Lie algebra

attachments/Pasted%20image%2020220221004536.png

On the cohomology of Lie algebras, from Brian Boe:

I haven’t been able to find a good reference relating the cohomology of Lie groups and Lie algebras. The closest I’ve found is the book "Lie groups, Lie algebras, and cohomology“ by Anthony W. Knapp, Princeton Univ Press, 1988. He focuses on Lie algebra cohomology, but does discuss some motivation in terms of vector field) on manifolds (Lie groups on manifolds (Lie groups), de Rham complex, etc. There are also a couple of chapters on homological algebra with applications to Lie algebras, which might be helpful in the context of my course this semester.

It’s also likely that some of the references in Knapp’s book would contain the sort of comparison information you’re looking for (maybe Borel & Wallach, or Brown?), but I’m not familiar with those.

Topics

What is the Cartan?
What is a regular weight?
What is a dominant weight?
What is a weight and a coweight?
What is a block of a category?
What is category O?
What is the Dynkin classification?
What is the adjoint representation?
What is the Springer correspondence?
What is a Borel subgroup?
What is a torus action?
What is the affine Grassmannian?
What is equivariant homology?
What is Borel-Moore cohomology?
What is the Bruhat order?
What is the Weyl group?
What is a Kac-Moody group?
What is the ADE classification?
What are the Lie algebra types?
What is a Coxeter group?

Results

What is the Borel–Weil–Bott theorem?
What is the PBW theorem?

Examples

\({\mathfrak{sl}}_2\): attachments/Pasted%20image%2020220208083155.png

\({\mathfrak{so}}(V)\): attachments/Pasted%20image%2020220211002817.png