In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra \mathfrak g 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 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. 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: Note that no assumption on the underlying base field is required. We note that Statement 2. for various \mathfrak g and V is equivalent to the statement This is the form of the theorem proven in. (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 \mathfrak g 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 \mathfrak g, there is some k such that. Then Engel's theorem implies the following theorem (also called Engel's theorem): when \mathfrak g has finite dimension, Indeed, if consists of nilpotent operators, then by 1. 2. applied to the algebra, there exists a flag such that. Since, this implies \mathfrak g 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 X(v) = 0 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. Step 2: Let. Then stabilizes W; i.e., X (v) \in W for each. Step 3: Finish up the proof by finding a nonzero vector that gets killed by.

Citations