Combinatorial Methods

Free Groups, Polynomials, and Free Algebras

By: Vladimir Shpilrain, Alexander Mikhalev, Jie-tai Yu

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Hardcover | 14 November 2003

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332 Pages

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24.13 x 17.15 x 1.91

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The main purpose of this book is to show how ideas from combinatorial group theory have spread to two other areas of mathematics: the theory of Lie algebras and affine algebraic geometry. Some of these ideas, in turn, came to combinatorial group theory from low-dimensional topology at the beginning of the 20th Century.

This book is divided into three fairly independent parts. Part I provides a brief exposition of several classical techniques in combinatorial group theory, namely, methods of Nielsen, Whitehead, and Tietze. Part II contains the main focus of the book. Here the authors show how the aforementioned techniques of combinatorial group theory found their way into affine algebraic geometry, a fascinating area of mathematics that studies polynomials and polynomial mappings. Part III illustrates how ideas from combinatorial group theory contributed to the theory of free algebras. The focus here is on Schreier varieties of algebras (a variety of algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free in the same variety of algebras).

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From the reviews: "This book is devoted to a combinatorial theory of three types of objects: (1) free groups, (2) polynomial algebras and free associative algebras, (3) free algebras of the so-called Nielsen-Schreier varieties of algebras. It considers problems related mainly to the groups of automorphisms of these objects... The authors have done a lot of work to show that the same problems and the same ideas are the moving forces of the three theories. The book contains a good background on the classical results (most of them without proof) and a detailed exposition of the recent results. A large portion of the exposition is devoted to topics in which the authors have made their own contribution." -- MATHEMATICAL REVIEWS "The book consists of three parts: groups, polynomial algebras and free Nielsen-Schreier algebras. ! The book contains very interesting material to which the authors have made a valuable contribution. The book includes many open and very important problems. ! The exposition of the material is made with care. So the book could be recommended for students even as a textbook." (Vyacheslav A. Artamonov, Zentralblatt MATH, Vol. 1039 (8), 2004)

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Preface	p. vii
Introduction	p. 1
Groups	p. 5
Introduction	p. 7
Classical Techniques of Combinatorial Group Theory	p. 9
Nielsen's Method	p. 9
Whitehead's Method	p. 11
Tietze's Method	p. 13
Free Differential Calculus	p. 14
Two applications	p. 17
Test Elements	p. 20
Nielsen's Commutator Test	p. 21
Recognizing Test Elements	p. 25
Test Sets	p. 30
The ""Double Jacobian"" Matrix	p. 33
Other Special Elements	p. 35
¿-Primitive Elements	p. 35
Almost Primitive and Generic Elements	p. 41
Automorphic Orbits	p. 45
Finite Orbits	p. 46
Abridged Orbits	p. 48
Counting primitive elements	p. 48
Complexity of Whitehead's algorithm	p. 53
Endomorphisms that Preserve Automorphic Orbits	p. 59
Polynomial Algebras	p. 65
Introduction	p. 67
The Jacobian Conjecture	p. 71
Polynomial Retracts and the Jacobian Conjecture	p. 72
Retracts of K[x, y]	p. 75
Applications to the Jacobian Conjecture	p. 76
The Cancellation Conjecture	p. 80
Equivalence and Stable Equivalence	p. 83
Around Danielewski's Example	p. 88
Nagata's Problem	p. 92
Gröbner Bases and Face Functions	p. 95
Lifting the Nagata Automorphism	p. 101
The Embedding Problem	p. 108
Embeddings of Curves in the Plane	p. 109
Elementary automorphisms and peak reduction	p. 111
A classification of parametric curves	p. 113
Embeddings of curves in the plane	p. 115
Embeddings of Hypersurfaces in Affine Space	p. 118
Invariants of isomorphic varieties	p. 122
Inequivalent isomorphic varieties	p. 124
The Embedding Conjecture for Free Associative Algebras	p. 127
Coordinate Polynomials	p. 129
Coordinates of Two-Variable Polynomial Algebras	p. 130
Proofs of main results	p. 132
Algorithm for detecting coordinate polynomials	p. 136
Relation to the Jacobian Conjecture	p. 137
Coordinates in Free Associative Algebra of Rank Two	p. 138
Fox derivatives in a free associative algebra	p. 141
Proofs and algorithms	p. 142
Density of Coordinates in Polynomial Algebras	p. 144
Coordinates in K[z][x,y]	p. 147
Preliminaries	p. 148
Characterization of tame and wild coordinates in K[z][x, y]	p. 151
New class of wild automorphisms	p. 158
Test Polynomials	p. 169
Test Polynomials in Polynomial Algebras	p. 171
Test Elements in Free Associative Algebras	p. 174
Open Problems	p. 180
Free Nielsen-Schreier Algebras	p. 183
Introduction	p. 185
Schreier Varieties of Algebras	p. 190
Main Types of Nielsen-Schreier Algebras	p. 191
Schreier Techniques	p. 196
Free Differential Calculus	p. 201
Free differential calculus and Schreier varieties	p. 201
Ranks of subalgebras	p. 207
Stable Equivalence	p. 208
The Rank of an Endomorphism	p. 210
Rank Theorems and Primitive Elements	p. 213
Basic Properties of Partial Derivatives	p. 214
Homogeneous Admissible Elements	p. 217
Elimination of Variables	p. 219
Rank Theorems	p. 225
Primitive Elements	p. 229
Primitive elements in free nonassociative algebras	p. 230
Primitive elements in free Lie superalgebras	p. 230
The Freiheitssatz and primitive elements	p. 235
Primitive Elements and Endomorphisms	p. 237
Inverse Images of Primitive Elements	p. 241
Generalized Primitive Elements	p. 244
Test Elements	p. 247
Retracts	p. 252
Endomorphisms and Automorphic Orbits	p. 254
Almost Primitive and Test Elements	p. 256
¿-Primitive Elements of Free Lie Algebras	p. 262
Generic Elements of Free Lie Algebras	p. 264
Free Leibniz Algebras	p. 270
Basic Notions	p. 271
Differential Separability of Subalgebras	p. 273
Residual Finiteness of Free Leibniz Algebras	p. 275
Automorphisms of Free Leibniz Algebras	p. 279
Simple automorphisms	p. 279
Recognizing tame automorphisms	p. 281
References	p. 283
Notation Index	p. 306
Author Index	p. 308
Subject Index	p. 313
Table of Contents provided by Publisher. All Rights Reserved.

Combinatorial Methods

Free Groups, Polynomials, and Free Algebras

At a Glance

Hardcover

Industry Reviews

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