---
date: 2022-04-05 23:42
modification date: Tuesday 5th April 2022 23:42:25
title: "Chevalley-Eilenberg complex"
aliases: [Chevalley-Eilenberg, CE, Lie algebra cohomology, "Kostant's theorem"]
created: 2023-03-31T15:22
updated: 2023-03-31T15:25
---

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [Representation Theory (Subject MOC)](Representation%20Theory%20(Subject%20MOC).md)

---

# Chevalley-Eilenberg complex

![](2023-03-31-50.png)

The straightforward observation to make here is that if $\mathfrak{g}$ is abelian and the action of $\mathfrak{g}$ on $V$ is trivial, then the differentials in this complex vanish, and $$\mathrm{H}_{\mathrm{CE}}^i(\mathfrak{g}, V) \cong \Lambda^i \mathfrak{g} \otimes_{\mathrm{k}} V$$


![](attachments/2021-10-03_14-50-57.png)

![](attachments/Pasted%20image%2020220319004210.png)
![](attachments/Pasted%20image%2020220319004234.png)

# Applications

To [Lie algebroids](Unsorted/Lie%20algebroid.md):

![](attachments/Pasted%20image%2020220422091239.png)

Kostant
![](2023-03-31-51.png)