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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2013
    • (edited May 3rd 2014)

    It just occurred to me that there is an immediate axiomatization of the Liouville-Poincaré 1-form (the canonical differential 1-form on a cotangent bundle) in differential cohesion.

    In fact, it is the special case of a much more general notion: for AA any type in differential cohesion the total space X𝒪 X(A)\underset{X}{\sum} \mathcal{O}_X(A) of the AA-valued structure sheaf over any XX carries a canonical AA-cocycle.

    For A=Ω 1A = \Omega^1 the sheaf of 1-forms and XX a manifold, this is the traditional Liouville-Poincaré 1-form on T *XT^* X.

    I made a quick note on that at differential cohesion – Liouville-Poincaré cocycle.

    Thanks to a conversation with Owen Gwilliam I now also understand how that construction gives the antibracket in the BV-BRST complex. I still need to write that out. Not today though.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJan 17th 2013

    traditional Liouville-Poincaré 1-form

    Google does not give any hits with phrase Liouville-Poincaré.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2013
    • (edited Jan 17th 2013)

    All sources on that 1-form say it is called the Liouville form or the Poincaré-form. I thought if it’s really due to both, one should give credit to both.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJan 17th 2013

    Thanks. I do know it as canonical form, indeed, Liouville-Poincaré sounds OK a priori, but it may repel some of the outside users – one has to be careful with new mixed names as they are expected to be different from the one with one name; for example Euler equation (in fluid dynamics) is completely different from say Euler-Lagrange equation (in variational calculus). I was just told be a student yesterday that he does not read nnLab any more, because the nnLab is like machine – always gives a correct answer but unrecognizable and hi-brow even when he thinks he knows the notion. This was sad to hear because it was about a good student.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2013

    Maybe somebody know or enjoys to dig out the original sources? Chances are that the idea is originally neither to Liouville nor to Poincaré! :-)

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeJan 17th 2013

    Yes, likely :) it could be instructive, how they were lead there, it is always instructive when it is about such great minds :)

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