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nLab > Latest Changes: equivariant de Rham cohomology

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 27th 2013

    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….)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2015
    • (edited Feb 3rd 2015)

    I have fixed a bunch of typographical glitches at equivariant de Rham cohomology . Also added paragraphs cross-linking with Weil algebra.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2019

    added pointer to

    • Oliver Goertsches, Leopold Zoller, Equivariant de Rham Cohomology: Theory and Applications, São Paulo J. Math. Sci. (2019) (doi:10.1007/s40863-019-00129-4)

    diff, v10, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2019

    added pointer to

    diff, v12, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2019

    added pointer also to

    and harmonized the list of references here with that at Weil algebra

    diff, v12, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2019

    finally spelled out the details of the Weil model: here

    diff, v14, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2019

    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

    d K =exp(−t a∧ι v a)∘(d dR+d W)∘exp(t a∧ι v a) =d dR+d W+[d dR+d W,t a∧ι v a] =d dR+d W+[d W,t a]∧ι v a−t a∧[d dR,ι v a] =d dR+d W+(r a−12f bc at b∧t c⏟)∧ι v a−t a∧ℒ v a \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…

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2019

    Oh, I see. Will edit…

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2019

    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

    • Henri Cartan, Sec. 6 of La transgression dans un groupe de Lie et dans un espace fibré principal, Colloque de topologie (espaces fibrés). Bruxelles, 1950

    If anyone has more publication details on this article of Cartan, in particular maybe a scan, we should add it.

    diff, v18, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2019

    added pointer to

    diff, v20, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2019

    added statement of the equivariant de Rham theorem (here)

    diff, v21, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2019

    added also these pointers:

    Generalization of the equivariant de Rham theorem to non-compact Lie groups is due to

    based on the simplicial de Rham complex

    see also

    diff, v22, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2019
    • (edited Jun 28th 2019)

    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 𝔤\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?

    diff, v24, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeOct 23rd 2020

    finally added publication data for:

    • Jaap Kalkman, BRST Model for Equivariant Cohomology and Representatives for the Equivariant Thom Class, Comm. Math. Phys. Volume 153, Number 3 (1993), 447-463. (euclid:1104252784)

    diff, v28, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeOct 28th 2020

    added pointer to:

    diff, v29, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2020

    added brief statement of the Cartan map (here)

    diff, v33, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2020

    Question: Does any author make explicit the zig-zag of quasi-isomorphisms between the Cartan model of a smooth GG-action on a manifold XX and the PL de Rham complex of the Borel space X⫽GX \sslash G?

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeOct 30th 2020

    added pointer to:

    diff, v36, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2021
    • (edited Aug 18th 2021)

    added pointer to:

    diff, v37, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2021
    • (edited Aug 18th 2021)

    added these pointers:

    diff, v38, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2021

    added pointer to:

    diff, v40, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2021

    added pointer to:

    This is a remarkably comprehensive textbook. Will be adding this pointer to some other related entries, too.

    diff, v41, current

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