Engel's theorem

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

${\displaystyle \operatorname {ad} (X)\colon {\mathfrak {g}}\to {\mathfrak {g}},}$

given by ${\displaystyle \operatorname {ad} (X)(Y)=[X,Y]}$, is a nilpotent endomorphism on ${\displaystyle {\mathfrak {g}}}$; i.e., ${\displaystyle \operatorname {ad} (X)^{k}=0}$ 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).

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). Engel's student K.A. Umlauf gave a complete proof in his 1891 dissertation, reprinted as (Umlauf 2010).

Statements

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

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

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

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

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

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

Proof

We prove the following form of the theorem:[2] if ${\displaystyle {\mathfrak {g}}\subset {\mathfrak {gl}}(V)}$ is a Lie subalgebra such that every ${\displaystyle X\in {\mathfrak {g}}}$ is a nilpotent endomorphism and if V has positive dimension, then there exists a nonzero vector v in V such that ${\displaystyle X(v)=0}$ for each X in ${\displaystyle {\mathfrak {g}}}$.

The proof is by induction on the dimension of ${\displaystyle {\mathfrak {g}}}$ 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 ${\displaystyle {\mathfrak {g}}}$ is positive.

Step 1: Find an ideal ${\displaystyle {\mathfrak {h}}}$ of codimension one in ${\displaystyle {\mathfrak {g}}}$.

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

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

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

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

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

See also

• Lie's theorem
• Heisenberg group

Notes

Citations

1. Fulton & Harris 1991, Exercise 9.10..
2. Fulton & Harris 1991, Theorem 9.9..

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. Volume 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. 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, MR 1771134
• 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) [First published 1891], Über Die Zusammensetzung Der Endlichen Continuierlichen Transformationsgruppen, Insbesondre Der Gruppen Vom Range Null, Inaugural-Dissertation, Leipzig (in German), Nabu Press, ISBN 978-1-141-58889-3