nLab
Cartan homotopy

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

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

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

i [a,b] 𝔤=[i a,l b] 𝔥and[i a,i b] 𝔥=0,for alla,b𝔤. \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\mathcal{T}_{X} is the tangent sheaf of a smooth manifold XX and Ω X *\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 Xnd *(Ω X *)[1]. \boldsymbol i\colon \mathcal{T}_{X}\to \mathcal{E}nd^{*}(\Omega ^{*}_{X})[-1].

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

References

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