# Engel's theorem

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Theorem in Lie representation theory

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

- ad ⁡ ( X ) : g → g , {\displaystyle \operatorname {ad} (X)\colon {\mathfrak {g}}\to {\mathfrak {g}},}

given by ad ⁡ ( X ) ( Y ) = [ X , Y ] {\displaystyle \operatorname {ad} (X)(Y)=[X,Y]} , is a [nilpotent endomorphism](/source/Nilpotent_endomorphism) on g {\displaystyle {\mathfrak {g}}} ; i.e., ad ⁡ ( X ) k = 0 {\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](/source/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](/source/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](/source/Subalgebra) of scalar matrices).

The theorem is named after the mathematician [Friedrich Engel](/source/Friedrich_Engel_(mathematician)), who sketched a proof of it in a letter to [Wilhelm Killing](/source/Wilhelm_Killing) dated 20 July 1890 ([Hawkins 2000](#CITEREFHawkins2000), p. 176). Engel's student K.A. Umlauf gave a complete proof in his 1891 dissertation, reprinted as ([Umlauf 2010](#CITEREFUmlauf2010)).

## Statements

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

1. Each X ∈ g {\displaystyle X\in {\mathfrak {g}}} is a nilpotent endomorphism on *V*.

1. There exists a flag V = V 0 ⊃ V 1 ⊃ ⋯ ⊃ V n = 0 , codim ⁡ V i = i {\displaystyle V=V_{0}\supset V_{1}\supset \cdots \supset V_{n}=0,\,\operatorname {codim} V_{i}=i} such that g ⋅ V i ⊂ V i + 1 {\displaystyle {\mathfrak {g}}\cdot V_{i}\subset V_{i+1}} ; i.e., the elements of g {\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 g {\displaystyle {\mathfrak {g}}} and *V* is equivalent to the statement

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

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

- g {\displaystyle {\mathfrak {g}}} is nilpotent if and only if ad ⁡ ( X ) {\displaystyle \operatorname {ad} (X)} is nilpotent for each X ∈ g {\displaystyle X\in {\mathfrak {g}}} .

Indeed, if ad ⁡ ( g ) {\displaystyle \operatorname {ad} ({\mathfrak {g}})} consists of nilpotent operators, then by 1. ⇔ {\displaystyle \Leftrightarrow } 2. applied to the algebra ad ⁡ ( g ) ⊂ g l ( g ) {\displaystyle \operatorname {ad} ({\mathfrak {g}})\subset {\mathfrak {gl}}({\mathfrak {g}})} , there exists a flag g = g 0 ⊃ g 1 ⊃ ⋯ ⊃ g n = 0 {\displaystyle {\mathfrak {g}}={\mathfrak {g}}_{0}\supset {\mathfrak {g}}_{1}\supset \cdots \supset {\mathfrak {g}}_{n}=0} such that [ g , g i ] ⊂ g i + 1 {\displaystyle [{\mathfrak {g}},{\mathfrak {g}}_{i}]\subset {\mathfrak {g}}_{i+1}} . Since C i g ⊂ g i {\displaystyle C^{i}{\mathfrak {g}}\subset {\mathfrak {g}}_{i}} , this implies g {\displaystyle {\mathfrak {g}}} is nilpotent. (The converse follows straightforwardly from the definition.)

## Proof

We prove the following form of the theorem:[2] *if g ⊂ g l ( V ) {\displaystyle {\mathfrak {g}}\subset {\mathfrak {gl}}(V)} is a Lie subalgebra such that every X ∈ g {\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 X ( v ) = 0 {\displaystyle X(v)=0} for each*X*in g {\displaystyle {\mathfrak {g}}} .*

The proof is by induction on the dimension of g {\displaystyle {\mathfrak {g}}} and consists of a few steps. (Note the structure of the proof is very similar to that for [Lie's theorem](/source/Lie's_theorem), which concerns a solvable algebra.) The basic case is trivial and we assume the dimension of g {\displaystyle {\mathfrak {g}}} is positive.

**Step 1**: Find an ideal h {\displaystyle {\mathfrak {h}}} of [codimension](/source/Codimension) one in g {\displaystyle {\mathfrak {g}}} .

