# nLab Cartan homotopy

Let $𝔤$ and $𝔥$ be differential graded Lie algebras and $i:𝔤\to 𝔥\left[-1\right]$ be a morphism of graded vector spaces. Define $l:𝔤\to 𝔥$ as

${l}_{a}={d}_{𝔥}{i}_{a}+{i}_{{d}_{𝔤}a}$\boldsymbol l_{a}= d_{\mathfrak{h}} \boldsymbol i_{a} + \boldsymbol i_{d_{\mathfrak{g}}a}

for any $a\in 𝔤$. The morphism $i$ is called a Cartan homotopy if it satisfies the two conditions

${i}_{\left[a,b{\right]}_{𝔤}}=\left[{i}_{a},{l}_{b}{\right]}_{𝔥}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\left[{i}_{a},{i}_{b}{\right]}_{𝔥}=0,\phantom{\rule{2em}{0ex}}\text{for all}\phantom{\rule{1em}{0ex}}a,b\in 𝔤.$\boldsymbol i_{[a,b]_{\mathfrak{g}}}= [\boldsymbol i_{a}, \boldsymbol l_{b}]_{\mathfrak{h}}\qquad \text{and} \qquad [\boldsymbol i_{a}, \boldsymbol i_{b}]_{\mathfrak{h}}=0,\qquad \text{for all}\quad a, b \in \mathfrak{g}.

This name has an evident geometric origin: if ${𝒯}_{X}$ is the tangent sheaf of a smooth manifold $X$ and ${\Omega }_{X}^{*}$ is the sheaf of complexes of differential forms, then the contraction of differential forms with vector fields is a Cartan homotopy

$i:{𝒯}_{X}\to ℰ{\mathrm{nd}}^{*}\left({\Omega }_{X}^{*}\right)\left[-1\right].$\boldsymbol i\colon \mathcal{T}_{X}\to \mathcal{E}nd^{*}(\Omega ^{*}_{X})[-1].

In this case, ${l}_{a}$ is the Lie derivative along the vector field $a$, and the conditions ${i}_{\left[a,b\right]}=\left[{i}_{a},{l}_{b}\right]$ and $\left[{i}_{a},{i}_{b}\right]=0$, together with the defining equation ${l}_{a}=\left[{d}_{{\Omega }_{X}^{*}},{i}_{a}\right]$ and with the equations ${l}_{\left[a,b\right]}=\left[{l}_{a},{l}_{b}\right]$ and $\left[{d}_{{\Omega }_{X}^{*}},{l}_{a}\right]=0$ expressing the fact that $l:{𝒯}_{X}\to ℰ{\mathrm{nd}}^{*}\left({\Omega }_{X}^{*}\right)$ is a dgla morphism, are nothing but the well-known Cartan identities involving contractions and Lie derivatives.

It is a straightforward computation to see that, if $i$ is a Cartan homotopy, then the degree zero morphism of graded vector spaces $l:𝔤\to 𝔥$ is actually a dgla morphism.

## References

Created on September 12, 2012 23:58:50 by Domenico Fiorenza (87.18.219.159)