---
date: 2022-04-05 23:42
modification date: Tuesday 5th April 2022 23:42:25
title: "Lie algebra"
aliases: [Lie algebra]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #lie-theory 
- Refs:
	- #todo/add-references
- Links:
	- [Lie group](Unsorted/Lie%20group.md)
	- [[Coxeter number]]
	- [adjoint representation](Unsorted/adjoint%20representation.md)
	- Weight/root theory:
		- [Unsorted/highest weight](Unsorted/highest%20weight.md)
		- [dominant weight](Unsorted/dominant%20weight.md)
		- [fundamental weights](Unsorted/fundamental%20weights.md)
		- [weight lattice](Unsorted/weight%20lattice.md)
		- [root](Unsorted/Weyl%20group.md)

---



# 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](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](vector field) 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](Cartan)?
- What is a [regular weight](regular%20weight)?
- What is a [dominant weight](dominant%20weight)?
- What is a [weight](weight) and a coweight?
- What is a [block](block) of a category?
- What is [category O](category%20O)?
- What is the [Dynkin classification](Dynkin%20classification)?
- What is the [adjoint representation](adjoint%20representation.md)?
- What is the [Springer correspondence](Springer%20correspondence)?
- What is a [Borel subgroup](Borel%20subgroup)?
- What is a [torus action](torus%20action)?
- What is the [affine Grassmannian](affine%20Grassmannian.md)?
- What is [equivariant homology](equivariant%20homology)?
- What is [Borel-Moore cohomology](Borel-Moore%20cohomology)?
- What is the [Bruhat order](Bruhat%20order)?
- What is the [Weyl group](Weyl%20group.md)?
- What is a [Kac-Moody group](Kac-Moody%20group)?
- What is the [ADE classification](ADE%20classification)?
- What are the [Lie algebra types](Lie%20algebra%20types.md)?
- What is a [Coxeter group](Coxeter%20group.md)?

# Results

- What is the [Borel–Weil–Bott theorem](Borel–Weil–Bott%20theorem)?
- What is the [PBW theorem](PBW%20theorem)?

# Examples

$\liesl_2$:
![](attachments/Pasted%20image%2020220208083155.png)

$\lieso(V)$:
![](attachments/Pasted%20image%2020220211002817.png)