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created equivariant de Rham cohomology with a brief note on the Cartan model.
(I seem to remember that we had discussion of this in the general context of Lie algebroids elsewhere already, several years back. But now I cannot find it….)
I have fixed a bunch of typographical glitches at equivariant de Rham cohomology . Also added paragraphs cross-linking with Weil algebra.
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and harmonized the list of references here with that at Weil algebra
I thought I’d follow Miettinen’s hep-th/9612209 for a concise and readable account of passing from Weil via Kalkman to Cartan. But if we accept the conjugation in his (14) don’t we get an extra quadratic term in (13)?
With the notation that I am using now in the entry (here), I get
$\begin{aligned} d_K & = \exp\big( -t^a \wedge \iota_{v^a} \big) \circ \big( d_{dR} + d_W \big) \circ \exp\big( t^a \wedge \iota_{v^a} \big) \\ & = d_{dR} + d_W + \big[ d_{dR} + d_W, t^a \wedge \iota_{v^a} \big] \\ & = d_{dR} + d_W + \big[ d_W, t^a \big] \wedge \iota_{v^a} - t^a \wedge \big[ d_{dR}, \iota_{v^a} \big] \\ & = d_{dR} + d_W + \big( r^a \underbrace{ - \tfrac{1}{2}f_{b c}^a t^b \wedge t^c } \big) \wedge \iota_{v^a} - t^a \wedge \mathcal{L}_{v^a} \end{aligned}$where the term over the brace is not in (13) of hep-th/9612209 .
Of course this is all elementary. Maybe I need to take a break…
Oh, I see. Will edit…
Okay, I have brought in statement and proof of a big lemma (here) explainng how the Cartan model is the result of solving in the Weil model the horizontality condition by applying the corresponding projection operator.
This is just an exegesis of Mathai-Quillen 86 around (5.6), which in turn follows the original
If anyone has more publication details on this article of Cartan, in particular maybe a scan, we should add it.
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Generalization of the equivariant de Rham theorem to non-compact Lie groups is due to
based on the simplicial de Rham complex
see also
I have added the proof idea of the equivariant de Rham theorem (here), explaining how the Weil model is the image in dgc-algebraic rational homotopy theory of the Borel construction.
To make this proof idea a proof one needs to add some facts about resolutions in $\mathfrak{g}$-rational homotopy theory. The usual sources all shy away from going this last step. What’s a source that goes all the way?
finally added publication data for:
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Question: Does any author make explicit the zig-zag of quasi-isomorphisms between the Cartan model of a smooth $G$-action on a manifold $X$ and the PL de Rham complex of the Borel space $X \sslash G$?
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Andreas Kübel, Andreas Thom, Equivariant characteristic forms in the Cartan model and Borel equivariant cohomology (arXiv:1508.07847)
Andreas Kübel, Andreas Thom, Equivariant Differential Cohomology, Trans. Amer. Math. Soc. 370 (2018), 8237-8283 (arXiv:1510.06392, doi:10.1090/tran/7315)
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