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nLab > Latest Changes: Cartan calculus

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2010

    stub for Cartan calculus

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJul 9th 2010

    I added some hints about related subjects.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2010

    Thanks, I was thinking of that, too. We should eventually have a separate entry on the Cartan model for equiv-cohomology…

    • CommentRowNumber4.
    • CommentAuthorKevin Lin
    • CommentTimeJul 20th 2010
    • (edited Jul 20th 2010)
    To be added (by me later, or by someone else sooner): Cartan calculus for polyvector fields and Schouten-Nijenhuis bracket?

    And here is a good reference for Cartan equivariant cohomology (as well as for other approaches).

    I noticed that the equivariant cohomology page doesn't yet have anything at all about the Cartan approach.
    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeJul 20th 2010

    Cartan’s model for equivariant cohomology is limited to rather special spaces, like differentiable manifolds, no ? On the other hand, equivariant cohomology is studied for more general spaces and under more general topological groups.

    • CommentRowNumber6.
    • CommentAuthorKevin Lin
    • CommentTimeJul 20th 2010
    Correct. But I don't consider manifolds to be a "limited" class of spaces...
    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeJul 20th 2010

    Come on, locally compact groups are so important (those beyond Lie groups). Even if you consider differentiable manifolds, the group variable is often so much more general.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2010

    Not sure what the issue is. Certainly Cartan’s model for equivariant cohomology deserves to be discussed or at least linked at the entry on equivariant cohomology.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeJul 20th 2010
    • (edited Jul 20th 2010)

    Surely it has to be a section. I was asking for it a cuple of weeks ago. But Kevin’s complaint

    I noticed that the equivariant cohomology page doesn’t yet have anything at all about the Cartan approach.

    has an appropriate answer: we preferred to center the entry about the general approach. Special features are for special sections or even separate entries. Kevin answers that manifolds are general enough. I still disagree.

    I would like to see also connection to the equivariant localization formulas. Very important in my nlab plans and need exactly the Cartan model for many aspects.

    • CommentRowNumber10.
    • CommentAuthorKevin Lin
    • CommentTimeJul 20th 2010
    • (edited Jul 20th 2010)
    OK. I understand. It's of course fine with me to put the bulk of the Cartan model in a separate entry. But we agree that it deserves a few sentences at least in the equivariant cohomology page.
    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeJul 20th 2010

    It deserves a large section at least…

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 30th 2020

    Removed the reference to Henri Cartan (there is still a pointer to the Cartan model, so no info is lost, but the impression avoided that Cartan calculus is named after Henri Cartan). What’s an actual reference to a publication by Élie Cartan that could go here?

    diff, v14, current

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 7th 2023

    Added the remaining identities.

    Added:

    Cartan calculus on diffeological spaces requires a nontrivial condition, which is explored and developed in

    diff, v15, current

1 to 13 of 13

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