Updated: 2024-04-25 07:28:45
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 […]
Updated: 2024-04-25 07:28:44
There are a few standard techniques we can use to generate new modules for a Lie algebra from old ones. We’ve seen direct sums already, but here are a few more. One way is to start with a module and then consider its dual space . I say that this can be made into an […]
Updated: 2024-04-25 07:28:44
As might be surmised from irreducible modules, a reducible module for a Lie algebra is one that contains a nontrivial proper submodule — one other than or itself. Now obviously if is a submodule we can form the quotient . This is the basic setup of a short exact sequence: The question is, does this […]
Updated: 2024-04-25 07:28:44
Sorry for the delay; it’s getting crowded around here again. Anyway, an irreducible module for a Lie algebra is a pretty straightforward concept: it’s a module such that its only submodules are and . As usual, Schur’s lemma tells us that any morphism between two irreducible modules is either or an isomorphism. And, as we’ve […]
Updated: 2024-04-25 07:28:43
It should be little surprise that we’re interested in concrete actions of Lie algebras on vector spaces, like we were for groups. Given a Lie algebra we define an -module to be a vector space equipped with a bilinear function — often written satisfying the relation Of course, this is the same thing as a […]
Updated: 2024-04-25 07:28:43
It turns out that all the derivations on a semisimple Lie algebra are inner derivations. That is, they’re all of the form for some . We know that the homomorphism is injective when is semisimple. Indeed, its kernel is exactly the center , which we know is trivial. We are asserting that it is also […]
Updated: 2024-04-25 07:28:42
We say that a Lie algebra is the direct sum of a collection of ideals if it’s the direct sum as a vector space. In particular, this implies that , meaning that the bracket of any two elements from different ideals is zero. Now, if is semisimple then there is a collection of ideals, each […]
Updated: 2024-04-25 07:28:42
Let’s go back to our explicit example of and look at its Killing form. We first recall our usual basis: which lets us write out matrices for the adjoint action: and from here it’s easy to calculate the Killing form. For example: We can similarly calculate all the other values of the Killing form on […]
Updated: 2024-04-25 07:28:42
The first and most important structural result using the Killing form regards its “radical”. We never really defined this before, but it’s not hard: the radical of a binary form on a vector space is the subspace consisting of all such that for all . That is, if we regard as a linear map , […]
