We can now define a symmetric bilinear form on our Lie algebra by the formula
It’s symmetric because the cyclic property of the trace lets us swap and and get the same value. It also satisfies another identity which is referred to as “associativity”, though it may not appear like the familiar version of that property at first:
Where we have used the trace identity from last time.
This is called the Killing form, named for Wilhelm Killing and not nearly so coincidentally as the Poynting vector. It will be very useful to study the structures of Lie algebras.
First, though, we want to show that the definition is well-behaved. Specifically, if is an ideal, then we can define to be the Killing form of . It turns out that is just the same as , but restricted to take its arguments in instead of all of .
A lemma: if is any subspace of a vector space and has its image contained in , then the trace of over is the same as its trace over . Indeed, take any basis of and extend it to one of ; the matrix of with respect to this basis has zeroes for all the rows that do not correspond to the basis of , so the trace may as well just be taken over .
Now the fact that is an ideal means that for any the mapping is an endomorphism of sending all of into . Thus its trace over is the same as its trace over all of , and the Killing form on applied to is the same as the Killing form on applied to the same two elements.
It’s obvious that if is nilpotent then will be solvable. And Engel’s theorem tells us that if each is ad-nilpotent, then is itself nilpotent. We can now combine this with our trace criterion to get a convenient way of identifying solvable Lie algebras.
If is a linear Lie algebra and for all and , then is solvable. We’d obviously like to use the trace criterion to show this, but we need a little massaging first.
The catch is that our consists of all the such that sends to . Clearly , but it may not be all of ; our hypothesis states that for all , but the criterion needs it to hold for all .
To get there, we use the following calculation, which is a useful lemma in its own right:
Now, if — so — and then
But since we know that , which means that the hypothesis kicks in: and so .
Then we know that all are nilpotent endomorphisms, which makes them ad-nilpotent. Engel’s theorem tells us that is nilpotent, which means is solvable.
We can also extend this out to abstract Lie algebras: if is any Lie algebra such that for all and , then is solvable. Indeed, we can apply the linear version to the image to see that this algebra is solvable. The kernel is just the center , which is abelian and thus automatically solvable. The image is thus the solvable quotient of by a solvable kernel, so we know that itself is solvable.
We’re going to need another way of identifying nilpotent endomorphisms. Let be two subspaces of endomorphisms on a finite-dimensional space , and let be the collection of such that sends into . If satisfies for all then is nilpotent.
The first thing we do is take the Jordan-Chevalley decomposition of — — and fix a basis that diagonalizes with eigenvalues . We define to be the -subspace of spanned by the eigenvalues. If we can prove that this space is trivial, then all the eigenvalues of must be zero, and thus itself must be zero.
We proceed by showing that any linear functional must be zero. Taking one, we define to be the endomorphism whose matrix with respect to our fixed basis is diagonal: . If is the corresponding basis of we can calculate that
Now we can find some polynomial such that ; there is no ambiguity here since if then the linearity of implies that
Further, picking we can see that , so has no constant term. It should be apparent that .
Now, we know that is the semisimple part of , so the Jordan-Chevalley decomposition lets us write it as a polynomial in with no constant term. But then we can write . Since maps into , so does , and our hypothesis tells us that
Hitting this with we find that the sum of the squares of the is also zero, but since these are rational numbers they must all be zero.
Thus, as we asserted, the only possible -linear functional on is zero, meaning that is trivial, all the eigenvalues of are zero, and is nipotent, as asserted.
Now that we’ve given the proof, we want to mention a few uses of the Jordan-Chevalley decomposition.
First, we let be any finite-dimensional -algebra — associative, Lie, whatever — and remember that contains the Lie algebra of derivations . I say that if then so are its semisimple part and its nilpotent part ; it’s enough to show that is.
Just like we decomposed in the proof of the Jordan-Chevalley decomposition, we can break down into the eigenspaces of — or, equivalently, of . But this time we will index them by the eigenvalue: consists of those such that for sufficiently large .
Now we have the identity:
which is easily verified. If a sufficiently large power of applied to and a sufficiently large power of applied to are both zero, then for sufficiently large one or the other factor in each term will be zero, and so the entire sum is zero. Thus we verify that .
If we take and then , and thus . On the other hand,
And thus satisfies the derivation property
so and are both in .
For the other side we note that, just as the adjoint of a nilpotent endomorphism is nilpotent, the adjoint of a semisimple endomorphism is semisimple. Indeed, if is a basis of such that the matrix of is diagonal with eigenvalues , then we let be the standard basis element of , which is isomorphic to using the basis . It’s a straightforward calculation to verify that
and thus is diagonal with respect to this basis.
So now if is the Jordan-Chevalley decomposition of , then is semisimple and is nilpotent. They commute, since
Since is the decomposition of into a semisimple and a nilpotent part which commute with each other, it is the Jordan-Chevalley decomposition of .
