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Engel's theorem

## Summary

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; 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, 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 implies the following 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 of 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 as ${\displaystyle X}$  is nilpotent; see Jordan decomposition in Lie algebras). Thus, by inductive hypothesis applied to the Lie subalgebra of ${\displaystyle {\mathfrak {gl}}({\mathfrak {g}}/{\mathfrak {h}})}$  generated by ${\displaystyle \operatorname {ad} ({\mathfrak {h}})}$ , 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 of codimension one. 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. ${\displaystyle \square }$

## 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. Vol. 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