Computational Aspects of Polynomial Identities

Polynomial identities are used to study the properties of algebras through polynomial conditions. Starting from simple properties such as commutativity a beautiful theory has evolved that studies algebras through the set of all their identities or classes of algebras satisfying a given set of identities. The goal of this book is to expose the more mature aspects of PI-theory to the general mathematical community, covering the important advances in the past 20 years.

Preface

xi

Symbols Commonly Used in Text

xvii

1 Basic Results

1

(40)

1.1 Definitions and Examples

1

(7)

1.2 Capelli Identities and Finite Dimensional Algebras

8

(3)

1.3 Graded Algebras and Grassman Algebras

11

(3)

1.4 Identities and Central Polynomials of Matrices

14

(8)

1.5 The Identity of Algebraicity

22

(2)

1.6 Review of Major Structure Theorems in PI-Theory

24

(4)

1.7 Representable Algebras

28

(11)

1.8 ACC in Ring Theory

39

(2)

2 Affine PI-algebras

41

(38)

2.1 Words Applied to Affine Algebras

43

(1)

2.2 Shirshov's Height Theorem

44

(8)

2.3 The Shirshov Program

52

(1)

2.4 The Trace Ring

52

(12)

2.5 The Razmyslov-Kemer-Braun Theorem

64

(5)

2.6 Shirshov's Lemma Revisited

69

(10)

3 T-Ideals and Relatively Free Algebras

79

(16)

3.1 Sets of Identities and T-Ideals

80

(1)

3.2 Relatively Free Algebras

81

(2)

3.3 Relatively Free Algebras Without 1, and Their T-Ideals

83

(4)

3.4 Recognizing T-Ideals

87

(1)

3.5 Specht's Problem Posed

88

(3)

3.6 Solution of Specht's Problem for Grassman Algebras

91

(4)

4 Specht's Problem in the Affine Case

95

(46)

4.1 Kemer's PI-Representability Theorem

95

(6)

4.2 The Kemer Index

101

(6)

4.3 Subdirectly Irreducible Algebras

107

(3)

4.4 Kemer's Lemmas

110

(16)

4.5 Specht's Problem Completed in the Affine Case

126

(2)

4.6 Appendix: Applying Kemer's Theory

128

(2)

4.7 Razmyslov-Zubrilin Traces Revisited

130

(5)

4.8 Pumping and d-Identities

135

(6)

5 Representations of Sn and Their Applications

141

(28)

5.1 Passage to F[Sn]

141

(3)

5.2 Review of the Representation Theory of Sn

144

(8)

5.3 Sn-Actions on Tn(V)

152

(2)

5.4 Codimensions and Regev's Exponential Bound Theorem

154

(9)

5.5 Multilinearization

163

(3)

5.6 Appendix: Representation Theory in Characteristic p > 0

166

(3)

6 Superidentities and Kemer's Main Theorem

169

(32)

6.1 Superidentities

169

(8)

6.2 Kemer's PI-Superrepresentability Theorem

177

(14)

6.3 Consequences of Kemer's theory

191

(10)

7 PI-Algebras in Characteristic p

201

(34)

7.1 Positive Results

201

(3)

7.2 The Extended Grassrnan Algebra

204

(5)

7.3 Non-Finitely Based T-Ideals in Characteristic 2

209

(5)

7.4 Non-Finitely Based T-Ideals in Odd Characteristic

214

(21)

8 Recent Structural Results

235

(10)

8.1 Left Noetherian PI-algebras Are Finitely Presented

235

(5)

8.2 Identities of Group Algebras and Enveloping Algebras

240

(5)

9 Poincare-Hilbert Series and Gelfand-Kirillov Dimension

245

(22)

9.1 The Hilbert Series of an Algebra

245

(3)

9.2 Gelfand-Kirillov Dimension

248

(11)

9.3 Upper Bounds for the Gelfand-Kirillov Dimension

259

(1)

9.4 Rationality of Certain Hilbert Series

260

(5)

9.5 The Multivariate Poincaré-Hilbert Series

265

(2)

10 More Representation Theory

267

(12)

10.1 Cocharacters

267

(3)

10.2 GL(V)-Representation Theory

270

(9)

11 Unified Theory of Identities

279

(14)

11.1 Identities in Universal Algebra

279

(2)

11.2 Varieties

281

(3)

11.3 Examples of Varieties of Algebras and their Theories

284

(9)

12 Trace Identities

293

(16)

12.1 Trace Polynomials and Identities

293

(9)

12.2 Finite Generation of Trace T-Ideals

302

(3)

12.3 Trace Codimensions

305

(4)

13 Exercises

309

(34)

14 Lists of Theorems and Examples

343

(12)

14.1 Major Theorems

343

(9)

14.2 Major Examples and Counterexamples

352

(3)

15 Some Open Questions

355

(4)

Bibliography

359

(12)

Index

371

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Computational Aspects of Polynomial Identities

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