希臘數學的最高成就是正多面體的分類，即五種所謂的柏拉圖體。最復雜的正多面體是二十面體。直到19世紀，數學中最重要的問題是解代數方程。在這本經典著作中，Klein展示了如何將這兩個看似無關的主題聯系起來，并將它們與另一個新的數學理論聯系在一起：超幾何函數和單值群。這清楚地表明了克萊因對數學統(tǒng)一性的高瞻遠矚。本書包括Peter Slodowy的評注和他關于Klein這本經典著作的解釋性論文，從而幫助讀者理解ADE的分類，以及它們在當前研究中的許多意想不到的聯系和應用。The highest achievement of the Greek mathematics is the classification of regular solids, the five so-called Platonic solids.The most complicated solid is the icosahedron. Up to and through the 19th century, the mostimportant problem in mathematics was to solvealgebraic equations. In this classic book, Klein showed how to relate these two seemingly unrelated topics and also tied them together with another new theory of mathematics: hypergeometric functions and monodromy groups. This clearly shows Klein's vision of the unity of mathematics.This book includes Peter Slodowy’s commentaries and his expository paper on Klein's book to help readers to understand the ADE classification, and their many unexpected connections and applications under current study.
The republication of Felix Klein's “Lectures on the icosahedron and the solution ofequations of the fifth degree” corresponds to the constantly growing demand for thiswork， which was published in Leipzig more than a hundred years ago. A good deal ofin-terest in Klein's book might be certainly due to the continuous relevance of the “icosa-hedral mathematics”， i.e. the mathematics， in which the geometry and symmetry ofthe icosahedron， as well as the other Platonic solids and the regular polygons， play an essential role. In this regard， the foUowing developments in the last twenty years are mentioned： the study of the so-called Klein singularities， also known as Du Val singu-larities， rational double points or simple singularities （see e.g. Du Val [1934l， M. Artin [1966J， Brieskorn [1968]， [1970]， Arnol'd [1972]or the survey articles ofArnol'd [1974]，Brieskorn [1976]， Durfee [1979]， Slowdowy [1983]， the investigation of certain ellip-tic and Hilbert-Blumenthal modular surfaces （see Hirzebruch [1976]， [1977]， Naruki [1978]， Burns [1983]）， the construction of an indecomposable vector bundle of rank 2 on P4 （Horrocks-Mumford [1973]and the analysis ofits properties （see Barth-Hulek-Moore [1984]， [1987]， Barth-Hulek [1985]， Decker-Schreyer [1986]， Hulek [1986]， [1987]，Hulek-Lange[1988]and the survey article of Hulek [1989]）. A particularly remarkable fact， especially with regard to Klein's thanks to Sophus Lie in the preface of his book，is the relationship between the Platonic solids， or more precisely the finite subgroups of SU（2，C）， and the complex simple Lie groups of types Ar， Dr， Er， which was discov-ered by Grothendieck and Brieskorn （see Brieskorn [1970]）. While this discovery built on deep studies on the theory of resolution and deformation of the singularities of sur-faces mentioned above and the geometry of the conjugation classes ofsimple algebraic groups， a more direct， although more formal derivation of this relationship was given by J. McKay， who showed how the irreducible characters of finite binary groups can be parameterized in a natural way by the vertices of the extended Coxeter-Witt-Dynkin diagrams ofthe corresponding Lie groups （see McKay [1980]， Ford-McKay [1979]）.
Preface of the Republication
Introduction to the Subject of the Icosahedral Book
Lectures on the Icosahedron and the Solution of
Equations of the Fifth Degree
Title page，Preface and Contents
Original Text
Table
Appendix
Comments on the Text
Further Developments
Literature
Appendix： The Icosahedrons and the Equations of the Fifth Degree
書單推薦
