Ideals of Lie Algebras
As we said, a homomorphism of Lie algebras is simply a linear mapping between them that preserves the bracket. I want to check, though, that this behaves in certain nice ways.
First off, there is a Lie algebra . That is, the trivial vector space can be given a (unique) Lie algebra structure, and every Lie algebra has a unique homomorphism and a unique homomorphism . This is easy.
Also pretty easy is the fact that we have kernels. That is, if is a homomorphism, then the set is a subalgebra of . Indeed, it’s actually an “ideal” in pretty much the same sense as for rings. That is, if and then . And we can check that
proving that is an ideal, and thus a Lie algebra in its own right.
Every Lie algebra has two trivial ideals: and . Another example is the “center” — in analogy with the center of a group — which is the collection of all such that for all . That is, those for which the adjoint action is the zero derivation — the kernel of — which is clearly an ideal.
If we say — again in analogy with groups — that is abelian; this is the case for the diagonal algebra , for instance. Abelian Lie algebras are rather boring; they’re just vector spaces with trivial brackets, so we can always decompose them by picking a basis — any basis — and getting a direct sum of one-dimensional abelian Lie algebras.
On the other hand, if the only ideals of are the trivial ones, and if is not abelian, then we say that is “simple”. These are very interesting, indeed.
As usual for rings, we can construct quotient algebras. If is an ideal, then we can define a Lie algebra structure on the quotient space . Indeed, if and are equivalence classes modulo , then we define
which is unambiguous since if and are two other representatives then and , and we calculate
and everything in the parens on the right is in .
Two last constructions in analogy with groups: the “normalizer” of a subspace is the subalgebra . This is the largest subalgebra of which contains as an ideal; if already is an ideal of then ; if we say that is “self-normalizing”.
The “centralizer” of a subset is the subalgebra . This is a subalgebra, and in particular we can see that .