Combinatorial methods

Free groups, polynomials, and free algebras

Jie-Tai yu, Vladimir shpilrain, Alexander a. mikhalev

315 pages, parution le 28/12/2003

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Résumé

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 in 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).

L'auteur - Jie-Tai yu

He received his Ph.D. degree from the University of Notre-Dame in 1994 and has held a position at the Univesity o Hong-Kong since then. His research interests include various aspects of commutative algebra and affine algebraic geometry.

L'auteur - Vladimir shpilrain

He received is Ph.D. degree from Moscow State University. He has been working at several universities in different countries, most recently at the University of Hong Kong (1997 - 1998) and the City College of New York (since 1998). His research interests include combinatorial group theory, affine algebraic geometry, cryptography, and theory of algorithms.

L'auteur - Alexander a. mikhalev

He received his Ph.D. and D.Sc. degrees in mathematics from Moscow State University, where he is now Professor in the department of Mechanics and Mathematics. His research focuses on Lie algebrass and superalgebras along with other types of nonassociative algebras.

Sommaire

Preface
Introduction
Groups: Introduction
- Classical Techniques
- Test Elements
- Other Special Elements
- Automorphic Orbits
Polynomial Algebras: Introduction
- The Jacobian Conjecture
- The Cancellation Conjecture
- Nagata's Problem
- The Embedding Problem
- Coordinate Polynomials
- Test Polynomials
Free Nielsen-Schreier Algebras: Introduction
- Schreier Varieties of Algebras
- Rank Theorems and Primitive Elements
- Generalized Primitive Elements
- Free Leibniz Algebras
References
Notations
Author Index
Subject Index

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	PAPIER
Éditeur(s)	Springer
Auteur(s)	Jie-Tai yu, Vladimir shpilrain, Alexander a. mikhalev
Parution	28/12/2003
Nb. de pages	315
Format	16 x 24
Couverture	Relié
Poids	610g
Intérieur	Noir et Blanc
EAN13	9780387405629
ISBN13	978-0-387-40562-9