Features
Presents a tighter formulation of Zubrilin’s theory
Contains a more direct proof of the Wehrfritz–Beidar theorem
Adds more details to the proof of Kemer’s difficult PI-representability theorem
Develops several newer techniques, such as the "pumping procedure"
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.
Author(s): Alexei Kanel-Belov, Yakov Karasik, Louis Halle Rowen
Series: Monographs and Research Notes in Mathematics
Edition: 2
Publisher: Chapman and Hall/CRC
Year: 2015
Language: English
Pages: C, XXVI, 418, B
Tags: Algebra;Pure Mathematics;Mathematics;Polynomials
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
Multilinearization
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
Superidentities
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
De-Multilinearization
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
Cocharacters
GL(V )-Representation Theory
Supplementary Material
List of Theorems
Some Open Questions
Bibliography
