If is a module of a Lie algebra , there is one submodule that turns out to be rather interesting: the submodule of vectors such that for all . We call these vectors “invariants” of .
As an illustration of how interesting these are, consider the modules we looked at last time. What are the invariant linear maps from one module to another ? We consider the action of on a linear map :
Or, in other words:
That is, a linear map is invariant if and only if it intertwines the actions on and . That is, .
Next, consider the bilinear forms on . Here we calculate
That is, a bilinear form is invariant if and only if it is associative, in the sense that the Killing form is:
There are a few constructions we can make, starting with the ones from last time and applying them in certain special cases.
First off, if and are two finite-dimensional -modules, then I say we can put an -module structure on the space of linear maps from to . Indeed, we can identify with : if is a basis for and is a basis for , then we can set up the dual basis of , such that . Then the elements form a basis for , and each one can be identified with the linear map sending to and all the other basis elements of to . Thus we have an inclusion , and a simple dimension-counting argument suffices to show that this is an isomorphism.
Now, since we have an action of on we get a dual action on . And because we have actions on and we get one on . What does this look like, explicitly? Well, we can write any such tensor as the sum of tensors of the form for some and . We calculate the action of on a vector :
In general we see that . In particular, the space of linear endomorphisms on is , and so it get an -module structure like this.
The other case of interest is the space of bilinear forms on a module . A bilinear form on is, of course, a linear functional on . And thus this space can be identified with . How does act on a bilinear form ? Well, we can calculate:
In particular, we can consider the case of bilinear forms on itself, where acts on itself by . Here we read