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Chevalley-Eilenberg cochain complex

Idea

A finite dimensional Lie algebra $g$ or degreewise finite-dimensional L-∞ algebra $g$ is encoded in a a differential $D:{\vee }^{•}g\to {\vee }^{•}g$ on the cofree co-commutative coalgebra generated by $g$.

The dual of this is a differential graded algebra $({\wedge }^{•}{g}^{*},d)$. The underlying cochain complex (forgetting the monoidal structure) is the Chevalley-Eilenberg cochain complex .

There is in fact a bijection between quasi-free cochain differential graded algebras in non-negative degree and L-∞ algebras.

The Chevalley-Eilenberg complex is usually defined a bit more generally for Lie algebras equipped with a Lie module? $g\to \mathrm{End}V$. In the above language this more general cochain complex is the one underlying the Lie ∞-algebroid that encodes this action in the sense of Lie ∞-algebroid representations.

Lie algebra cohomology

The cohomology of the Chevalley-Eilenberg complex is called Lie algebra cohomology.