# Cohomology with coefficient in a Lie algebra

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

• If the álgebra is the commutator lie algebra of an associative álgebra, you get just the commutator Lie álgebra of the usual cohomology ring. – Mariano Suárez-Álvarez Dec 6 '13 at 13:24

## 1 Answer

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 $\deg(x)=2$.

• Thank you, could you please give me some references? Another question:should I underestand from your answer that lie algebra coefficients does not contain new information to study the homotopy type of a topological space? – Ali Taghavi Dec 6 '13 at 13:34
• Sorry I don't have a precise reference. I would look at the standard algebraic topology books (note that this is true for any algebra, not only Lie). And yes, $k$-algebra coefficients do not bring any new information. – abx Dec 6 '13 at 13:47