Chevalley–Eilenberg complex

Summary

Let $\mathfrak{g}$ be a Lie algebra and $M$ a $\mathfrak{g}$-module. The Chevalley–Eilenberg complex of $\mathfrak{g}$ with coefficients in $M$ is a cochain complex of alternating multilinear maps from $\mathfrak{g}$ to $M$. Its differential is built from the action and the bracket. Its cohomology is the Lie algebra cohomology $H^*(\mathfrak{g}, M)$. It gives an explicit way to compute this cohomology. For the Lie algebra of a compact connected Lie group, with trivial coefficients, it computes the de Rham cohomology of the group.