- This is the most difficult step. Let h {\displaystyle {\mathfrak {h}}} be a maximal (proper) subalgebra of g {\displaystyle {\mathfrak {g}}} , which exists by finite-dimensionality. We claim it is an ideal of codimension one. For each X ∈ h {\displaystyle X\in {\mathfrak {h}}} , it is easy to check that (1) ad ⁡ ( X ) {\displaystyle \operatorname {ad} (X)} induces a linear endomorphism g / h → g / h {\displaystyle {\mathfrak {g}}/{\mathfrak {h}}\to {\mathfrak {g}}/{\mathfrak {h}}} and (2) this induced map is nilpotent (in fact, ad ⁡ ( X ) {\displaystyle \operatorname {ad} (X)} is nilpotent as X {\displaystyle X} is nilpotent; see [Jordan decomposition in Lie algebras](/source/Jordan%E2%80%93Chevalley_decomposition#Preservation_under_representations)). Thus, by inductive hypothesis applied to the Lie subalgebra of g l ( g / h ) {\displaystyle {\mathfrak {gl}}({\mathfrak {g}}/{\mathfrak {h}})} generated by ad ⁡ ( h ) {\displaystyle \operatorname {ad} ({\mathfrak {h}})} , there exists a nonzero vector *v* in g / h {\displaystyle {\mathfrak {g}}/{\mathfrak {h}}} such that ad ⁡ ( X ) ( v ) = 0 {\displaystyle \operatorname {ad} (X)(v)=0} for each X ∈ h {\displaystyle X\in {\mathfrak {h}}} . That is to say, if v = [ Y ] {\displaystyle v=[Y]} for some *Y* in g {\displaystyle {\mathfrak {g}}} but not in h {\displaystyle {\mathfrak {h}}} , then [ X , Y ] = ad ⁡ ( X ) ( Y ) ∈ h {\displaystyle [X,Y]=\operatorname {ad} (X)(Y)\in {\mathfrak {h}}} for every X ∈ h {\displaystyle X\in {\mathfrak {h}}} . But then the subspace h ′ ⊂ g {\displaystyle {\mathfrak {h}}'\subset {\mathfrak {g}}} spanned by h {\displaystyle {\mathfrak {h}}} and *Y* is a Lie subalgebra in which h {\displaystyle {\mathfrak {h}}} is an ideal of codimension one. Hence, by maximality, h ′ = g {\displaystyle {\mathfrak {h}}'={\mathfrak {g}}} . This proves the claim.

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

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

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

- Write g = h + L {\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, Y {\displaystyle Y} is a nilpotent endomorphism (by hypothesis) and so Y k ( v ) ≠ 0 , Y k + 1 ( v ) = 0 {\displaystyle Y^{k}(v)\neq 0,Y^{k+1}(v)=0} for some *k*. Then Y k ( v ) {\displaystyle Y^{k}(v)} is a required vector as the vector lies in *W* by Step 2. ◻ {\displaystyle \square }

## See also

- [Lie's theorem](/source/Lie's_theorem)

- [Heisenberg group](/source/Heisenberg_group)

## Notes

### Citations

1. **[^](#cite_ref-FOOTNOTEFultonHarris1991Exercise_9.10._1-0)** [Fulton & Harris 1991](#CITEREFFultonHarris1991), Exercise 9.10..

1. **[^](#cite_ref-FOOTNOTEFultonHarris1991Theorem_9.9._2-0)** [Fulton & Harris 1991](#CITEREFFultonHarris1991), Theorem 9.9..

## Works cited

- [Erdmann, Karin](/source/Karin_Erdmann); Wildon, Mark (2006). *Introduction to Lie Algebras* (1st ed.). Springer. [ISBN](/source/ISBN_(identifier)) [1-84628-040-0](https://en.wikipedia.org/wiki/Special:BookSources/1-84628-040-0).

- [Fulton, William](/source/William_Fulton_(mathematician)); [Harris, Joe](/source/Joe_Harris_(mathematician)) (1991). *Representation theory. A first course*. [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics), Readings in Mathematics. Vol. 129. New York: Springer-Verlag. [doi](/source/Doi_(identifier)):[10.1007/978-1-4612-0979-9](https://doi.org/10.1007%2F978-1-4612-0979-9). [ISBN](/source/ISBN_(identifier)) [978-0-387-97495-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-97495-8). [MR](/source/MR_(identifier)) [1153249](https://mathscinet.ams.org/mathscinet-getitem?mr=1153249). [OCLC](/source/OCLC_(identifier)) [246650103](https://search.worldcat.org/oclc/246650103).

- Hawkins, Thomas (2000), [*Emergence of the theory of Lie groups*](https://books.google.com/books?isbn=978-0-387-98963-1), Sources and Studies in the History of Mathematics and Physical Sciences, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [ISBN](/source/ISBN_(identifier)) [978-0-387-98963-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-98963-1), [MR](/source/MR_(identifier)) [1771134](https://mathscinet.ams.org/mathscinet-getitem?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*](https://books.google.com/books?isbn=978-1141588893), Inaugural-Dissertation, Leipzig (in German), Nabu Press, [ISBN](/source/ISBN_(identifier)) [978-1-141-58889-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-141-58889-3)

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Adapted from the Wikipedia article [Engel's theorem](https://en.wikipedia.org/wiki/Engel's_theorem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Engel's_theorem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.