---
tag: HKZABKRPN6
kind: concept
name: Cohomology of Lie groups
msc: [57T10, 17B56]
sources:
  - url: https://en.wikipedia.org/wiki/Lie_algebra_cohomology
---

## Summary

The cohomology of a [[Lie group|7JBRGCVJRB]] $G$ is the [[singular cohomology|6N7BC70CX0]] of its underlying manifold, with the extra structure given by the group multiplication.
The multiplication makes it a Hopf algebra, which strongly restricts its form.
For a compact connected group, real cohomology can be computed from invariant differential forms, that is, from the cohomology of the Lie algebra.
This turns a topological question into a question of linear algebra.

## Explanations

### L1 human

Cohomology counts the holes of a space in each dimension.
A Lie group is a space, so one can ask about its holes.
The circle has one hole of dimension $1$, and the group $\mathrm{SU}(2)$, a $3$-sphere, has one hole of dimension $3$.
Because a Lie group looks the same at every point, its holes can be detected by forms that are invariant under the group.
Those forms are determined by their values at the identity, where they become algebra on the Lie algebra.