**Mathematical Surveys and Monographs**

Volume: 122;
2005;
352 pp;
Hardcover

MSC: Primary 16; 17;

**Print ISBN: 978-0-8218-3829-7
Product Code: SURV/122**

List Price: $104.00

AMS Member Price: $83.20

MAA Member Price: $93.60

**Electronic ISBN: 978-1-4704-1349-1
Product Code: SURV/122.E**

List Price: $98.00

AMS Member Price: $78.40

MAA Member Price: $88.20

#### Supplemental Materials

# Polynomial Identities and Asymptotic Methods

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*Antonio Giambruno; Mikhail Zaicev*

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 PI-algebra is any algebra satisfying at least one nontrivial
polynomial identity. This includes the polynomial rings in one or
several variables, the Grassmann algebra, finite-dimensional algebras,
and many other algebras occurring naturally in mathematics. The core
of the book is the proof that the sequence of codimensions of any
PI-algebra has integral exponential growth – the PI-exponent 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.

#### Readership

Graduate students and research mathematicians interested in polynomial identity algebras.

#### Reviews & Endorsements

Written by two of the leading experts in the theory of PI-algebras, the book is interesting and useful.

-- Vesselin Drensky for Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Polynomial Identities and Asymptotic Methods

- Contents v6 free
- Preface ix10 free
- Chapter 1. Polynomial Identities and PI-Algebras 116 free
- 1.1. Basic definitions and examples 116
- 1.2. T-ideals and varieties of algebras 318
- 1.3. Homogeneous and multilinear polynomials 520
- 1.4. Stable identities and generic elements 1025
- 1.5. Special types of identities 1227
- 1.6. Symmetric functions 1530
- 1.7. Identities of matrix algebras 1631
- 1.8. A theorem of Lewin 2035
- 1.9. Identities of block-triangular matrices 2439
- 1.10. Central polynomials in matrix algebras 2641
- 1.11. Structure theorems 2944
- 1.12. Some applications of the structure theorems 3550
- 1.13. The Gelfand-Kirillov dimension of a PI-algebra 3651

- Chapter 2. S[sub(n)]-Representations 4358
- Chapter 3. Group Gradings and Group Actions 6176
- 3.1. Group-graded algebras 6176
- 3.2. Abelian gradings and group actions 6378
- 3.3. G-actions, G-gradings and free algebras 6580
- 3.4. Wedderburn decompositions 6984
- 3.5. Finite dimensional simple superalgebras 7489
- 3.6. Involutions on matrix algebras 7792
- 3.7. Superalgebras and Grassmann envelopes 8095
- 3.8. Supercommutative envelopes 8398

- Chapter 4. Codimension and Colength Growth 87102
- 4.1. Codimensions and colengths 87102
- 4.2. An exponential upper bound for the codimensions 94109
- 4.3. Identities of graded algebras 97112
- 4.4. Robinson-Schensted correspondence 101116
- 4.5. Cocharacters of PI-algebras 104119
- 4.6. Capelli polynomials and the strip theorem 107122
- 4.7. Amitsur polynomials and hooks 108123
- 4.8. Finitely generated superalgebras 110125
- 4.9. Colength growth: a polynomial upper bound 115130

- Chapter 5. Matrix Invariants and Central Polynomials 119134
- 5.1. S[sub(n)]-action on tensor space 119134
- 5.2. Trace identities 122137
- 5.3. A primer of matrix invariants 124139
- 5.4. The discriminant 125140
- 5.5. Invariants and central polynomials 128143
- 5.6. Constructing S[sub(k)]-maps 131146
- 5.7. Computing central polynomials 132147
- 5.8. Cocharacters and trace cocharacters 135150
- 5.9. Multialternating polynomials 137152
- 5.10. Asymptotics for the codimensions of k x k matrices 139154

- Chapter 6. The PI-Exponent of an Algebra 143158
- Chapter 7. Polynomial Growth and Low PI-exponent 165180
- Chapter 8. Classifying Minimal Varieties 193208
- Chapter 9. Computing the Exponent of a Polynomial 215230
- 9.1. The exponent of standard and Capelli polynomials 215230
- 9.2. An upper bound for the exponent of a polynomial 219234
- 9.3. Powers of standard polynomials 225240
- 9.4. Essential hooks and reduced algebras 238253
- 9.5. The exponent of Amitsur polynomials 242257
- 9.6. The exponent of a Lie monomial 245260
- 9.7. Evaluating polynomials 247262
- 9.8. Asymptotics for the standard and the Capelli identities 251266

- Chapter 10. G-Identities and G [omitted] [S[sub(n)]-Action 255270
- 10.1. G-identities, G-codimensions and G [omitted] S[sub(n)]-action 255270
- 10.2. Decomposable monomials 259274
- 10.3. Essential G-identities. Amitsur's theorem on *-identities 261276
- 10.4. Representations of wreath products 264279
- 10.5. Graded identities and polynomial growth 267282
- 10.6. The Z[sub(2)] [omitted] S[sub(n)]-action 272287
- 10.7. Finite dimensional algebras with [omitted]-action 274289
- 10.8. The Z[sub(2)]-exponent of a finite dimensional algebra 276291
- 10.9. Simple and semisimple [omitted]-algebras 280295

- Chapter 11. Super algebras, *-Algebras and Codimension Growth 283298
- 11.1. Notation and more 283298
- 11.2. *-varieties of almost polynomial growth 285300
- 11.3. Supervarieties of almost polynomial growth 289304
- 11.4. Capelli identities on superalgebras 292307
- 11.5. Superalgebras and polynomial growth 294309
- 11.6. *-algebras and the Nagata-Higman theorem 296311
- 11.7. Polynomial growth of the *-codimensions 298313
- 11.8. Supervarieties of exponent 2 301316
- 11.9. Further properties 304319

- Chapter 12. Lie Algebras and Non-associative Algebras 307322
- 12.1. Introduction to Lie algebras 307322
- 12.2. Identities of Lie algebras 309324
- 12.3. Codimension growth of Lie algebras 314329
- 12.4. Exponents of Lie algebras 323338
- 12.5. Overexponential codimension growth 327342
- 12.6. Lie superalgebras, alternative and Jordan algebras 328343
- 12.7. The general non-associative case 330345

- Appendix A. The Generalized-Six-Square Theorem 333348
- Bibliography 341356
- Index 349364 free