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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

    Thank you,
    Francisco Kordon,
    franciscokordon at
    • 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 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

    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 :) 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.