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], [15], [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. In this chapter, we give all preliminaries in finite-dimensional Lie algebras which are necessary to develop all other chapters in this book. We start from the definition of Lie algebras, structure constants, subalgebras, ideals, quotient Lie algebras, simple Lie algebras, and semisimple Lie algebras with examples. Homomorphism and isomorphism theorems, Killing form, derivation, representations of semisimple Lie algebras and also that of sl(2,C) are also explained. Moreover, the rootspace decomposition of semisimple Lie algebras, Dynkin diagrams, Cartan matrices, rank and dimensions of simple Lie algebras, Weyl groups, universal enveloping algebras, construction of semisimple Lie algebras by generators and relations, Cartan-Weyl basis, and character of finite-dimensional representations are also explained. Cartan matrices of all classical simple algebras, structures of Weyl groups of simple algebras, Weyl groups, root systems highest short and long roots of all classical simple algebras are given. Basic properties of Lie algebras of vector fields are also given. Throughout this chapter, all Lie algebras are finite dimensional unless otherwise stated.
Publication Year: 2016
Publication Date: 2016-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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Cited By Count: 1
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