Engel’s Theorem
When we say that a Lie algebra is nilpotent, another way of putting it is that for any sufficiently long sequence
of elements of
the nested adjoint
is zero for all . In particular, applying
enough times will eventually kill any element of
. That is, each
is ad-nilpotent. It turns out that the converse is also true, which is the content of Engel’s theorem.
But first we prove this lemma: if is a linear Lie algebra on a finite-dimensional, nonzero vector space
that consists of nilpotent endomorphisms, then there is some nonzero
for which
for all
.
If then
is spanned by a single nilpotent endomorphism, which has only the eigenvalue zero, and must have an eigenvector
, proving the lemma in this case.
If is any nontrivial subalgebra of
then
is nilpotent for all
. We also get an everywhere-nilpotent action on the quotient vector space
. But since
, the induction hypothesis gives us a nonzero vector
that gets killed by every
. But this means that
for all
, while
. That is,
is strictly contained in the normalizer
.
Now instead of just taking any subalgebra, let be a maximal proper subalgebra in
. Since
is properly contained in
, we must have
, and thus
is actually an ideal of
. If
then we could find an even larger subalgebra of
containing
, in contradiction to our assumption, so as vector spaces we can write
for any
.
Finally, let consist of those vectors killed by all
, which the inductive hypothesis tells us is a nonempty collection. Since
is an ideal,
sends
back into itself:
. Picking a
as above, its action on
is nilpotent, so it must have an eigenvector
with
. Thus
for all
.
So, now, to Engel’s theorem. We take a Lie algebra consisting of ad-nilpotent elements. Thus the algebra
consists of nilpotent endomorphisms on the vector space
, and there is thus some nonzero
for which
. That is,
has a nontrivial center —
.
The quotient thus has a lower dimension than
, and it also consists of ad-nilpotent elements. By induction on the dimension of
we assume that
is actually nilpotent, which proves that
itself is nilpotent.
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