Engel’s Theorem and Lie’s Theorem

Abstract

A useful result in linear algebra states that if V is a finite-dimensional vector space and x : V → V is a nilpotent linear map, then there is a basis of V in which x is represented by a strictly upper triangular matrix.

To understand Lie algebras, we need a much more general version of this result. Instead of considering a single linear transformation, we consider a Lie subalgebra L of gl(V). We would like to know when there is a basis of V in which every element of L is represented by a strictly upper triangular matrix.

As a strictly upper triangular matrix is nilpotent, if such a basis exists then every element of L must be a nilpotent map. Surprisingly, this obvious necessary condition is also sufficient; this result is known as Engel’s Theorem.

It is also natural to ask the related question: When is there a basis of V in which every element of L is represented by an upper triangular matrix? If there is such a basis, then L is isomorphic to a subalgebra of a Lie algebra of upper triangular matrices, and so L is solvable. Over C at least, this necessary condition is also sufficient. We prove this result, Lie’s Theorem, in §6.4 below.