nLab
Cartan homotopy

Let $𝔤$ and $𝔥$ be differential graded Lie algebras and $i:𝔤\to 𝔥[-1]$ 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

$$i_{[a,b{]}_{𝔤}}=[i_a,l_b{]}_{𝔥}\text{and}[i_a,i_b{]}_{𝔥}=0,\text{for all}a,b\in 𝔤.$$\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

$$i:{𝒯}_X\to ℰ{\mathrm{nd}}^{*}({\Omega }_X^{*})[-1].$$\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 $i_{[a,b]}=[i_a,l_b]$ and $[i_a,i_b]=0$, together with the defining equation $l_a=[d_{{\Omega }_X^{*}},i_a]$ and with the equations $l_{[a,b]}=[l_a,l_b]$ and $[d_{{\Omega }_X^{*}},l_a]=0$ expressing the fact that $l:{𝒯}_X\to ℰ{\mathrm{nd}}^{*}({\Omega }_X^{*})$ 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 $i$ is a Cartan homotopy, then the degree zero morphism of graded vector spaces $l:𝔤\to 𝔥$ is actually a dgla morphism.

References

• D. Fiorenza, M. Manetti. L-∞ algebras, Cartan homotopies and period maps; arXiv:math/0605297

• D. Fiorenza, M. Manetti. A period map for generalized deformations. Journal of Noncommutative Geometry, Vol. 3, No. 4 (2009), 579-597; arXiv:0808.0140.

• D. Fiorenza, E. Martinengo. A short note on ∞-groupoids and the period map for projective manifolds Publications of the nLab. Vol. 2 (2012); arXiv:0911.3845