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Let $G$ be a Lie group and $A$ a smooth $G$-module. Define $C^n(G,A)={ C^n(G,A)=\{ f: G^n \to A|~f~ is ~smooth }$ A|~f~\text{is smooth}\}$ and $\partial^n: C^n \to C^{n+1}$ by the standard formula as used in the cohomology of abstract groups. I think this cohomology must be well studied. Can somebody provide me some references for this cohomology. A similar cohomology for topological groups has been studied by Hu and Heller using continuous cochains.

Also, can somebody tell me what kind of Lie group extensions $H^2(G,A)$ correspoind to.
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smooth cohomology of Lie groups

Let $G$ be a Lie group and $A$ a smooth $G$-module. Define $C^n(G,A)={ f: G^n \to A|~f~ is ~smooth }$ and $\partial^n: C^n \to C^{n+1}$ by the standard formula as used in the cohomology of abstract groups. I think this cohomology must be well studied. Can somebody provide me some references for this cohomology. A similar cohomology for topological groups has been studied by Hu and Heller using continuous cochains.

Also, can somebody tell me what kind of Lie group extensions $H^2(G,A)$ correspoind to.