TITLE: Chevalley Eilenberg complex definitions?
QUESTION [21 upvotes]: In Weibel's An Introduction to Homological Algebra, the Chevalley-Eilenberg complex of a Lie algebra $g$ is defined as $\Lambda^*(g) \otimes Ug$ where $Ug$ is the universal enveloping algebra of $g$.  The differential here has degree -1.
I have been told that the Chevalley-Eilenberg complex for $g$ is 
    $C^*(g) = \text{Sym}(g^*[-1])$,
the free graded commutative algebra on the vector space dual of $g$ placed in degree 1.  The bracket $[,]$ is a map $\Lambda^2 g \to g$ so its dual $d : = [,]* \colon g^* \to \Lambda^2 g^*$ is a map from $C^1(g) \to C^2(g)$.  Since $C^*(g)$ is free, this defines a derivation, also called $d$, from $C^*(g)$ to itself.  This derivation satisfies $d^2 = 0$ precisely because $[,]$ satisfies the Jacobi identity.
Finally, Kontsevich and Soibelman in Deformation Theory I leave it as an exercise to construct the Chevalley-Eilenberg complex in analogy to the way that the Hochschild complex is constructed for an associative algebra by considering formal deformations.
The first is a chain complex, the second a cochain complex, and what do either have to do with formal deformations of $g$?

REPLY [10 votes]: Just to add to what Mariano has written and focussing only on the deformations aspect, the exercise in Kontsevich-Soibelman hints at the "principle" that deformations of algebraic structures are always governed by a cohomology theory and that if you didn't know which, you would discover it by analysing the conditions defining infinitesimal deformations, trivial infinitesimal deformations and obstructions to integrating infinitesimal deformations.
I think that this is a very good exercise and it helps motivate the classical formulas for the differentials, at least in the case of cohomology with values in the adjoint module.
I would add another reference to the one of Gerstenhaber et al, and that is the older paper by Nijenhuis and Richardson Deformations of Lie algebra structures.  In general there are a number of classic papers of Nijenhuis and Richardson on this topic.  In particular they define a graded Lie algebra structure on the deformation complex which makes very clear the nature of the obstructions.

REPLY [10 votes]: The other comment revealed most of the story. Let me add some points. First, the Chevalley-Eilenberg complex is defined for the most general case of $H^\bullet(g,M)$ --- cohomology with coefficients in a module M. In the case M=k the trivial module, you get your second complex. In the case $M=g$ the adjoint representation, you get the comples Kontsevich and Soibelman ask to construct. 
The first complex is some version of Koszul complex $(A^!)^* \otimes A$ for quadratic algebras: one can think of the Koszul dual of the dg-commutative algebra $\Lambda^\bullet( g^* )$ as the quadratic-linear algebra $U(\mathfrak{g})$. It is acyclic, and gives a resolution of the trivial module by free $U(g)$-modules, so you can use is to compute Ext groups $Ext_{U(g)}^\bullet(k,M)=H^\bullet(g,M)$.
Finally, another way to think of your second complex is as follows. If $g$ is the Lie algebra of a Lie group $G$, one can consider the subcomplex of the de Rham complex consisting of left-invariant differential forms. A left-invariant form is defined by its behaviour at the unit of the group, 1-forms are dual to the tangent space and form $ g^* $ , 2-forms give $\Lambda^2(g^*)$ etc., and you get precisely the CE complex, where the differential is what the de Rham differential induces. If $G$ is compact, it is easy to show that this complex has the same cohomology as the de Rham complex, so we compute the de Rham cohomology of the group in this case!