希腊数学的最高成就是正多面体的分类,即五种所谓的柏拉图体。最复杂的正多面体是二十面体。直到19世纪,数学中最重要的问题是解代数方程。在这本经典著作中,Klein展示了如何将这两个看似无关的主题联系起来,并将它们与另一个新的数学理论联系在一起:超几何函数和单值群。这清楚地表明了克莱因对数学统一性的高瞻远瞩。本书包括Peter Slodowy的评注和他关于Klein这本经典著作的解释性论文,从而帮助读者理解ADE的分类,以及它们在当前研究中的许多意想不到的联系和应用。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