1 - Finite-dimensional Lie algebras

Abstract

History of the development of finite-dimensional Lie algebras is described in the preface itself. Lie theory has its name from the work of Sophus Lie [6], who studied certain transformation groups, that is, the groups of symmetries of algebraic or geometric objects that are now called Lie groups. Using the researches of Sophus Lie and Wilhelm Killing, Cartan [9] in his 1894 thesis, completed the classification of finite-dimensional simple Lie algebras over C. The nine types of this classification (consisting of the four classes of classical simple Lie algebras and five exceptional simple Lie algebras) correspond to the nine types of finite Cartan matrices and to the nine types of Dynkin Diagrams [10, 11]. Chevalley [12] and Harish-Chandra [13] constructed a scheme that began with a finite Cartan matrix and produced finite-dimensional simple Lie algebra. During 1976, Serre [14–16] proved the defining relations on the generators and Cartan integers (elements of the Cartan matrix) of the finite-dimensional complex semi-simple Lie algebras.