The ultimate aim in the series on Lie algebras I am posting here is to cover the representation theory of semisimple Lie algebras. To get there, we first need to discuss some technical tools—for instance, invariant bilinear forms.
Generalities on representations
Fix a Lie algebra . Given representations , we clearly have a representation ; given a morphism of representations , i.e. one which respects the action of , the kernel and image are themselves representations.
Proposition 1 The category of finite-dimensional representations of is an abelian category.
There are a couple of easy technical facts to check (e.g. a monomorphism is the kernel of its cokernel), but we have essentially proved this by the above discussion.
We know, by the embedding theorem, that a small abelian category is a full subcategory of the category of left -modules for some non-unique ring . In our case, we could take .
There are a few other ways we can build new representations from old ones:
The fact that this is a representation can be directly checked. Moreover, the reason for this somewhat odd choice comes from the theory of Lie groups: If the Lie group acts on , then it acts on by
and differentiating this gives the action of the Lie algebra.
Again, the choice comes from the theory of Lie groups, where if acts on , then acts on by
There is a key idea here:
Proposition 4 is left invariant by , i.e. for all , if and only if is an -homomorphism.
This in turn follows from (3).
In general, if we have some representation of , then an invariant element is one annihilated by : In the setting of Lie groups, this corresponding to being fixed by the group.
Given -representations , the set of bilinear forms is isomorphic functorially to and consequently has an action of , defined by
so an invariant bilinear form is one satisfying
On the -module with the adjoint representation, there is a useful way of constructing invariant bilinear forms. Let be an -representation, and denote for any by the linear transformation obtained by multiplication by .
Define the form
This is a symmetric bilinear form.
Proposition 5 is -invariant, where is given the adjoint action on itself.
Proof: This is a simple computation, using the definition :
Since (in general, for matrices ), the claim follows.