= Lie algebra cohomology
{c}
{title2=$H^k(\mathfrak g,V)$}
For a <Lie algebra> $\mathfrak g$ over a <field> $k$ and a <Lie algebra representation> $V$, set $C^n(\mathfrak g,V)=\operatorname{Hom}_k(\Lambda^n\mathfrak g,V)$. These <alternating multilinear maps> form a <cochain complex> with differential
$$
(df)(x_0,\ldots,x_n)=\sum_i(-1)^i x_i f(x_0,\ldots,\widehat{x_i},\ldots,x_n)+\sum_{i<j}(-1)^{i+j} f([x_i,x_j],x_0,\ldots,\widehat{x_i},\ldots,\widehat{x_j},\ldots,x_n).
$$
The hat means that the argument is omitted. The <Jacobi identity> and the <Lie algebra representation> identity give $d^2=0$. The Lie algebra cohomology is the <quotient vector space> $H^n(\mathfrak g,V)=\ker(d:C^n\to C^{n+1})/\operatorname{im}(d:C^{n-1}\to C^n)$. In degree zero it consists of the <invariant vectors of a Lie algebra representation>; with a <trivial Lie algebra representation> as coefficients, $H^1(\mathfrak g,k)=\operatorname{Hom}_k(\mathfrak g/[\mathfrak g,\mathfrak g],k)$.<Lie Groups and Lie Algebras by Pavel Etingof (2024)>[Section 48.1, standard complex and low-degree cohomology]{ref}
For trivial real coefficients, $H^2(\mathfrak g,\mathbb R)$ classifies <central extensions of a Lie algebra> by a fixed one-dimensional center, up to equivalences inducing the identity on the quotient and center. If $\mathfrak g$ is abelian and two-dimensional, every <alternating bilinear form> is a <Lie algebra two-cocycle> and every coboundary is zero; a nonzero such form produces the <Heisenberg Lie algebra>.
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