= Chevalley–Eilenberg cochains of a graded Lie algebra
{c}
{title2=$C^*(L)$}
= Chevalley–Eilenberg cochain algebra
{c}
{synonym}
For a positively <graded Lie algebra> $L$ over $\mathbb Q$, finite-dimensional in each degree, its cochain algebra is free graded commutative on the graded dual of $sL$, with differential dual to the Lie bracket and its Koszul signs. The Jacobi identity gives $d^2=0$. A Lie element of degree $r$ contributes a cochain generator of degree $r+1$. Its <cohomology> computes $\operatorname{Ext}_{U(L)}(\mathbb Q,\mathbb Q)$ with total grading; the standard free enveloping-algebra resolution gives this identification. For a <free graded Lie algebra>, the tensor-algebra augmentation has a length-one free resolution, explaining the square-zero <cohomology> of a rational <wedge sum> of spheres.
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