Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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