Handbook of Linear Algebra, 2nd Edition

Chapter 87

Lie Algebras

Robert Wilson

Rutgers University

A Lie algebra is a (nonassociative) algebra satisfying x2 = 0 for all elements x of the algebra (which implies anticommutativity) and the Jacobi identity. Lie algebras arise nat urally 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, classi fication, 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 ...