In its simplest form, a flag is simply a strictly-increasing sequence of subspaces of a given finite-dimensional vector space. And we almost always say that a flag starts with and ends with . In the middle we have some other subspaces, each one strictly including the one below it. We say that a flag is “complete” if — and thus — and for our current purposes all flags will be complete unless otherwise mentioned.
The useful thing about flags is that they’re a little more general and “geometric” than ordered bases. Indeed, given an ordered basis we have a flag on : define to be the span of . As a partial converse, given any (complete) flag we can come up with a not-at-all-unique basis: at each step let be the preimage in of some nonzero vector in the one-dimensional space .
We say that an endomorphism of “stabilizes” a flag if it sends each back into itself. In fact, we saw something like this in the proof of Lie’s theorem: we build a complete flag on the subspace , building the subspace up one basis element at a time, and then showed that each stabilized that flag. More generally, we say a collection of endomorphisms stabilizes a flag if all the endomorphisms in the collection do.
So, what do Lie’s and Engel’s theorems tell us about flags? Well, Lie’s theorem tells us that if is solvable then it stabilizes some flag in . Equivalently, there is some basis with respect to which the matrices of all elements of are upper-triangular. In other words, is isomorphic to some subalgebra of . We see that not only is solvable, it is in a sense the archetypal solvable Lie algebra.
The proof is straightforward: Lie’s theorem tells us that has a common eigenvector . We let this span the one-dimensional subspace and consider the action of on the quotient . Since we know that the image of in will again be solvable, we get a common eigenvector . Choosing a pre-image with we get our second basis vector. We can continue like this, building up a basis of such that at each step we can write for all and some .
For nilpotent , the same is true — of course, nilpotent Lie algebras are automatically solvable — but Engel’s theorem tells us more: the functional $\lambda$ must be zero, and the diagonal entries of the above matrices are all zero. We conclude that any nilpotent is isomorphic to some subalgebra of . That is, not only is nilpotent, it is the archetype of all nilpotent Lie algebras in just the same way as is the archetypal solvable Lie algebra.
More generally, if is any solvable (nilpotent) Lie algebra and is any finite-dimensional representation of , then we know that the image is a solvable (nilpotent) linear Lie algebra acting on , and thus it must stabilize some flag of . As a particular example, consider the adjoint action ; a subspace of invariant under the adjoint action of is just the same thing as an ideal of , so we find that there must be some chain of ideals:
where . Given such a chain, we can of course find a basis of with respect to which the matrices of the adjoint action are all in ().
In either case, we find that is nilpotent. Indeed, if is already nilpotent this is trivial. But if is merely solvable, we see that the matrices of the commutators for lie in
But since is a homomorphism, this is the matrix of acting on , and obviously its action on the subalgebra is nilpotent as well. Thus each element of is ad-nilpotent, and Engel’s theorem then tells us that is a nilpotent Lie algebra.
The lemma leading to Engel’s theorem boils down to the assertion that there is some common eigenvector for all the endomorphisms in a nilpotent linear Lie algebra on a finite-dimensional nonzero vector space . Lie’s theorem says that the same is true of solvable linear Lie algebras. Of course, in the nilpotent case the only possible eigenvalue was zero, so we may find things a little more complicated now. We will, however, have to assume that is algebraically closed and that no multiple of the unit in is zero.
We will proceed by induction on the dimension of using the same four basic steps as in the lemma: find an ideal of codimension one, so we can write for some ; find common eigenvectors for ; find a subspace of such common eigenvectors stabilized by ; find in that space an eigenvector for .
First, solvability says that properly includes , or else the derived series wouldn’t be able to even start heading towards . The quotient must be abelian, with all brackets zero, so we can pick any subspace of this quotient with codimension one and it will be an ideal. The preimage of this subspace under the quotient projection will then be an ideal of codimension one.
Now, is a subalgebra of , so we know it’s also solvable, so induction tells us that there’s a common eigenvector for the action of . If is zero, then must be one-dimensional abelian, in which case the proof is obvious. Otherwise there is some linear functional defined by
Of course, is not the only such eigenvector; we define the (nonzero) subspace by
Next we must show that sends back into itself. To see this, pick and and check that
But if , then we’d have ; we need to verify that . In the nilpotent case — Engel’s theorem — the functional was constantly zero, so this was easy, but it’s a bit harder here.
Fixing and we pick to be the first index where the collection is linearly independent — the first one where we can express as the linear combination of all the previous . If we write for the subspace spanned by the first of these vectors, then the dimension of grows one-by-one until we get to , and from then on.
I say that each of the are invariant under each . Indeed, we can prove the congruence
that is, acts on by multiplication by , plus some “lower-order terms”. For this is the definition of ; in general we have
for some .
