Abstract Lie Algebras

One purpose of this book is to give a solid but compact account of the theory of Lie algebras over fields of characteristic 0, with emphasis on the basic simplicity of the theory and on new approaches to the major theorems. Another is to give a general and extensive treatment of Cartan and related sub-algebras of Lie algebras over arbitrary fields. The first two chapters present preliminary material on modules and non-associative algebras. This is followed by a compact self-contained development of the theory of Lie algebras of characteristic 0, covering the following topics: Solvable and nilpotent Lie algebras and the theorems of Lie and Cartan; Cartan subalgebras and Cartan's criteria for solvability and semisimplicity; Levi's radical splitting theorem and the complete reducibility of representations of semisimple Lie algebras; The theory of abstract root systems and the classification of semisimple Lie algebras; The isomorphism theorem for semisimple Lie algebras and their irreducible modules; and automorphisms of Lie algebras and the conjugacy of Cartan subalgebras and Borel subalgebras. The last chapter develops an extensive theory of Cartan and related subalgebras of Lie algebras over arbitrary fields, covering the following topics: Lie algebras graded by a group; nilpotent Lie algebras and their representations; Engel subalgebras and Fitting subalgebras; Cartan subalgebras; and tori in Lie p-algebras.