Facts About Solvability and Nilpotence
Solvability is an interesting property of a Lie algebra , in that it tends to “infect” many related algebras. For one thing, all subalgebras and quotient algebras of
are also solvable. For the first count, it should be clear that if
then
. On the other hand, if
is a quotient epimorphism then any element in
has a representative in
, so if the derived series of
bottoms out at
then so must the derived series of
.
As a sort of converse, suppose that is a solvable quotient of
by a solvable ideal
; then
is itself solvable. Indeed, if
and
is the quotient epimorphism then
, as we saw above. That is,
, but since
is solvable this means that
— as a subalgebra — is solvable, and thus
is as well.
Finally, if and
are solvable ideals of
then so is
. Here, we can use the third isomorphism theorem to establish an isomorphism
. The right hand side is a quotient of
, and so it’s solvable, which makes
a solvable quotient by a solvable ideal, meaning that
is itself solvable.
As an application, let be any Lie algebra and let
be a maximal solvable ideal, contained in no larger solvable ideal. If
is any other solvable ideal, then
is solvable as well, and it obviously contains
. But maximality then tells us that
, from which we conclude that
. Thus we conclude that the maximal solvable ideal
is unique; we call it the “radical” of
, written
.
In the case that the radical of is zero, we say that
is “semisimple”. In particular, a simple Lie algebra is semisimple, since the only ideals of
are itself and
, and
is not solvable.
In general, the quotient is semisimple, since if it had a solvable ideal it would have to be of the form
for some
containing
. But if
is a solvable quotient of
by a solvable ideal, then
must be solvable, which means it must be contained in the radical of
. Thus the only solvable ideal of
is
, as we said.
We also have some useful facts about nilpotent algebras. First off, just as for solvable algebras all subalgebras and quotient algebras of a nilpotent algebra are nilpotent. Even the proof is all but identical.
Next, if — where
is the center of
— is nilpotent then
is as well. Indeed, to say that
is to say that
for some
. But then
.
Finally, if is nilpotent, then
. To see this, note that if
is the first term of the descending central series that equals zero, then
, since the brackets of everything in
with anything in
are all zero.