nLab
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 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 $H_{\mathrm{Lie}}^{*}(𝔤,M)={\mathrm{Ext}}_{U(𝔤)}(𝔤,M)={\mathrm{Hom}}_{U(𝔤)}(V(𝔤),M)\cong {\mathrm{Hom}}_k({\Lambda }^{*}𝔤,M)$ where $U(𝔤)$ is the universal enveloping of $𝔤$ and $V(𝔤)=U(𝔤)\otimes {\Lambda }^{*}𝔤$ (with the appropriate differential) is the Chevalley-Eilenberg chain complex. Now if $𝔤$ is finite-dimensional then ${\mathrm{Hom}}_{U(𝔤)}(V(𝔤),M)\cong \mathrm{CE}(𝔤,M)$ and $\mathrm{CE}(𝔤)=\mathrm{CE}(𝔤,k)={\Lambda }^{*}{𝔤}^{*}$.

References

• MathOverflow: definitions of Chevalley-Eilenberg complex

• C. Chevalley, S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63, (1948). 85–124.