And so we conclude that, using the obvious basis of the action of on this subspace is in the form of an upper-triangular matrix with down the diagonal. The trace of this matrix is . And in particular, the trace of the action of on is . But and both act as endomorphisms of — the one by design and the other by the above proof — and the trace of any commutator is zero! Since must have an inverse we conclude that .
Okay so that checks out that the action of sends back into itself. We finish up by picking some eigenvector of , which we know must exist because we’re working over an algebraically closed field. Incidentally, we can then extend to all of by using .
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.
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.
There are two big types of Lie algebras that we want to take care of right up front, and both of them are defined similarly. We remember that if and are ideals of a Lie algebra , then — the collection spanned by brackets of elements of and — is also an ideal of . And since the bracket of any element of with any element of is back in , we can see that . Similarly we conclude , so .
Now, starting from we can build up a tower of ideals starting with and moving down by . We call this the “derived series” of . If this tower eventually bottoms out at we say that is “solvable”. If is abelian we see that , so is automatically solvable. At the other extreme, if is simple — and thus not abelian — the only possibility is , so the derived series never gets down to , and thus is not solvable.
We can build up another tower, again starting with , but this time moving down by . We call this the “lower central series” or “descending central series” of . If this tower eventually bottoms out at we say that is “nilpotent”. Just as above we see that abelian Lie algebras are automatically nilpotent, while simple Lie algebras are never nilpotent.
It’s not too hard to see that for all . Indeed, to start. Then if then
so the assertion follows by induction. Thus we see that any nilpotent algebra is solvable, but solvable algebras are not necessarily nilpotent.
As some explicit examples, we look back at the algebras and . The second, as we might guess, is nilpotent, and thus solvable. The first, though, is merely solvable.
First, let’s check that is nilpotent. The obvious basis consists of all the matrix entries with , and we can know that
We have an obvious sense of the “level” of an element: the difference , which is well-defined on each basis element. We can tell that the bracket of two basis elements gives either zero or another basis element whose level is the sum of the levels of the first two basis elements. The ideal is spanned by all the basis elements of level . The ideal is then spanned by basis elements of level . And so it goes, each spanned by basis elements of level . But this must run out soon enough, since the highest possible level is . In terms of the matrix, elements of are zero everywhere on or below the diagonal; elements of are also zero one row above the diagonal; and so on, each step pushing the nonzero elements “off the edge” to the upper-right of the matrix. Thus is nilpotent, and thus solvable as well.
Turning to , we already know that , which we just showed to be solvable! We see that , which will eventually bottom out at , thus is solvable as well. However, we can also calculate that
and so the derived series of stops after the first term and never reaches . Thus this algebra is solvable, but not nilpotent.
Let’s pause and catch our breath with an actual example of some of the things we’ve been talking about. Specifically, we’ll consider — the special linear Lie algebra on a two-dimensional vector space. This is a nice example not only because it’s nicely representative of some general phenomena, but also because the algebra itself is three-dimensional, which helps keep clear the distinction between as a Lie algebra and the adjoint action of on itself, particularly since these are both thought of in terms of matrix multiplications.
Now, we know a basis for this algebra:
which we will take in this order. We want to check each of the brackets of these basis elements:
Writing out each bracket of basis elements as a (unique) linear combination of basis elements specifies the bracket completely, by linearity. We call the coefficients the “structure constants” of , and they determine the algebra up to isomorphism.
Okay, now we want to use this basis of the vector space and write down matrices for the action of on :
Now, both and are nilpotent. In the case of we can see that sends the line spanned by to the line spanned by , the line spanned by to the line spanned by , and the line spanned by to zero. So we can calculate the powers:
and the exponential:
Similarly we can calculate the exponential of :
So now it’s a simple matter to write down the following element of :
In other words, , , and .
We can also see that and themselves are also nilpotent, as endomorphisms of the vector space . We can calculate their exponentials:
and the product:
It’s easy to check from here that conjugation by has the exact same effect as the action of : .
This is a very general phenomenon: if is any linear Lie algebra and is nilpotent, then conjugation by the exponential of is the same as applying the exponential of the adoint of .
Indeed, considering , we can write it as
where and are left- and right-multiplication by in . Since these two commute with each other and both are nilpotent we can write
That is, the action of is the same as left-multiplication by followed by right-multiplication by . All we need now is to verify that this is the inverse of , but the expanded Leibniz identity from last time tells us that , thus proving our assertion.
We can also tell at this point that the nilpotency of and and that of and are not unrelated. Indeed, if is nilpotent then is, too. Indeed, since and are commuting nilpotents, their difference — — is again nilpotent.
We must be careful to note that the converse is not true. Indeed, is ad-nilpotent, but itself is certainly not nilpotent.