Ulteriori informazioni
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.
Sommario
Preface.- Introduction.- I. Groups : Introduction. Classical Techniques. Test Elements. Other Special Elements. Automorphic Orbits.- II. Polynomial Algebras : Introduction. The Jacobian Conjecture. The Cancellation Conjecture. Nagata's Problem. The Embedding Problem. Coordinate Polynomials. Test Polynomials.- III. 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.
Riassunto
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.
Testo aggiuntivo
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)
Relazione
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)
