Combinatorial methods: free groups, polynomials, and free algebras / Alexander A. Mikhalev, Vladimir Shpilrain, Jie-Tai Yu

Auteur: Mikhalev, Alexander Aleksandrovich (1965-) - AuteurCo-auteur: Shpilrain, Vladimir (1960-) - Auteur ; Yu, Jie-Tai (1954-) - AuteurType de document: MonographieCollection: CMS books in mathematics ; 19Langue: anglaisPays: AllemagneÉditeur: Berlin : Springer, 2004Description: 1 vol. (XII-315 p.) ; 24 cm ISBN: 9780387405629 ; rel. Résumé: 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. For many years the theories of these three types of objects have been developed without having much influence on each other. Combinatorial group theory is the oldest one and was created at the beginning of the 20th century to address the needs of topology. The combinatorial theory of polynomials was inspired mainly by algebraic geometry and the need to perform computations with concrete commutative algebras; combinatorial theory of free associative algebras often mimicked combinatorial commutative algebra. Finally, the theory of free Nielsen-Schreier algebras generalized combinatorial theory of free Lie algebras which was influenced by the theory of free groups. Of course, there have been some more or less occasional interactions between these three theories. The real and systematic cooperation between the theories of free groups, polynomial and free associative algebras and free Lie and free Nielsen-Schreier algebras started much later, maybe in the 1970's and 1980's. The book treats combinatorial theory of free groups, polynomial algebras and free Nielsen-Schreier algebras from the same point of view. 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. I believe that the algebraic community will find the book interesting and useful. The text is suitable both for beginners and experts. Each of the parts of the book (free groups, polynomial algebras, free Nielsen-Schreier algebras) could serve as a course on combinatorial theory of the corresponding objects, but the lecturer would need to select some chapters and sections of the corresponding part (and perhaps to extend the exposition of the course with some other text). Another possibility is to use the book for a course on topics similar to those included in each part of the book or to include chapters of the book in a graduate course on combinatorial algebra. It will also serve as a good reference book. ... (MathSciNet).Bibliographie: Notes bibliogr. Index. Sujets MSC: 16R10 Associative rings and algebras -- Rings with polynomial identity -- T-ideals, identities, varieties of rings and algebras
16S36 Associative rings and algebras -- Rings and algebras arising under various constructions -- Ordinary and skew polynomial rings and semigroup rings
17A32 Nonassociative rings and algebras -- General nonassociative rings -- Leibniz algebras
20E05 Group theory and generalizations -- Structure and classification of infinite or finite groups -- Free nonabelian groups
20E36 Group theory and generalizations -- Structure and classification of infinite or finite groups -- Automorphisms of infinite groups
En-ligne: Springerlink | Zentralblatt | MathSciNet
Location Call Number Status Date Due
Salle R 05008-01 / 16 MIK (Browse Shelf) Available

Notes bibliogr. Index

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.
For many years the theories of these three types of objects have been developed without having much influence on each other. Combinatorial group theory is the oldest one and was created at the beginning of the 20th century to address the needs of topology. The combinatorial theory of polynomials was inspired mainly by algebraic geometry and the need to perform computations with concrete commutative algebras; combinatorial theory of free associative algebras often mimicked combinatorial commutative algebra. Finally, the theory of free Nielsen-Schreier algebras generalized combinatorial theory of free Lie algebras which was influenced by the theory of free groups. Of course, there have been some more or less occasional interactions between these three theories. The real and systematic cooperation between the theories of free groups, polynomial and free associative algebras and free Lie and free Nielsen-Schreier algebras started much later, maybe in the 1970's and 1980's.
The book treats combinatorial theory of free groups, polynomial algebras and free Nielsen-Schreier algebras from the same point of view. 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.
I believe that the algebraic community will find the book interesting and useful. The text is suitable both for beginners and experts. Each of the parts of the book (free groups, polynomial algebras, free Nielsen-Schreier algebras) could serve as a course on combinatorial theory of the corresponding objects, but the lecturer would need to select some chapters and sections of the corresponding part (and perhaps to extend the exposition of the course with some other text). Another possibility is to use the book for a course on topics similar to those included in each part of the book or to include chapters of the book in a graduate course on combinatorial algebra. It will also serve as a good reference book. ... (MathSciNet)

There are no comments for this item.

Log in to your account to post a comment.
Languages: English | Français | |