Combinatorial Methods: Free Groups, Polynomials, and Free Algebras [Paperback]

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 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 Gr?bner 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.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.

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