nLab Cartan homotopy

Let $\mathfrak{g}$ and $\mathfrak{h}$ be differential graded Lie algebras and $\boldsymbol i\colon \mathfrak{g}\to \mathfrak{h}[-1]$ be a morphism of graded vector spaces. Define $\boldsymbol l:\mathfrak{g}\to \mathfrak{h}$ as

$\boldsymbol l_{a}= d_{\mathfrak{h}} \boldsymbol i_{a} + \boldsymbol i_{d_{\mathfrak{g}}a}$

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

$\boldsymbol i_{[a,b]_{\mathfrak{g}}}= [\boldsymbol i_{a}, \boldsymbol l_{b}]_{\mathfrak{h}}\qquad \text{and} \qquad [\boldsymbol i_{a}, \boldsymbol i_{b}]_{\mathfrak{h}}=0,\qquad \text{for all}\quad a, b \in \mathfrak{g}.$

This name has an evident geometric origin: if $\mathcal{T}_{X}$ 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

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

In this case, $\boldsymbol l_{a}$ is the Lie derivative along the vector field $a$, and the conditions $\boldsymbol i_{[a,b]}= [\boldsymbol i_{a}, \boldsymbol l_{b}]$ and $[\boldsymbol i_{a}, \boldsymbol i_{b}]=0$, together with the defining equation $\boldsymbol l_{a}=[d_{\Omega ^{*}_{X}},\boldsymbol i_{a}]$ and with the equations $\boldsymbol l_{[a,b]}=[\boldsymbol l_{a},\boldsymbol l_{b}]$ and $[d_{\Omega ^{*}_{X}},\boldsymbol l_{a}]=0$ expressing the fact that $\boldsymbol l\colon \mathcal{T}_{X}\to \mathcal{E}nd^{*}(\Omega ^{*}_{X})$ 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 $\boldsymbol i$ is a Cartan homotopy, then the degree zero morphism of graded vector spaces $\boldsymbol l\colon \mathfrak{g}\to \mathfrak{h}$ is actually a dgla morphism.