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.

Continue reading →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.

Continue reading →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.