Computational Aspects of Polynomial Identities: Volume l, Kemer's Theorems, 2nd Edition, 2nd Edition (Hardback) book cover

Computational Aspects of Polynomial Identities

Volume l, Kemer's Theorems, 2nd Edition, 2nd Edition

By Alexei Kanel-Belov, Yakov Karasik, Louis Halle Rowen

Chapman and Hall/CRC

418 pages

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Computational Aspects of Polynomial Identities: Volume l, Kemer’s Theorems, 2nd Edition presents the underlying ideas in recent polynomial identity (PI)-theory and demonstrates the validity of the proofs of PI-theorems. This edition gives all the details involved in Kemer’s proof of Specht’s conjecture for affine PI-algebras in characteristic 0.

The book first discusses the theory needed for Kemer’s proof, including the featured role of Grassmann algebra and the translation to superalgebras. The authors develop Kemer polynomials for arbitrary varieties as tools for proving diverse theorems. They also lay the groundwork for analogous theorems that have recently been proved for Lie algebras and alternative algebras. They then describe counterexamples to Specht’s conjecture in characteristic p as well as the underlying theory. The book also covers Noetherian PI-algebras, Poincaré–Hilbert series, Gelfand–Kirillov dimension, the combinatoric theory of affine PI-algebras, and homogeneous identities in terms of the representation theory of the general linear group GL.

Through the theory of Kemer polynomials, this edition shows that the techniques of finite dimensional algebras are available for all affine PI-algebras. It also emphasizes the Grassmann algebra as a recurring theme, including in Rosset’s proof of the Amitsur–Levitzki theorem, a simple example of a finitely based T-ideal, the link between algebras and superalgebras, and a test algebra for counterexamples in characteristic p.

Table of Contents

Basic Associative PI-Theory

Basic Results

Preliminary Definitions

Noncommutative Polynomials and Identities

Graded Algebras

Identities and Central Polynomials of Matrices

Review of Major Structure Theorems in PI Theory

Representable Algebras

Sets of Identities

Relatively Free Algebras

Generalized Identities

A Few Words Concerning Affine PI-Algebras: Shirshov’s Theorem

Words Applied to Affine Algebras

Shirshov’s Height Theorem

Shirshov’s Lemma

The Shirshov Program

The Trace Ring

Shirshov’s Lemma Revisited

Appendix A: The Independence Theorem for Hyperwords

Appendix B: A Subexponential Bound for the Shirshov Height

Representations of Sn and Their Applications

Permutations and identities

Review of the Representation Theory of Sn

Sn-Actions on Tn(V )

Codimensions and Regev’s Theorem


Affine PI-Algebras

The Braun-Kemer-Razmyslov Theorem

Structure of the Proof

A Cayley-Hamilton Type Theorem

The Module M over the Relatively Free Algebra C{X, Y,Z} of cn+1

The Obstruction to Integrality Obstn(A) ⊆ A

Reduction to Finite Modules

Proving that Obstn(A) · (CAPn(A))2 = 0

The Shirshov Closure and Shirshov Closed Ideals

Kemer’s Capelli Theorem

First Proof (Combinatoric)

Second Proof (Pumping plus Representation Theory)

Specht’s Conjecture

Specht’s Problem and Its Solution in the Affine Case (Characteristic 0)

Specht’s Problem Posed

Early Results on Specht’s Problem

Kemer’s PI-Representability Theorem

Multiplying Alternating Polynomials, and the First Kemer Invariant

Kemer’s First Lemma

Kemer’s Second Lemma

Significance of Kemer’s First and Second Lemmas

Manufacturing Representable Algebras

Kemer’s PI-Representability Theorem Concluded

Specht’s Problem Solved for Affine Algebras

Pumping Kemer Polynomials

Appendix: Strong Identities and Specht’s Conjecture

Superidentities and Kemer’s Solution for Non-Affine Algebras


Kemer’s Super-PI Representability Theorem

Kemer’s Main Theorem Concluded

Consequences of Kemer’s Theory

Trace Identities

Trace Polynomials and Identities

Finite Generation of Trace T-Ideals

Trace Codimensions

Kemer’s Matrix Identity Theorem in Characteristic p

PI-Counterexamples in Characteristic p


The Extended Grassmann Algebra

Non-Finitely Based T-Ideals in Characteristic

Non-Finitely Based T-Ideals in Odd Characteristic

Other Results for Associative PI-Algebras

Recent Structural Results

Left Noetherian PI-Algebras

Identities of Group Algebras

Identities of Enveloping Algebras

Poincaré-Hilbert Series and Gelfand-Kirillov Dimension

The Hilbert Series of an Algebra

The Gelfand-Kirillov Dimension

Rationality of Certain Hilbert Series

The Multivariate Poincaré-Hilbert Series

More Representation Theory


GL(V )-Representation Theory

Supplementary Material

List of Theorems

Some Open Questions


About the Authors

Alexei Kanel-Belov is a professor in the Department of Mathematics at Bar-Ilan University. His research interests include ring theory, semigroup theory, polynomial automorphisms, quantization, symbolical dynamic combinatorial geometry and its mechanical applications, elementary mathematics, and mathematical education.

Yakov Karasik completed his doctorate at the Department of Mathematics at Technion - Israel Institute of Technology.

Louis Halle Rowen is a professor in the Department of Mathematics at Bar-Ilan University. His research interests include noncommutative algebra, finite dimensional division algebras, the structure theory of rings, and tropical algebras.

About the Series

Chapman & Hall/CRC Monographs and Research Notes in Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Algebra / General