Engel's theorem

Short description: Theorem in Lie representation theory

In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra [math]\displaystyle{ \mathfrak g }[/math] is a nilpotent Lie algebra if and only if for each [math]\displaystyle{ X \in \mathfrak g }[/math], the adjoint map

[math]\displaystyle{ \operatorname{ad}(X)\colon \mathfrak{g} \to \mathfrak{g}, }[/math]

given by [math]\displaystyle{ \operatorname{ad}(X)(Y) = [X, Y] }[/math], is a nilpotent endomorphism on [math]\displaystyle{ \mathfrak{g} }[/math]; i.e., [math]\displaystyle{ \operatorname{ad}(X)^k = 0 }[/math] 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 (Hawkins 2000). Engel's student K.A. Umlauf gave a complete proof in his 1891 dissertation, reprinted as (Umlauf 2010).

Statements

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

1. Each [math]\displaystyle{ X \in \mathfrak{g} }[/math] is a nilpotent endomorphism on V.
2. There exists a flag [math]\displaystyle{ V = V_0 \supset V_1 \supset \cdots \supset V_n = 0, \, \operatorname{codim} V_i = i }[/math] such that [math]\displaystyle{ \mathfrak g \cdot V_i \subset V_{i+1} }[/math]; i.e., the elements of [math]\displaystyle{ \mathfrak g }[/math] are simultaneously strictly upper-triangulizable.

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

We note that Statement 2. for various [math]\displaystyle{ \mathfrak g }[/math] and V is equivalent to the statement

• For each nonzero finite-dimensional vector space V and a subalgebra [math]\displaystyle{ \mathfrak g \subset \mathfrak{gl}(V) }[/math], there exists a nonzero vector v in V such that [math]\displaystyle{ X(v) = 0 }[/math] for every [math]\displaystyle{ X \in \mathfrak g. }[/math]

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 [math]\displaystyle{ \mathfrak g }[/math] is said to be nilpotent if the lower central series of it vanishes in a finite step; i.e., for [math]\displaystyle{ C^0 \mathfrak g = \mathfrak g, C^i \mathfrak g = [\mathfrak g, C^{i-1} \mathfrak g] }[/math] = (i+1)-th power of [math]\displaystyle{ \mathfrak g }[/math], there is some k such that [math]\displaystyle{ C^k \mathfrak g = 0 }[/math]. Then Engel's theorem implies the following theorem (also called Engel's theorem): when [math]\displaystyle{ \mathfrak g }[/math] has finite dimension,

• [math]\displaystyle{ \mathfrak g }[/math] is nilpotent if and only if [math]\displaystyle{ \operatorname{ad}(X) }[/math] is nilpotent for each [math]\displaystyle{ X \in \mathfrak g }[/math].

Indeed, if [math]\displaystyle{ \operatorname{ad}(\mathfrak g) }[/math] consists of nilpotent operators, then by 1. [math]\displaystyle{ \Leftrightarrow }[/math] 2. applied to the algebra [math]\displaystyle{ \operatorname{ad}(\mathfrak g) \subset \mathfrak{gl}(\mathfrak g) }[/math], there exists a flag [math]\displaystyle{ \mathfrak g = \mathfrak{g}_0 \supset \mathfrak{g}_1 \supset \cdots \supset \mathfrak{g}_n = 0 }[/math] such that [math]\displaystyle{ [\mathfrak g, \mathfrak g_i] \subset \mathfrak g_{i+1} }[/math]. Since [math]\displaystyle{ C^i \mathfrak g\subset \mathfrak g_i }[/math], this implies [math]\displaystyle{ \mathfrak g }[/math] is nilpotent. (The converse follows straightforwardly from the definition.)

Proof

We prove the following form of the theorem: if [math]\displaystyle{ \mathfrak{g} \subset \mathfrak{gl}(V) }[/math] is a Lie subalgebra such that every [math]\displaystyle{ X \in \mathfrak{g} }[/math] is a nilpotent endomorphism and if V has positive dimension, then there exists a nonzero vector v in V such that [math]\displaystyle{ X(v) = 0 }[/math] for each X in [math]\displaystyle{ \mathfrak{g} }[/math].

The proof is by induction on the dimension of [math]\displaystyle{ \mathfrak{g} }[/math] 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 [math]\displaystyle{ \mathfrak{g} }[/math] is positive.

Step 1: Find an ideal [math]\displaystyle{ \mathfrak{h} }[/math] of codimension one in [math]\displaystyle{ \mathfrak{g} }[/math].

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

Step 2: Let [math]\displaystyle{ W = \{ v \in V | X(v) = 0, X \in \mathfrak{h} \} }[/math]. Then [math]\displaystyle{ \mathfrak{g} }[/math] stabilizes W; i.e., [math]\displaystyle{ X (v) \in W }[/math] for each [math]\displaystyle{ X \in \mathfrak{g}, v \in W }[/math].

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

Step 3: Finish up the proof by finding a nonzero vector that gets killed by [math]\displaystyle{ \mathfrak{g} }[/math].

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

See also

• Lie's theorem
• Heisenberg group

Works cited

• Erdmann, Karin; Wildon, Mark (2006). Introduction to Lie Algebras (1st ed.). Springer. ISBN 1-84628-040-0. 
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. OCLC 246650103. 
• Hawkins, Thomas (2000), Emergence of the theory of Lie groups, Sources and Studies in the History of Mathematics and Physical Sciences, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98963-1, https://books.google.com/books?isbn=978-0-387-98963-1 
• Hochschild, G. (1965). The Structure of Lie Groups. Holden Day. 
• Humphreys, J. (1972). Introduction to Lie Algebras and Representation Theory. Springer. 
• Umlauf, Karl Arthur (2010) (in German), Über Die Zusammensetzung Der Endlichen Continuierlichen Transformationsgruppen, Insbesondre Der Gruppen Vom Range Null, Inaugural-Dissertation, Leipzig, Nabu Press, ISBN 978-1-141-58889-3, https://books.google.com/books?isbn=978-1141588893