Engel's theorem

In representation theory,[] a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra is a nilpotent Lie algebra if and only if for each, the adjoint map
given by, is a nilpotent endomorphism on ; i.e., for some k. It is a consequence of the theorem, also called Engel's theorem, which says that if a Lie algebra of matrices consists of nilpotent matrices, then the matrices can all be simultaneously brought to a strictly upper triangular form. Note that if we merely have a Lie algebra of matrices which is nilpotent as a Lie algebra, then this conclusion does not follow.
The theorem is named after the mathematician Friedrich Engel, who sketched a proof of it in a letter to Wilhelm Killing dated 20 July 1890. Engel's student K.A. Umlauf gave a complete proof in his 1891 dissertation, reprinted as.

Statements

Let be the Lie algebra of the endomorphisms of a finite-dimensional vector space V and a subalgebra. Then Engel's theorem states the following are equivalent:

Each is a nilpotent endomorphism on V.
There exists a flag such that ; i.e., the elements of are simultaneously strictly upper-triangulizable.

Note that no assumption on the underlying base field is required.
We note that Statement 2. for various and V is equivalent to the statement

For each nonzero finite-dimensional vector space V and a subalgebra, there exists a nonzero vector v in V such that for every

This is the form of the theorem proven in #Proof.
In general, a Lie algebra is said to be nilpotent if the lower central series of it vanishes in a finite step; i.e., for = -th power of, there is some k such that. Then Engel's theorem implies the following theorem : when has finite dimension,

is nilpotent if and only if is nilpotent for each.

Indeed, if consists of nilpotent operators, then by 1. 2. applied to the algebra, there exists a flag such that. Since, this implies is nilpotent.

Proof

We prove the following form of the theorem: if is a Lie subalgebra such that every is a nilpotent endomorphism and if V'' has positive dimension, then there exists a nonzero vector v in V such that for each X in.
The proof is by induction on the dimension of and consists of a few steps. The basic case is trivial and we assume the dimension of is positive.
Step 1: Find an ideal of codimension one in.
Step 2: Let. Then stabilizes W''; i.e., for each.
Step 3: Finish up the proof by finding a nonzero vector that gets killed by.