In representation theory, a branch of mathematics, Engel's theorem states that a finite-dimensional Lie algebra
is a nilpotent Lie algebra if and only if for each
, the adjoint map
![{\displaystyle \operatorname {ad} (X)\colon {\mathfrak {g}}\to {\mathfrak {g}},}](//wikimedia.org/api/rest_v1/media/math/render/svg/19767fe9c5bad1e6d777e382a70ccc2147fedb7d)
given by
, is a nilpotent endomorphism on
; i.e.,
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).
Let
be the Lie algebra of the endomorphisms of a finite-dimensional vector space V and
a subalgebra. Then Engel's theorem states the following are equivalent:
- Each
is a nilpotent endomorphism on V.
- There exists a flag
such that
; i.e., the elements of
are simultaneously strictly upper-triangulizable.
Note that no assumption on the underlying base field is required.
We note that Statement 2. for various
and V is equivalent to the statement
- For each nonzero finite-dimensional vector space V and a subalgebra
, there exists a nonzero vector v in V such that
for every ![{\displaystyle X\in {\mathfrak {g}}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/9c279903cc01a257b8f37cfbc24d95cfba3f4f1d)
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
is said to be nilpotent if the lower central series of it vanishes in a finite step; i.e., for
= (i+1)-th power of
, there is some k such that
. Then Engel's theorem implies the following theorem (also called Engel's theorem): when
has finite dimension,
is nilpotent if and only if
is nilpotent for each
.
Indeed, if
consists of nilpotent operators, then by 1.
2. applied to the algebra
, there exists a flag
such that
. Since
, this implies
is nilpotent. (The converse follows straightforwardly from the definition.)
We prove the following form of the theorem: if
is a Lie subalgebra such that every
is a nilpotent endomorphism and if V has positive dimension, then there exists a nonzero vector v in V such that
for each X in
.
The proof is by induction on the dimension of
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
is positive.
Step 1: Find an ideal
of codimension one in
.
- This is the most difficult step. Let
be a maximal (proper) subalgebra of
, which exists by finite-dimensionality. We claim it is an ideal of codimension one. For each
, it is easy to check that (1)
induces a linear endomorphism
and (2) this induced map is nilpotent (in fact,
is nilpotent as
is nilpotent; see Jordan decomposition in Lie algebras). Thus, by inductive hypothesis applied to the Lie subalgebra of
generated by
, there exists a nonzero vector v in
such that
for each
. That is to say, if
for some Y in
but not in
, then
for every
. But then the subspace
spanned by
and Y is a Lie subalgebra in which
is an ideal of codimension one. Hence, by maximality,
. This proves the claim.
Step 2: Let
. Then
stabilizes W; i.e.,
for each
.
- Indeed, for
in
and
in
, we have:
since
is an ideal and so
. Thus,
is in W.
Step 3: Finish up the proof by finding a nonzero vector that gets killed by
.
- Write
where L is a one-dimensional vector subspace. Let Y be a nonzero vector in L and v a nonzero vector in W. Now,
is a nilpotent endomorphism (by hypothesis) and so
for some k. Then
is a required vector as the vector lies in W by Step 2. ![{\displaystyle \square }](//wikimedia.org/api/rest_v1/media/math/render/svg/455831d58fa08f311b934d324adcff89a868b4e4)
- 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