The Category of Lie Algebras is (not quite) Abelian
We’d like to see that the category of Lie algebras is Abelian. Unfortunately, it isn’t, but we can come close. It should be clear that it’s an -category, since the homomorphisms between any two Lie algebras form a vector space. Direct sums are also straightforward: the Lie algebra is the direct sum as vector spaces, with for and and the regular brackets on and otherwise.
We’ve seen that the category of Lie algebras has a zero object and kernels; now we need cokernels. It would be nice to just say that if is a homomorphism then is the quotient of by the image of , but this image may not be an ideal. Luckily, ideals have a few nice closure properties.
First off, if and are ideals of , then — the subspace spanned by brackets of elements of and — is also an ideal. Indeed, we can check that which is back in . Similarly, the subspace sum is an ideal. And, most importantly for us now, the intersection is an ideal, since if then both and , so as well. In fact, this is true of arbitrary intersections.
This is important, because it means we can always expand any subset to an ideal. We take all the ideals of that contain and intersect them. This will then be another ideal of containing , and it is contained in all the others. And we know that this intersection is nonempty, since there’s always at least the ideal .
So while may not be an ideal of , we can expand it to an ideal and take the quotient. The projection onto this quotient will be the largest epimorphism of that sends everything in to zero, so it will be the cokernel of .
Where everything falls apart is normality. The very fact that we have ideals as a separate concept from subalgebras is the problem. Any subalgebra is the image of a monomorphism — the inclusion, if nothing else. But not all these subalgebras are themselves kernels of other morphisms; only those that are ideals have this property.
Still, the category is very nice, and these properties will help us greatly in what follows.