Engel's theorem


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

given by , is a nilpotent endomorphism on ; i.e., 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).



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:

  1. Each   is a nilpotent endomorphism on V.
  2. 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  

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:[2] 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.  

See also





  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