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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2009
    • (edited Nov 2nd 2009)
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 22nd 2013
    • (edited Jan 22nd 2013)

    At Lie-Rinehart pair in Revision 8 somebody added the words

    CAUTION: Elsewhere in the literature, it is : a Lie algebra morphism from 𝔤Der(A)\mathfrak{g} \to Der(A)

    I don’t understand what this addition is good for. That homomorphism is stated precisely this way just three lines above.

    Therefore I am removing that addition now. But please let me know if I am missing something.

    • CommentRowNumber3.
    • CommentAuthorGuest
    • CommentTimeFeb 24th 2020
    In the original reference [Rinehart] it is not asked that the anchor map be a morphism of Lie algebras.

    A sufficient condition for that to happen is that the annihilator A_L={a \in A : aX=0}
    be trivial. See Lemma 2.2 in https://arxiv.org/abs/2002.05718

    Thank you,
    Francisco Kordon,
    franciscokordon at uca.fr
    • CommentRowNumber4.
    • CommentAuthorPraphulla
    • CommentTimeApr 27th 2024
    It may be useful (for me) if some one can discuss about the notion of morphism of Lie-Rinehart algebras here.

    As the notion of morphisms of Lie algebroids, I am expecting this notion to be not so straightforward.

    I came across two notions of morphisms of Lie-Rinehart algebras

    1) something to do with "pullback", by Madeliene Jotz Lean in the work https://www.uni-math.gwdg.de/mjotz/JotzLean18c.pdf This seem to be natural and reminds me the notion of morphism of ringed spaces where we pushforward the sheaf on X to sheaf on Y when talking about morphisms of ringed spaces (X,O_X)--->(Y,O_Y). In my opinion the word "pushforward" should be used in the paper instead of "pullback".

    2) notion of morphism in Camille Laurent-Gengoux and Ruben Louis work https://hal.science/hal-03462506/document.

    I am trying to see if these two are related. I do not think they are.
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2024

    The first question one would want to check is whether either definition reduces — in the case where the base ring is C (X)C^\infty(X) — to the ordinary homomorphisms of Lie algebroids (under the equivalence between such LR-pairs and Lie algebroids over XX)?

    (That said, I have not had the leisure to look closely at the articles, and may not find the time.)

    • CommentRowNumber6.
    • CommentAuthorPraphulla
    • CommentTimeApr 28th 2024
    Yes. I tried that before asking here :)

    https://youtu.be/SMBriA04EOw?feature=shared talks about the above two notions (as far as I understand) and calls one of them to be a morphism, the other as comorphism.

    When I understand more and get some confidence, I will try to add it here.
    • CommentRowNumber7.
    • CommentAuthorBryceClarke
    • CommentTimeJul 3rd 2024

    Added a reference to the thesis of Joel Couchman

    diff, v21, current