommutative algebra

Universal algebra

Symmetric functions and matrix invariants

Polynomial maps

Azumaya algebras and irreducible representations

Tensor symmetry

Combinatorial aspects of polynomial identities

Growth

Shirshov’s theorem

$2\times 2$ matrices

The structure theorems

Matrix identities

Structure theorems

Invariants and trace identities

Involutions and matrices

A geometric approach

Spectrum and dimension

The relatively free algebras

The nilpotent radical

Finitedimensional and affine PI algebras

The relatively free algebras

Identities and superalgebras

The Specht problem

The PIexponent

Codimension growth for matrices

Codimension growth for algebras satisfying a Capelli identity

The Golod–Shafarevich counterexamples


Additional Material

Reviews

There is a substantial difference between this book and the other texts on the subject, including fairly recent ones. Here the authors, while giving a comprehensive treatment of the theory, stress the connection with invariant theory and its geometric aspects.
... In conclusion, apart from some recent results in positive characteristics and the extensions to nonassociative algebras, this book gives a rather complete and uptodate presentation of the theory, and it can be used for several different graduate courses on this subject.
Daniela La Mattina


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A polynomial identity for an algebra (or a ring) \(A\) is a polynomial in noncommutative variables that vanishes under any evaluation in \(A\). An algebra satisfying a nontrivial polynomial identity is called a PI algebra, and this is the main object of study in this book, which can be used by graduate students and researchers alike.
The book is divided into four parts. Part 1 contains foundational material on representation theory and noncommutative algebra. In addition to setting the stage for the rest of the book, this part can be used for an introductory course in noncommutative algebra. An expert reader may use Part 1 as reference and start with the main topics in the remaining parts. Part 2 discusses the combinatorial aspects of the theory, the growth theorem, and Shirshov's bases. Here methods of representation theory of the symmetric group play a major role. Part 3 contains the main body of structure theorems for PI algebras, theorems of Kaplansky and Posner, the theory of central polynomials, M. Artin's theorem on Azumaya algebras, and the geometric part on the variety of semisimple representations, including the foundations of the theory of Cayley–Hamilton algebras. Part 4 is devoted first to the proof of the theorem of Razmyslov, Kemer, and Braun on the nilpotency of the nil radical for finitely generated PI algebras over Noetherian rings, then to the theory of Kemer and the Specht problem. Finally, the authors discuss PI exponent and codimension growth. This part uses some nontrivial analytic tools coming from probability theory. The appendix presents the counterexamples of Golod and Shafarevich to the Burnside problem.
Graduate students and researchers interested in the theory of (non)commutative rings and their application.

Chapters

Introduction

Foundations

Noncommutative algebra

Universal algebra

Symmetric functions and matrix invariants

Polynomial maps

Azumaya algebras and irreducible representations

Tensor symmetry

Combinatorial aspects of polynomial identities

Growth

Shirshov’s theorem

$2\times 2$ matrices

The structure theorems

Matrix identities

Structure theorems

Invariants and trace identities

Involutions and matrices

A geometric approach

Spectrum and dimension

The relatively free algebras

The nilpotent radical

Finitedimensional and affine PI algebras

The relatively free algebras

Identities and superalgebras

The Specht problem

The PIexponent

Codimension growth for matrices

Codimension growth for algebras satisfying a Capelli identity

The Golod–Shafarevich counterexamples

There is a substantial difference between this book and the other texts on the subject, including fairly recent ones. Here the authors, while giving a comprehensive treatment of the theory, stress the connection with invariant theory and its geometric aspects.
... In conclusion, apart from some recent results in positive characteristics and the extensions to nonassociative algebras, this book gives a rather complete and uptodate presentation of the theory, and it can be used for several different graduate courses on this subject.
Daniela La Mattina