= Lie algebra cohomology
{c}
{title2=$H^k(\mathfrak g,V)$}
The Chevalley-Eilenberg complex consists of alternating maps from $\mathfrak g^k$ to a <Lie algebra representation> $V$, with differential built from the bracket and module action. For trivial real coefficients, a two-cocycle is an alternating bilinear form $\kappa$ satisfying $\kappa([X,Y],Z)+\kappa([Y,Z],X)+\kappa([Z,X],Y)=0$. Coboundaries have the form $b([X,Y])$, up to the differential's sign convention. Their quotient is $H^2(\mathfrak g,\mathbb R)$ and classifies <central extensions of a Lie algebra>.
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