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Combinatorial Methods

Free Groups, Polynomials, and Free Algebras

  • Book
  • © 2004

Overview

Authors:
  1. Alexander A. Mikhalev

    1. Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

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  2. Vladimir Shpilrain

    1. Department of Mathematics, The City College of New York, New York, USA

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  3. Jie-Tai Yu

    1. Department of Mathematics, The University of Hong Kong, Hong Kong, China

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Part of the book series: CMS Books in Mathematics (CMSBM)

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    1. Introduction

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 1-3

  3. Groups

    1. Front Matter

      Pages 5-8
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    2. Classical Techniques of Combinatorial Group Theory

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 9-19

    3. Test Elements

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 20-34

    4. Other Special Elements

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 35-44

    5. Automorphic Orbits

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 45-64

  4. Polynomial Algebras

    1. Front Matter

      Pages 65-70
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    2. The Jacobian Conjecture

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 71-79

    3. The Cancellation Conjecture

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 80-91

    4. Nagata’s Problem

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 92-107

    5. The Embedding Problem

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 108-128

    6. Coordinate Polynomials

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 129-168

    7. Test Elements of Polynomial and Free Associative Algebras

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 169-181

  5. Free Nielsen-Schreier Algebras

    1. Front Matter

      Pages 183-189
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    2. Schreier Varieties of Algebras

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 190-212

    3. Rank Theorems and Primitive Elements

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 213-243

    4. Generalized Primitive Elements

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 244-269

    5. Free Leibniz Algebras

      • Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu
      Pages 270-282

  6. Back Matter

    Pages 283-315
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Keywords

About this book

This book is about three seemingly independent areas of mathematics: combinatorial group theory, the theory of Lie algebras and affine algebraic geometry. Indeed, for many years these areas were being developed fairly independently. Combinatorial group theory, the oldest of the three, was born in the beginning of the 20th century as a branch of low-dimensional topology. Very soon, it became an important area of mathematics with its own powerful techniques. In the 1950s, combinatorial group theory started to influence, rather substantially, the theory of Lie algebrasj thus combinatorial theory of Lie algebras was shaped, although the origins of the theory can be traced back to the 1930s. In the 1960s, B. Buchberger introduced what is now known as Gröbner bases. This marked the beginning of a new, "combinatorial", era in commu­ tative algebra. It is not very likely that Buchberger was directly influenced by ideas from combinatorial group theory, but his famous algorithm bears resemblance to Nielsen's method, although in a more sophisticated form.

Reviews

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)

Authors and Affiliations

  • Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

    Alexander A. Mikhalev

  • Department of Mathematics, The City College of New York, New York, USA

    Vladimir Shpilrain

  • Department of Mathematics, The University of Hong Kong, Hong Kong, China

    Jie-Tai Yu

Bibliographic Information

  • Book Title: Combinatorial Methods

  • Book Subtitle: Free Groups, Polynomials, and Free Algebras

  • Authors: Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu

  • Series Title: CMS Books in Mathematics

  • DOI: https://doi.org/10.1007/978-0-387-21724-6

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 2004

  • Hardcover ISBN: 978-0-387-40562-9Published: 14 November 2003

  • Softcover ISBN: 978-1-4419-2344-8Published: 14 December 2011

  • eBook ISBN: 978-0-387-21724-6Published: 12 November 2012

  • Series ISSN: 1613-5237

  • Series E-ISSN: 2197-4152

  • Edition Number: 1

  • Number of Pages: XII, 315

  • Topics: Algebraic Geometry, Non-associative Rings and Algebras

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