We’ll end this series (for now) by talking about two things that we really should talk about: Lie algebra homology, and relative Lie algebra (co)homology. (We will work over a field k, but really we are thinking only of the case k=\mathbb{C}).

Continuing on from last time, let’s now take a look at some actual computational methods for Lie algebra cohomology, as well as some applications and important results. We’ll study the cohomology of semisimple Lie algebras, finite dimensional nilpotent Lie algebras, and then take a little detour to talk about the Borel–Weil–Bott theorem.

As part of a reading group I’m in on Kac–Moody groups (something I know nothing about), I had to give a talk introducing the basics of Lie algebra cohomology (something I know very little about), along with some sort of motivation and intuition, and some worked examples. Since I’ve written these notes anyway, I figured I might as well put them up on this blog, and then, when I inevitably forget all I ever once knew, I can refer back to here.

In the previous post of this series, I talked a bit about basic loop space stuff and how this gave birth to the idea of ‘homotopically-associative algebras’. I’m going to detour slightly from what I was going to delve into next and speak about delooping for a bit first. Then I’ll introduce spectra as sort of a generalisation of infinite deloopings. I’ll probably leave the stuff about E_\infty-algebras for another post, but will definitely at least mention about how it ties in to all this stuff.