Engel's theorem

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

$\mathrm{ad}(X):\mathfrak{𝔤}\to \mathfrak{𝔤},$

given by $\mathrm{ad}(X)(Y)=[X,Y]$, is a nilpotent endomorphism on $\mathfrak{𝔤}$; i.e., $\mathrm{ad}(X{)}^k=0$ 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 (Hawkins 2000, p. 176). Engel's student K.A. Umlauf gave a complete proof in his 1891 dissertation, reprinted as (Umlauf 2010).

Statements

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

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

Note that no assumption on the underlying base field is required. We note that Statement 2. for various $\mathfrak{𝔤}$ and V is equivalent to the statement

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

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

• $\mathfrak{𝔤}$ is nilpotent if and only if $\mathrm{ad}(X)$ is nilpotent for each $X\in \mathfrak{𝔤}$.

Indeed, if $\mathrm{ad}(\mathfrak{𝔤})$ consists of nilpotent operators, then by 1. $\iff$ 2. applied to the algebra $\mathrm{ad}(\mathfrak{𝔤})\subset \mathfrak{𝔤}\mathfrak{𝔩}(\mathfrak{𝔤})$, there exists a flag $\mathfrak{𝔤}={\mathfrak{𝔤}}_0\supset {\mathfrak{𝔤}}_1\supset \cdots \supset {\mathfrak{𝔤}}_n=0$ such that $[\mathfrak{𝔤},{\mathfrak{𝔤}}_i]\subset {\mathfrak{𝔤}}_{i+1}$. Since $C^i\mathfrak{𝔤}\subset {\mathfrak{𝔤}}_i$, this implies $\mathfrak{𝔤}$ is nilpotent. (The converse follows straightforwardly from the definition.)

Proof

We prove the following form of the theorem: if $\mathfrak{𝔤}\subset \mathfrak{𝔤}\mathfrak{𝔩}(V)$ is a Lie subalgebra such that every $X\in \mathfrak{𝔤}$ is a nilpotent endomorphism and if V has positive dimension, then there exists a nonzero vector v in V such that $X(v)=0$ for each X in $\mathfrak{𝔤}$. The proof is by induction on the dimension of $\mathfrak{𝔤}$ 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 $\mathfrak{𝔤}$ is positive. Step 1: Find an ideal $\mathfrak{𝔥}$ of codimension one in $\mathfrak{𝔤}$.

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

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

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

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

Write $\mathfrak{𝔤}=\mathfrak{𝔥}+L$ where L is a one-dimensional vector subspace. Let Y be a nonzero vector in L and v a nonzero vector in W. Now, $Y$ 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. $◻$

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. 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{{citation}}: CS1 maint: unrecognized language (link)