Engel's theorem

From Groupprops
Jump to: navigation, search
This fact is related to: Lie algebras
View other facts related to Lie algebrasView terms related to Lie algebras |

Statement

Here are two versions. The first is the basic statement, and the second follows:

  1. Let V be a finite-dimensional vector space (over any field), and L a Lie subalgebra of gl(V) such that every element of L induces a nilpotent linear transformation on V. Then, there is a basis for V in which all elements of L are strictly upper triangular matrices.
  2. If L is a finite-dimensional Lie algebra over a field such that for any x \in L, ad \, x is nilpotent, then L is a nilpotent Lie algebra. In other words, for finite-dimensional Lie algebras over a field, being an Engel Lie ring is equivalent to being a nilpotent Lie ring.

Related facts

Facts

Engel's theorem does not say that any nilpotent Lie algebra inside gl(V) can be given a basis where it is upper triangular. For instance, a one-dimensional Lie algebra is Abelian and hence nilpotent, yet one may not be able to change basis to make it strictly upper triangular.