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 (i.e. the naÃÂ¯ve 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 . 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. (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 = (i+1)-th power of , there is some k such that . Then Engel's theorem implies the following theorem (also called Engel's 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. (The converse follows straightforwardly from the definition.)

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. (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 is positive.

Step 1: Find an ideal of codimension one in .

This is the most difficult step. Let be a maximal (proper) subalgebra of , which exists by finite-dimensionality. We claim it is an ideal of codimension one. For each , it is easy to check that (1) induces a linear endomorphism and (2) this induced map is nilpotent (in fact, is nilpotent as is nilpotent; see Jordan decomposition in Lie algebras). Thus, by inductive hypothesis applied to the Lie subalgebra of generated by , there exists a nonzero vector v in such that for each . That is to say, if for some Y in but not in , then for every . But then the subspace spanned by and Y is a Lie subalgebra in which is an ideal of codimension one. Hence, by maximality, . This proves the claim.

Step 2: Let . Then stabilizes W; i.e., for each .

Indeed, for in and in , we have: since is an ideal and so . Thus, is in W.

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

Write where L is a one-dimensional vector subspace. Let Y be a nonzero vector in L and v a nonzero vector in W. Now, is a nilpotent endomorphism (by hypothesis) and so for some k. Then is a required vector as the vector lies in W by Step 2.

Notes

Citations

Works cited