Lie algebra cohomology

Summary

Lie algebra cohomology is a sequence of vector spaces $H^n(\mathfrak{g}, M)$ attached to a Lie algebra $\mathfrak{g}$ and a $\mathfrak{g}$-module $M$. It measures obstructions in Lie theory: $H^1$ classifies derivations up to inner ones, and $H^2$ classifies abelian extensions. For a compact connected Lie group, the cohomology of its Lie algebra with real coefficients is the de Rham cohomology of the group. It is the Lie algebra analogue of group cohomology.