TITLE: Which is the correct universal enveloping algebra in positive characteristic?
QUESTION [15 upvotes]: This is an extension of this question about symmetric algebras in positive characteristic.  The title is also a bit tongue-in-cheek, as I am sure that there are multiple "correct" answers.
Let $\mathfrak g$ be a Lie algebra over $k$.  One can define the universal enveloping algebra $U\mathfrak g$ in terms of the adjunction: $$\text{Hom}_{\rm LieAlg}(\mathfrak g, A) = \text{Hom}_{\rm AsAlg}(U\mathfrak g, A)$$ for any associative algebra $A$.  Then it's easy enough to check that $U\mathfrak g$ is the quotient of the free tensor algebra generated by $\mathfrak g$ by the ideal generated by elements of the form $xy - yx - [x,y]$.  (At least, I'm sure of this when the characteristic is not $2$.  I don't have a good grasp in characteristic $2$, though, because I've heard that the correct notion of "Lie algebra" is different.)
But there's another good algebra, which agrees with $U\mathfrak g$ in characteristic $0$.  Namely, if $\mathfrak g$ is the Lie algebra of some algebraic group $G$, then I think that the algebra of left-invariant differential operators is some sort of "divided-power" version of $U\mathfrak g$.
So, am I correct that this notions diverge in positive characteristic?  If so, does the divided-power algebra have a nice generators-and-relations description?  More importantly, which rings are used for what?

REPLY [5 votes]: This question popped up on the feed recently, and I wanted to add another, "brave new math"-style answer. Namely, the point of Lie algebras is that, in characteristic $0$, a Lie algebra (resp., an $L_\infty$-algebra) gives necessary and sufficient information to define a formal Lie group (resp., a cogroup object in CDGA). Global sections of this group form a commutative, co-associative Hopf algebra and the (topological/filtered/etc.) dual will be the universal enveloping algebra.
There is an analogous story to the $L_\infty$ story in characteristic $p$, proved by Akhil Mathew and Lukas Brantner, https://arxiv.org/abs/1904.07352. They prove that the structure of a derived connected formal group scheme (which determines a formal moduli problem) is determined by a so-called partition Lie algebra structure on the tangent space. This is no longer an "additive" structure (not determined by a multilinear operad unlike $L_\infty$), but it is "just" some nicely combinatorial algebraic structure on the tangent space. If you modify your definition of Lie algebra using this picture then to each (partition) Lie algebra you can associate a formal moduli problem, and to every formal moduli problem you can associate a universal enveloping algebra by taking the dual to a universal Hopf algebra associated to your formal moduli problem. From what I understand, the "$\pi_0$ operations" inherent in this structure are exactly Lie algebra operations and divided powers.