On the nlab [page](https://ncatlab.org/nlab/show/Chevalley-Eilenberg+algebra#DefForLieAlg) for Chevalley–Eilenberg algebras, it defines $\operatorname{CE}(\mathfrak g)$ for $\mathfrak g$ finite dimensional, and then says "This has a more or less evident generalization to infinite-dimensional Lie algebras", and provides no more details.


Please could someone outline this generalisation, and if the theory follows through (e.g. is there still a contravariant equivalence of categories?).



CE(g) for g infinite dimensional
A definition that always works and does agree with that one in the finite-dimensional case is the following: put
$$
C^k(\mathfrak{g})=({\Lambda}^k\mathfrak{g})^\*=\operatorname{Hom}({\Lambda}^k\mathfrak{g}, \mathbb{F}).
 $$
(Here $\mathbb{F}$ is the ground field, of course.)
The differential is given by the formula
$$
(d\phi)(g\_1\wedge\dotsb\wedge g\_{k+1})=\sum\_{1\le i<j\le k+1}(-1)^{i+j-1}\phi([g\_i,g\_j]\wedge g\_1\wedge\dotsb \wedge\widehat{g\_i}\wedge\dotsb\wedge \widehat{g\_j}\dotsb\wedge g\_{k+1}),
 $$
where the hat means missing the corresponding factor. If you define the product of cochains by the usual shuffle product formula, you will see that the differential is a derivation, and so all nice properties still hold. The main thing that breaks is precisely the identification $({\Lambda}^k\mathfrak{g})^\*\cong {\Lambda}^k(\mathfrak{g}^\*)$ used in the nlab page.


(See, for example, the classical reference *[Cohomology of Infinite-Dimensional Lie Algebras](https://doi.org/10.1007/978-1-4684-8765-7)* by D.B. Fuks.)


The statement about equivalence of categories does not extend, however, precisely because the cochain algebra is not generated by degree one elements. There are many different situations where things can be patched: many infinite-dimensional algebras have additional gradings with finite-dimensional components (and you can work in the corresponding monoidal category), sometimes all you need is continuous cochains on algebras that have some topology etc. Again, the book of Fuks details many of those situations.


