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

for some k.[1] 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 (i.e. the naïve replacement in Lie's theorem of "solvable" with "nilpotent", and "upper triangular" with "strictly upper triangular", is false; this already fails for the one-dimensional Lie subalgebra of scalar matrices).

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 (Hawkins 2000, p. 176).

Umlauf gave a complete proof in his 1891 dissertation, reprinted as (Umlauf 2010).

be the Lie algebra of the endomorphisms of a finite-dimensional vector space V and

Then Engel's theorem states the following are equivalent: 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 This is the form of the theorem proven in #Proof.

(This statement is trivially equivalent to Statement 2 since it allows one to inductively construct a flag with the required property.)

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

Then Engel's theorem implies the following theorem (also called Engel's theorem): when

has finite dimension, Indeed, if

consists of nilpotent operators, then by 1.

2. applied to the algebra

{\displaystyle \operatorname {ad} ({\mathfrak {g}})\subset {\mathfrak {gl}}({\mathfrak {g}})}

, there exists a flag

(The converse follows straightforwardly from the definition.)

We prove the following form of the theorem:[2] if

is a nilpotent endomorphism and if V has positive dimension, then there exists a nonzero vector v in V such that

The proof is by induction on the dimension of

and consists of a few steps.

(Note the structure of the proof is very similar to that for Lie's theorem, which concerns a solvable algebra.)

The basic case is trivial and we assume the dimension of

Step 1: Find an ideal

Step 3: Finish up the proof by finding a nonzero vector that gets killed by