Combinatorial Methods: Free Groups, Polynomials, and Free Algebras
Combinatorial Methods: Free Groups, Polynomials, and Free Algebras
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DOI：
10.1007/978-0-387-21724-6
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发表时间：
2003-11
期刊：
影响因子：
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通讯作者：
A. Mikhalev;V. Shpilrain;Jietai Yu
中科院分区：
文献类型：
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作者：
A. Mikhalev;V. Shpilrain;Jietai Yu
本书的主要目的是展示组合群论的思想如何传播到数学的其他两个领域：李代数理论和仿射代数几何。其中一些想法又在 20 世纪初从低维拓扑演变为组合群论。本书分为三个相当独立的部分。第一部分简要阐述了组合群论中的几种经典技术，即 Nielsen、Whitehead 和 Tietze 的方法。第二部分包含了本书的主要重点。在这里，作者展示了上述组合群论技术如何进入仿射代数几何，仿射代数几何是研究多项式和多项式映射的一个令人着迷的数学领域。第三部分说明了组合群论的思想如何对自由代数理论做出了贡献。这里的重点是代数的 Schreier 簇（如果该簇的自由代数的任何子代数在同一代数簇中是自由的，则称该代数簇是 Schreier）。
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).