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When I did net-surfing at home, I met some geometric backgrounds of Lie algebras and encountered the concept of Lie algebroids. In differential geometry, a Lie algebroid seems to be defined as motivated via a vector bundle on manifolds. Is this important in "algebraic" geometry too (How about positive-characteristic)? Does anyone know some references about Lie algebroid in algebraic aspects?

(In the first place, how Lie algebra of vector fields on algebraic variety are nice?)

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  • $\begingroup$ In char 0, it is covered in Gaitsgory–Robenblyum, A study of ..., Vol 2. In char p, you could look at Brantner–Waldron for purely inseparable extensions. $\endgroup$
    – Z. M
    Commented May 13, 2022 at 7:12
  • $\begingroup$ Lie algebras of vector fields, namely foliations, behave poorly in positive characteristic, but it is a feature. For example, the relative tangent bundle of $k[x]\subset k[x,y]$ recovers only $k[x,y^p]\subset k[x,y]$. But one can work it out to ones advantage. <wink> I would give a reference for this, but I am writing it down right now. <laugh> $\endgroup$
    – P. Grabowski
    Commented May 31, 2022 at 19:03
  • $\begingroup$ @Z.M Thanks for your comment! I searched the documents, but it seems to be a very high level for self-studying :( I might use it for reference. $\endgroup$
    – Frank Voigt
    Commented Jun 17, 2022 at 5:37
  • $\begingroup$ @P.Grabowski Thanks for your comment! I asked this question because I wanted some application of restricted Lie algebras for geometric topics or others. And I saw a fact that Lie algebras of vector fields are closed for p-th power in positive characteristic. So, if such Lie algebras have geometric aspects, how affect its p-th power in geometry? and I'm here. By the way, according to your comment, it seems to get some technic for getting information from it... I look forward to your writing :) $\endgroup$
    – Frank Voigt
    Commented Jun 17, 2022 at 6:07

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I suggest having a look at

Beĭlinson, A.; Bernstein, J. A proof of Jantzen conjectures.

MR: Matches for: MR=1237825

§1.2 .

https://people.math.harvard.edu/~gaitsgde/grad_2009/BB%20-%20Jantzen.pdf

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