Lie algebra cohomology

Remarks

The complex is the Chevalley–Eilenberg complex. In low degrees, $H^0(\mathfrak{g}, M) = \set{m \in M : x \cdot m = 0 \text{ for all } x}$, the invariants. $H^1(\mathfrak{g}, M)$ is the space of linear maps $D \colon \mathfrak{g} \to M$ with $D[x, y] = x \cdot D(y) - y \cdot D(x)$, modulo the maps $x \mapsto x \cdot m$. Example: let $\mathfrak{g} = F x$ be one-dimensional; then $C^n = 0$ for $n \ge 2$ and $d m = (x \mapsto x \cdot m)$ on $C^0$. So $H^0(\mathfrak{g}, M) = \ker(x)$ and $H^1(\mathfrak{g}, M) \cong M / xM$; for trivial $M = F$ both are $F$. Non-example: dropping the bracket terms from $d$ gives a map that does not square to zero. For $m \in M$, the result would be $(x_0, x_1) \mapsto x_0 x_1 m - x_1 x_0 m = [x_0, x_1] \cdot m$, which is nonzero for $\mathfrak{sl}_2(F)$ acting on $F^2$; the bracket term $-\omega([x_0, x_1])$ cancels it. Equivalently, $H^n(\mathfrak{g}, M) \cong \operatorname{Ext}^n_{U(\mathfrak{g})}(F, M)$, where $U(\mathfrak{g})$ is the universal enveloping algebra.