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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 2nd 2009
   • (edited Nov 2nd 2009)
   • PermaLink
   Author: Urs
   Format: Textcreated [[Lie-Rinehart pair]]

   created Lie-Rinehart pair
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 22nd 2013
   • (edited Jan 22nd 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAt _[[Lie-Rinehart pair]]_ in [Revision 8](http://www.ncatlab.org/nlab/revision/diff/Lie-Rinehart+pair/8) somebody added the words > CAUTION: Elsewhere in the literature, it is : > a Lie algebra morphism from $\mathfrak{g} \to Der(A)$ 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.

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

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

   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.

3. • CommentRowNumber3.
   • CommentAuthorGuest
   • CommentTimeFeb 24th 2020
   • PermaLink
   Author: Guest
   Format: TextIn 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

   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
4. • CommentRowNumber4.
   • CommentAuthorPraphulla
   • CommentTimeApr 27th 2024
   • PermaLink
   Author: Praphulla
   Format: TextIt 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.

   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.
5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 27th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe first question one would want to check is whether either definition reduces --- in the case where the base ring is $C^\infty(X)$ --- to the ordinary homomorphisms of Lie algebroids (under the equivalence between such LR-pairs and Lie algebroids over $X$)? (That said, I have not had the leisure to look closely at the articles, and may not find the time.)

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

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

6. • CommentRowNumber6.
   • CommentAuthorPraphulla
   • CommentTimeApr 28th 2024
   • PermaLink
   Author: Praphulla
   Format: TextYes. 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.

   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.
7. • CommentRowNumber7.
   • CommentAuthorBryceClarke
   • CommentTimeJul 3rd 2024
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   Author: BryceClarke
   Format: MarkdownItexAdded a reference to the thesis of Joel Couchman <a href="https://ncatlab.org/nlab/revision/diff/Lie-Rinehart+pair/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lie-Rinehart+pair/21">v21</a>, <a href="https://ncatlab.org/nlab/show/Lie-Rinehart+pair">current</a>

   Added a reference to the thesis of Joel Couchman

   diff, v21, current