Engel's theorem explained

In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra

ak g is a nilpotent Lie algebra if and only if for each X \in ak g , the adjoint map\operatorname{ad}(X)\colon ak{g} \to ak{g}, given by \operatorname{ad}(X)(Y) = [X, Y] , is a nilpotent endomorphism on ak{g} ; i.e., \operatorname{ad}(X) ^{k} = 0 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

ak{gl}(V) be the Lie algebra of the endomorphisms of a finite-dimensional vector space V and ak g \subset ak{gl}(V) a subalgebra. Then Engel's theorem states the following are equivalent:

1. Each

X \in ak{g} is a nilpotent endomorphism on V.

1. There exists a flag

V = V _{0} \supset V _{1} \supset … \supset V _{n} = 0 , \operatorname{codim} V _{i} = i such that ak g ⋅ V _{i} \subset V _{i+1} ; i.e., the elements of ak g are simultaneously strictly upper-triangulizable.

Note that no assumption on the underlying base field is required.

We note that Statement 2. for various

ak g and V is equivalent to the statement

• For each nonzero finite-dimensional vector space V and a subalgebra

ak g \subset ak{gl}(V) , there exists a nonzero vector v in V such that X(v) = 0 for every X \in ak g.

This is the form of the theorem proven in

1. 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

ak g is said to be nilpotent if the lower central series of it vanishes in a finite step; i.e., for C ^{0} ak g = ak g, C ^{i} ak g = [ ak g, C ^{i-1} ak g] = (i+1)-th power of ak g , there is some k such that C ^{k} ak g = 0 . Then Engel's theorem implies the following theorem (also called Engel's theorem): when ak g has finite dimension,ak g is nilpotent if and only if \operatorname{ad}(X) is nilpotent for each X \in ak g .Indeed, if \operatorname{ad}( ak g) consists of nilpotent operators, then by 1. \Leftrightarrow 2. applied to the algebra \operatorname{ad}( ak g) \subset ak{gl}( ak g) , there exists a flag ak g = ak{g} _{0} \supset ak{g} _{1} \supset … \supset ak{g} _{n} = 0 such that [ ak g, ak g _{i]} \subset ak g _{i+1} . Since C ^{i} ak g\subset ak g _{i} , this implies ak g is nilpotent. (The converse follows straightforwardly from the definition.)

Proof

We prove the following form of the theorem: if

ak{g} \subset ak{gl}(V) is a Lie subalgebra such that every X \in ak{g} 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 ak{g} .

The proof is by induction on the dimension of

ak{g} 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 ak{g} is positive.

Step 1: Find an ideal

ak{h} of codimension one in ak{g} .

This is the most difficult step. Let

ak{h} be a maximal (proper) subalgebra of ak{g} , which exists by finite-dimensionality. We claim it is an ideal of codimension one. For each X \in ak h , it is easy to check that (1) \operatorname{ad}(X) induces a linear endomorphism ak{g}/ ak{h} \to ak{g}/ ak{h} and (2) this induced map is nilpotent (in fact, \operatorname{ad}(X) is nilpotent as X is nilpotent; see Jordan decomposition in Lie algebras). Thus, by inductive hypothesis applied to the Lie subalgebra of ak{gl}( ak{g}/ ak{h}) generated by \operatorname{ad}( ak{h}) , there exists a nonzero vector v in ak{g}/ ak{h} such that \operatorname{ad}(X)(v) = 0 for each X \in ak{h} . That is to say, if v = [Y] for some Y in ak{g} but not in ak h , then [X, Y] = \operatorname{ad}(X)(Y) \in ak{h} for every X \in ak{h} . But then the subspace ak{h}' \subset ak{g} spanned by ak{h} and Y is a Lie subalgebra in which ak{h} is an ideal of codimension one. Hence, by maximality, ak{h}' = ak g . This proves the claim.

Step 2: Let

W = \{ v \in V | X(v) = 0 , X \in ak{h} \} . Then ak{g} stabilizes W; i.e., X (v) \in W for each X \in ak{g}, v \in W .

Indeed, for

Y in ak{g} and X in ak{h} , we have: X(Y(v)) = Y(X(v)) + [X, Y](v) = 0 since ak{h} is an ideal and so [X, Y] \in ak{h} . Thus, Y(v) is in W.

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

ak{g} .

Write

ak{g} = ak{h} + L where L is a one-dimensional vector subspace. Let Y be a nonzero vector in L and v a nonzero vector in W. Now, Y is a nilpotent endomorphism (by hypothesis) and so Y ^{k(v)} \ne 0 , Y ^{k+1} (v) = 0 for some k. Then Y ^{k(v)} is a required vector as the vector lies in W by Step 2. \square

See also

• Lie's theorem
• Heisenberg group

Notes

Citations

Works cited

• Book: Introduction to Lie Algebras . 1st . Erdmann . Karin . Wildon . Mark . Karin Erdmann . 2006 . Springer . 1-84628-040-0.
• Book: Representation theory. A first course . Fulton . William . Harris . Joe . William Fulton (mathematician) . Joe Harris (mathematician) . 1991 . Springer-Verlag . New York . 129 . Graduate Texts in Mathematics, Readings in Mathematics . 10.1007/978-1-4612-0979-9 . 978-0-387-97495-8 . 1153249 . 246650103.
  • Book: Hochschild, G. . The Structure of Lie Groups . 1965 . Holden Day.
• Book: Humphreys, J. . Introduction to Lie Algebras and Representation Theory . 1972 . Springer.