# The meaning of a “subcomplex” of the Cartan-Eilenberg of a Lie algebra

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra, and $\mathfrak{g}^*$ be the dual vector space. We have the standard Cartan-Eilenberg complex $(\wedge^{\cdot} \mathfrak{g}^* ,\text{d}_{CE})$ where the differential is given by the dual of the Lie bracket.

Now assume that we have a Lie subalgebra $\mathfrak{k}\subset \mathfrak{g}$ and a $\mathfrak{k}$ invariant complement $\mathfrak{p}\subset \mathfrak{g}$. That is: $$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$$ as vector spaces, $[\mathfrak{k},\mathfrak{k}]\subset \mathfrak{k}$ and $[\mathfrak{k},\mathfrak{p}]\subset \mathfrak{p}$. Remember that in general $[\mathfrak{p},\mathfrak{p}]\nsubseteq$$\mathfrak{p}. We can construct \wedge^{\cdot} \mathfrak{p}^* and define a derivative on it as follows. For any \theta \in \wedge^k \mathfrak{p}^* , use the split \mathfrak{g}^* =\mathfrak{k}^* \oplus\mathfrak{p}^* we can treat \theta as an element in \wedge^k \mathfrak{g}^* , take the Cartan-Eilenberg differential, we get \text{d}_{CE}\theta \in \wedge^{k+1} \mathfrak{g}^* . Again use \mathfrak{g}^* =\mathfrak{k}^* \oplus\mathfrak{p}^* we get$$ \wedge^{k+1} \mathfrak{g}^* = \bigoplus_{i=0}^{k+1} \wedge^i \mathfrak{k}^* \otimes \wedge^{k+1-i} \mathfrak{p}^*$$Let$\text{pr}$denote the projection onto the$i=0$component of the above decomposition. Then our derivation$\overline{\text{d}}$is defined to be$\text{pr}\circ \text{d}_{CE}: \wedge^{\cdot} \mathfrak{p}^* \rightarrow \wedge^{\cdot+1}\mathfrak{p}^* $. It is not difficult to check that$\overline{\text{d}}$is a derivation but$\overline{\text{d}}\circ \overline{\text{d}}\neq 0$. It measures the failure of$[\mathfrak{p},\mathfrak{p}]\subseteq\mathfrak{p}$. My question is: what does this$\overline{\text{d}}\circ \overline{\text{d}}$stand for? Is it a kind of curvature? Is there any nice expression of it? In particular, does it have anything to do with the connection on the principle bundle$K\rightarrow G \rightarrow G/K\$?

I have the impression that Kostant has done some work of this kind but I just cannot find them.

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Take a look at Chapter X in the third volume of "Connections, curvature and cohomology" by Greub, Halperin and Vanstone. You will find that indeed what you have can be interpreted as a curvature of an algebraic connection. – José Figueroa-O'Farrill Aug 8 '12 at 14:11