Now that I’ve discussed some of the basic definitions in the theory of Lie algebras, it’s time to look at specific subclasses: nilpotent, solvable, and eventually semisimple Lie algebras. Today, I want to focus on nilpotence and its applications.
Engel’s Theorem
To start with, choose a Lie algebra for some finite-dimensional
-vector space
; recall that
is the Lie algebra of linear transformations
with the bracket
. The previous definition was in terms of matrices, but here it is more natural to think in terms of linear transformations without initially fixing a basis.
Engel’s theorem is somewhat similar in its statement to the fact that commuting diagonalizable operators can be simultaneously diagonalized.
Theorem 1 (Engel) Let
consist of nilpotent operators. Then there exists a vector
,
, such that
for all
.
The theorem can also be stated in the form where is a representation for
where the elemetns of
act via nilpotent operators; in that case there is a vector annihilated by
. I will tacitly use this equivalence below.
The corollary makes the analogy between commuting diagonalizable operators clearer:
Corollary 2 Under the same hypothesis, there exists a basis of
with which each element of
is represented as a strictly upper-triangular matrix.
The proof of the theorem goes by induction on (and not on
!). When
, the result is immediate. Otherwise, if
, we have a proper Lie subalgebra
. Choose
to be a maximal subalgebra.
Claim 1
has codimension one in
and
is an ideal.
Proof: [Proof of the claim] Well, acts via
on both
and
. In the latter case, we know by Engel’s theorem for
, and the inductive hypothesis, that there exists a nonzero
such that
for
; lifting
to
, it follows that
. Moreover
. Both these imply
is a Lie subalgebra of
, and contains
as an ideal. By maximality of
, it follows that
, so we’re done.
Now, to prove the theorem, choose as before, and use the inductive hypothesis again to see that the vector space
is nonzero.
Claim 2
is stable under
.
In other words, if ,
, then
for any
. But
and since
is an ideal, so
. This is essentially a new version of the “fundamental calculation” in discussing representations of
.
Now, we have for some
; we know that
is a linear operator on
by the previous claim, and is by assumption nilpotent. So
. Choose
nonzero in that intersection; then both
annihilate
, proving Engel’s theorem.
Proof: [Proof of the corollary] By linear algebra, the statement of the corollary is equivalent to the statement: There exists a flag of -stable spaces
such that
for each
, and
acts trivially on the quotients
. To construct the flag, proceed as follows. Choose
to be spanned by a vector
as in Engel’s theorem. Then
acts as a family of nilpotent operators on
. Choose
such that
is generated by an element
annihilated by
, by Engel again. Repeat until the process terminates, since
is finite.
Nilpotence
The notion of nilpotence in Lie algebras is slightly more complicated than simply considering strictly upper-triangular matrices. The idea instead is to say that a Lie algebra is nilpotent if it is at least “almost” commutative, in that taking successive brackets eventually yields zero.
Formally:
Definition 3 The lower central series of a Lie algebra
is defined by
. A Lie algebra is said to be nilpotent if its lower central series eventually becomes zero.
If a Lie algebra is nilpotent, then must be nilpotent for any
. Indeed,
maps
since
.
The converse is also true:
Theorem 4 Suppose
is a Lie algebra such that each
is nilpotent. Then
is nilpotent.
Induction on , as usual. The image
is a Lie subalgebra (since
is a homomorphism of Lie algebras). Each element of
consists of a nilpotent operator on
by assumption. So by Engel, there exists
such that
. In other words,
“commutes” with all of
, or lies in the center:
Definition 5 The center of a Lie algebra
is the set of all
such that
if
.
The center is actually a Lie ideal of . We have just shown that under the hypotheses of the theorem, the center
of
is nonzero. Now
still satisfies the conditions of the theorem, and by the inductive hypothesis is nilpotent. So we are reduced to:
Lemma 6 Suppose
is a Lie algebra,
its center, and
is nilpotent. Then
is nilpotent.
Indeed, it follows by the nilpotence of that some
, which implies
.
July 26, 2009 at 9:12 pm
[…] theory. Tags: derived series, Lie algebras, Lie's theorem, solvability trackback I talked a bit earlier about nilpotent Lie algebras and Engel’s theorem. There is an analog for solvable Lie […]