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 1The 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:

Definition 2If are representations of , then is a representation under the action

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.

Definition 3If is a representation of , then the dual space is a representation of under

Again, the choice comes from the theory of Lie groups, where if acts on , then acts on by

So, we now know how to make and into -representations for any . Thus, we can make into a -module. The action is given by

which follows from unwinding the definitions (1) and (2).

There is a key idea here:

Proposition 4is leftinvariantby , 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.

** Bilinear forms **

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 5is -invariant, where is given the adjoint action on itself.

*Proof:* This is a simple computation, using the definition :

Similarly

Since (in general, for matrices ), the claim follows.

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