ABSTRACT
A Lie algebra is a (nonassociative) algebra satisfying x 2= 0 for all elements x of the algebra (which implies anticommutativity) and the Jacobi identity. Lie algebras arise naturally as (vector) subspaces of associative algebras closed under the commutator operation [a, b] = ab − ba. The finite-dimensional simple Lie algebras over algebraically closed fields of characteristic zero occur in many applications. This chapter outlines the structure, classification, and representation theory of these algebras. We also give examples of other types of algebras, e.g., one class of infinite-dimensional simple algebras and one class of finite-dimensional simple Lie algebras over fields of prime characteristic. Section 70.1 is devoted to general definitions about Lie algebras. Section 70.2 discusses semisimple and simple algebras. This section includes the classification of finite-dimensional simple Lie algebras over algebraically closed fields of characteristic zero. Section 70.3 discusses module theory and includes the classification of finite-dimensional irreducible modules for the aforementioned algebras as well as the explicit construction of some of these modules. Section 70.4 discusses graded algebras and modules and uses this formalism to present results on dimensions of irreducible modules.
