Hardcover ISBN:  9780821838297 
Product Code:  SURV/122 
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Product Code:  SURV/122.E 
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Hardcover ISBN:  9780821838297 
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Product Code:  SURV/122.B 
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Hardcover ISBN:  9780821838297 
Product Code:  SURV/122 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470413491 
Product Code:  SURV/122.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821838297 
eBook ISBN:  9781470413491 
Product Code:  SURV/122.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 122; 2005; 352 ppMSC: Primary 16; 17
This book gives a state of the art approach to the study of polynomial identities satisfied by a given algebra by combining methods of ring theory, combinatorics, and representation theory of groups with analysis. The idea of applying analytical methods to the theory of polynomial identities appeared in the early 1970s and this approach has become one of the most powerful tools of the theory.
A PIalgebra is any algebra satisfying at least one nontrivial polynomial identity. This includes the polynomial rings in one or several variables, the Grassmann algebra, finitedimensional algebras, and many other algebras occurring naturally in mathematics. The core of the book is the proof that the sequence of codimensions of any PIalgebra has integral exponential growth – the PIexponent of the algebra. Later chapters further apply these results to subjects such as a characterization of varieties of algebras having polynomial growth and a classification of varieties that are minimal for a given exponent. Results are extended to graded algebras and algebras with involution.
The book concludes with a study of the numerical invariants and their asymptotics in the class of Lie algebras. Even in algebras that are close to being associative, the behavior of the sequences of codimensions can be wild.
The material is suitable for graduate students and research mathematicians interested in polynomial identity algebras.
ReadershipGraduate students and research mathematicians interested in polynomial identity algebras.

Table of Contents

Chapters

1. Polynomial identities and PIalgebras

2. $S_n$representations

3. Group gradings and group actions

4. Codimension and colength growth

5. Matrix invariants and central polynomials

6. The PIexponent of an algebra

7. Polynomial growth and low PIexponent

8. Classifying minimal varieties

9. Computing the exponent of a polynomial

10. $G$identities and $G \wr S_n$action

11. Super algebras, *algebras and codimension growth

12. Lie algebras and nonassociative algebras


Additional Material

Reviews

Written by two of the leading experts in the theory of PIalgebras, the book is interesting and useful.
Vesselin Drensky for Zentralblatt MATH


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This book gives a state of the art approach to the study of polynomial identities satisfied by a given algebra by combining methods of ring theory, combinatorics, and representation theory of groups with analysis. The idea of applying analytical methods to the theory of polynomial identities appeared in the early 1970s and this approach has become one of the most powerful tools of the theory.
A PIalgebra is any algebra satisfying at least one nontrivial polynomial identity. This includes the polynomial rings in one or several variables, the Grassmann algebra, finitedimensional algebras, and many other algebras occurring naturally in mathematics. The core of the book is the proof that the sequence of codimensions of any PIalgebra has integral exponential growth – the PIexponent of the algebra. Later chapters further apply these results to subjects such as a characterization of varieties of algebras having polynomial growth and a classification of varieties that are minimal for a given exponent. Results are extended to graded algebras and algebras with involution.
The book concludes with a study of the numerical invariants and their asymptotics in the class of Lie algebras. Even in algebras that are close to being associative, the behavior of the sequences of codimensions can be wild.
The material is suitable for graduate students and research mathematicians interested in polynomial identity algebras.
Graduate students and research mathematicians interested in polynomial identity algebras.

Chapters

1. Polynomial identities and PIalgebras

2. $S_n$representations

3. Group gradings and group actions

4. Codimension and colength growth

5. Matrix invariants and central polynomials

6. The PIexponent of an algebra

7. Polynomial growth and low PIexponent

8. Classifying minimal varieties

9. Computing the exponent of a polynomial

10. $G$identities and $G \wr S_n$action

11. Super algebras, *algebras and codimension growth

12. Lie algebras and nonassociative algebras

Written by two of the leading experts in the theory of PIalgebras, the book is interesting and useful.
Vesselin Drensky for Zentralblatt MATH