nLab Chevalley-Eilenberg cochain complex

for more see at Chevalley-Eilenberg algebra

Idea

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

The dual of this is a differential graded algebra $(\wedge^\bullet g^*, d)$. 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 $g \to End V$. 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 cochain complex agrees with the Lie algebra cohomology with trivial coefficients. The Lie algebra is however defined also for infinite-dimensional Lie algebras and arbitrary module $M$ coefficients. Namely the Lie algebra cohomology is $H^*_{Lie}(\mathfrak{g},M) = Ext_{U(\mathfrak{g})}(\mathfrak{g},M) = H^*(Hom_{U(\mathfrak{g})}(V(\mathfrak{g}),M))\cong H^*(Hom_k(\Lambda^*\mathfrak{g},M))$ where $U(\mathfrak{g})$ is the universal enveloping of $\mathfrak{g}$ and $V(\mathfrak{g}) = U(\mathfrak{g})\otimes \Lambda^* \mathfrak{g}$ (with the appropriate differential) is the Chevalley-Eilenberg chain complex. Now if $\mathfrak{g}$ is finite-dimensional then $Hom_{U(\mathfrak{g})}(V(\mathfrak{g}),M)\cong CE(\mathfrak{g},M)$ and $CE(\mathfrak{g}) = CE(\mathfrak{g},k) = \Lambda^* \mathfrak{g}^*$.