Opis: Computational Aspects of Polynomial Identities: Volume l - Louis Halle Rowen, Yaakov Karasik, Belov Alexey

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, Poincare-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.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 Poincare-Hilbert Series and Gelfand-Kirillov Dimension The Hilbert Series of an Algebra The Gelfand-Kirillov Dimension Rationality of Certain Hilbert Series The Multivariate Poincare-Hilbert Series More Representation Theory Cocharacters GL(V )-Representation Theory Supplementary Material List of Theorems Some Open Questions Bibliography

Szczegóły: Computational Aspects of Polynomial Identities: Volume l - Louis Halle Rowen, Yaakov Karasik, Belov Alexey

Tytuł: Computational Aspects of Polynomial Identities: Volume l
Autor: Louis Halle Rowen, Yaakov Karasik, Belov Alexey
Producent: Productivity Press Inc
ISBN: 9781498720083
Rok produkcji: 2015
Ilość stron: 444
Oprawa: Twarda
Waga: 0.8 kg

Recenzje: Computational Aspects of Polynomial Identities: Volume l - Louis Halle Rowen, Yaakov Karasik, Belov Alexey