Table Of ContentComputational Aspects of
Polynomial Identities
Research Notes in Mathematics
Volume 9
Computational Aspects of
Polynomial Identities
Alexei Kanei-Belov
Hebrew University of jerusalem, Israel
Moscow Institute of Open Education, Russia
Louis Halle Rowen
Bar-llan University, Ramat-Gan, Israel
Boca Raton London New York
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Library of Congress Cataloging-in-Publication Data
Rowen, Louis Halle.
Computational aspects of polynomial identities I Alexei Kanel-Belov, Louis Halle Rowen.
p. em.-- (Research notes in mathematics; 9)
Includes bibliographical references and index.
ISBN 1-56881-163-2
1. PI-algebras. 2. Polynomial rings. 3. Combinatorial analysis. I. Kanel-Belov, Alexei,
1963-II. Title. III. Research notes in mathematics (Boston, Mass.); 9.
QA25l.R687 2004
512'.4--dc22
2004053474
This work was carried out under the auspices of the
Israel Academy of Science Center for Excellence
at Bar-Man University.
Contents
Preface xi
Symbols Commonly Used in Text xvii
1 Basic Results 1
1.1 Definitions and Examples 1
1.2 Capelli Identities and Finite Dimensional Algebras 8
1.3 Graded Algebras and Grassman Algebras 11
1.4 Identities and Central Polynomials of Matrices 14
1.5 The Identity of Algebraicity 22
1.6 Review of Major Structure Theorems in Pi-Theory 24
1.7 Representable Algebras 28
1.8 ACC in Ring Theory 39
2 Affine Pl-algebras 41
2.1 Words Applied to Affine Algebras 43
2.2 Shirshov's Height Theorem 44
2.3 The Shirshov Program 52
2.4 The Trace Ring 52
2.5 The Razmyslov-Kemer-Braun Theorem 64
2.6 Shirshov's Lemma Revisited 69
3 T-ldeals and Relatively Free Algebras 79
3.1 Sets of Identities and T-Ideals 80
3.2 Relatively Free Algebras 81
3.3 Relatively Free Algebras Without 1, and Their T-Ideals . . 83
3.4 Recognizing T-Ideals 87
3.5 Specht's Problem Posed 88
3.6 Solution of Specht's Problem for Grassman Algebras . . .. 91
vn
vi ¡i Contents
4 Specht's Problem in the Affine Case 95
4.1 Kemer's PI-Representability Theorem 95
4.2 The Kemer Index 101
4.3 Subdirectly Irreducible Algebras 107
4.4 Kemer's Lemmas 110
4.5 Specht's Problem Completed in the Affine Case 126
4.6 Appendix: Applying Kemer's Theory 128
4.7 Razmyslov-Zubrilin Traces Revisited 130
4.8 Pumping and d-Identities 135
5 Representations of S and Their Applications 141
n
5.1 Passage to F[S ] 141
n
5.2 Review of the Representation Theory of S 144
n
5.3 S -Actions on Tn(V) 152
n
5.4 Codimensions and Regev's Exponential Bound Theorem . . 154
5.5 Multilinearization 163
5.6 Appendix: Representation Theory in Characteristic p > 0 . 166
6 Superidentities and Kemer's Main Theorem 169
6.1 Superidentities 169
6.2 Kemer's PI-Superrepresentability Theorem 177
6.3 Consequences of Kemer's theory 191
7 Pi-Algebras in Characteristic p 201
7.1 Positive Results 201
7.2 The Extended Grassman Algebra 204
7.3 Non-Finitely Based T-Ideals in Characteristic 2 209
7.4 Non-Finitely Based T-Ideals in Odd Characteristic 214
8 Recent Structural Results 235
8.1 Left Noetherian PI-algebras Are Finitely Presented 235
8.2 Identities of Group Algebras and Enveloping Algebras . . . 240
9 Poincare-Hilbert Series and Gelfand-Kirillov Dimension 245
9.1 The Hubert Series of an Algebra 245
9.2 Gelfand-Kirillov Dimension 248
9.3 Upper Bounds for the Gelfand-Kirillov Dimension 259
9.4 Rationality of Certain Hubert Series 260
9.5 The Multivariate Poincare-Hilbert Series 265
Contents ix
10 More Representation Theory 267
10.1 Cocharacters 267
10.2 GL(V>Representation Theory 270
11 Unified Theory of Identities 279
11.1 Identities in Universal Algebra 279
11.2 Varieties 281
11.3 Examples of Varieties of Algebras and their Theories . . .. 284
12 Trace Identities 293
12.1 Trace Polynomials and Identities 293
12.2 Finite Generation of Trace T-Ideals 302
12.3 Trace Codimensions 305
13 Exercises 309
14 Lists of Theorems and Examples 343
14.1 Major Theorems 343
14.2 Major Examples and Counterexamples 352
15 Some Open Questions 355
Bibliography 359
Index 371
