For a topological space X we can consider the coefficient of singular cohomology in a Lie algebra A. Then we obtain a graded Lie algebra, that is [x,y]=(-1)^i+j-1 [y,x], for homogeneous elements x and y of degree i and j. 


My question :
Is there an example of two (nice) topological spaces X and y, with different homotopy type, such that they have the same homotopy group, homology group and cohomology rings. But their graded lie Algebra cohomologies are not isomorphic. By this question, I mean to what extent "cohomology with coefficient in Lie algebras" is useful?


Another question what is the graded lie algebra structure for this type of cohomology for X=CP^n with coefficient algebra A=M\_{n}(C). 
Can one introduce me some related references?
Thank you 



Cohomology with coefficient in a Lie algebra
I assume your Lie algebra $\mathfrak{g}$ is over a field $k$. Then what you are looking at is just the graded algebra $H^\*(X,k)\otimes\_k \mathfrak{g}$. If the cohomology algebras $H^\*(X,k)$ and $H^\*(Y,k)$ are isomorphic, so are their tensor products with $\mathfrak{g}$.


In particular, if $X=\mathbb{C}\mathbb{P}^n$, you get the Lie algebra $k[x]/x^{n+1}\otimes\_k \mathfrak{g}$, with $\deg(x)=2$.


