Relation between de Rham cohomology group of Lie group as a manifold and group cohomology of Lie group Is there some relation between De Rham cohomology group of Lie group as a manifold and group cohomology of Lie group?
At first glance, they are two different things. De Rham Cohomology group is defined by differential form on manifold. While group cohomology is used to classify the group extension.
My question:
1.For group cohomology $H^n_\sigma(G,A)$, we need group $G$, abelian group $A$ and $\sigma : G\rightarrow \mathrm{Aut}(A)$. If I fixed $G$ is some Lie group,  $A=\mathbb{R}$ and $\sigma$ as trivial homomorphism. Is there some relation between group cohomology $H^n_0(G,\mathbb{R})$ and De Rham cohomology $H^{n}_{\mathrm{dR}}(G)$?
 A: The answer to the question is: not really.
Consider $G=\mathbb{R}^n\,.$ This has nontrivial group cohomology in all degrees $\le n$ (by the van Est isomorphism theorem, for example), while the classifying space is the point $*\,,$ which has vanishing de Rham cohomology in positive degrees. Therefore, there is certainly no surjection from the de Rham cohomology to the group cohomology.
On the other extreme, consider $G=T^n\,.$ This has vanishing group cohomology in positive degrees (since it's compact), while it has nontrivial de Rham cohomology in all degrees $\le n\,,$ therefore there is no such injection either.
The only relation I know of is that if your group $G$ is compact and has vanishing homotopy groups up to and including degree $n\,,$ then both cohomologies vanish up to degree $n$ (again, by the van Est isomorphism theorem). However, if $n\ge 3$ then this implies that $G$ is a point, by the result in this question: https://math.stackexchange.com/questions/960372/a-question-about-contractibility-of-a-lie-group
There isn't even a strong relationship between group cohomology of $G$ and the cohomology of $BG\,.$ There is one, in a sense, but it's more complicated. If $G$ is discrete however, the group cohomology of $G$ is isomorphic to the cohomology of $BG$ with coefficients in the discrete group $\mathbb{R}\,.$ For nondiscrete Lie groups $G$, there is an analogue to this isomorphism, but it's more complex and involves $EG\,.$
Note: when I say group cohomology I am strictly speaking of the cohomology of group cocycles valued in the Lie group $\mathbb{R}\,,$ however you can consider group cohomology valued in any abelian group.
