Let $G$ be a group and let $k$ be a field (characteristic 0 if you want). Let $L$ be the graded Lie ring associated to the lower central series of $G$, that is, $L$, as a graded abelian group is $\oplus\_{i \geq 1} G\_{i}/G\_{i+1}$ where $G\_1 = G$, $G\_i = (G\_{i-1}, G)$, and the Lie bracket is induced by the commutator $(a,b)$ on $G$. Tensor $L$ with $\_{\mathbb Z}k$ to get a Lie algebra $\hat L$ over $k$. 


Is there any relationship between the cohomology algebra of $G$, $H^{\*}(G, k)$, and the cohomology of the (graded) Lie algebra $\hat L$, $H^{\*}(\hat L, k)$? 


I realize I am not being precise about what category these cohomologies are being computed in, as in graded versus ungraded. I am hoping someone can tell me there is a relationship or that there is no connection. Perhaps there are conditions on $G$ that ensure these cohomology algebras are isomorphic? References?



Relationship between the cohomology of a group and the cohomology of its associated Lie algebra
The
*continuous cohomology* of a group $\Gamma$ is the direct limit
$$H^\*\_{\text{cts}}(\Gamma;\mathbb Q)=\lim\_{\longrightarrow}\ H^\*(\Gamma/K;\mathbb Q)$$
of the cohomology of all its finitely generated nilpotent quotients
$\Gamma/K$. The basic properties of continuous cohomology are
established in Hain, "Algebraic cycles and extensions of variations of mixed Hodge structure", 175–221 in *Complex geometry and Lie theory*, Proc. Sympos. Pure Math 53.


There is an obvious comparison map
$H^\*\_{\text{cts}}(\Gamma;\mathbb Q)\to H^\*(\Gamma;\mathbb Q)$, which is always an
isomorphism on $H^0$ and $H^1$, and is always injective on $H^2$. A
finitely generated group $\Gamma$ is called *pseudo-nilpotent* if
this map is an isomorphism in every degree.


Nomizu's theorem implies that for finitely generated groups,
$H^\*\_{\text{cts}}(\Gamma;\mathbb Q)$ coincides with the continuous cohomology
$H^\*\_{\text{cts}}(\mathfrak{g};\mathbb Q)$ of the Malcev Lie algebra
$\mathfrak{g}$ of $\Gamma$. The Malcev Lie algebra is a certain
pronilpotent $\mathbb Q$–Lie algebra associated to $\Gamma$ mentioned above by Tom Goodwillie. It has the property that its associated graded $\text{gr}(\mathfrak{g})$ is isomorphic to your $\hat{L}=\text{gr}(\Gamma)$, the associated graded of the group $\Gamma$. Moreover, in many (possibly all?) cases there is an isomorphism
$H^\*\_{\text{cts}}(\mathfrak{g};\mathbb Q)\approx H^\*(\hat L;\mathbb Q)$.


Some examples of pseudo-nilpotent groups—that is, groups for which the cohomology of the group and its associated Lie algebra coincide—are free groups, fundamental groups of Riemann surfaces, and pure braid groups. (This property is closely related to the property of a space $X$ being a *rational $K(\pi,1)$*, in the sense that the localization-at-0/rationalization of $X$ is aspherical.) Definitely not all groups have this property, however. For example, the reason that the pure braid group is pseudo-nilpotent is that it is the fundamental group of the complement of a particularly nice hyperplane arrangement. But without some condition on the arrangement, Falk showed that there are aspherical-hyperplane-complement-groups that are not pseudo-nilpotent. 


I recommend reading Hain's excellent article for more information; unfortunately I could never find it online, though you can read snippets on Google Books. In the interest of full disclosure: much of this answer was taken from my paper "Representation theory and homological stability", with Benson Farb (arXiv:1008.1368, pages 61-62).


