This is an extension of [this question](https://mathoverflow.net/questions/7080/definition-of-the-symmetric-algebra-in-arbitrary-characteristic-for-graded-vector/) 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?



Which is the correct universal enveloping algebra in positive characteristic?
The notions do indeed diverge in positive characteristic: there is the enveloping algebra, and then (in the case that $\mathfrak g$ is the Lie algebra of an algebraic group G) there is also the hyperalgebra of G, which is the divided-power version you mention. In characteristic 0 these two algebras coincide, but in positive characteristic they differ very much. In particular, the hyperalgebra is not finitely-generated in positive characteristic; see Takeuchi's paper "Generators and Relations for the Hyperalgebras of Reductive Groups" for the reductive case. There is also a good exploration of the hyperalgebra in Haboush's paper "Central Differential Operators of Split Semisimple Groups Over Fields of Positive Characteristic."


One can obtain the hyperalgebra as follows. I don't know in what generality the following construction holds, so let's say that $\mathfrak g$ is the Lie algebra of an algebraic group G defined over $\mathbb Z$. Then there is a $\mathbb Z$-form of the enveloping algebra of G (the Kostant $\mathbb Z$-form) formed by taking divided powers, and upon base change this algebra becomes the hyperalgebra. Alternatively, one can take an appropriate Hopf-algebra dual of the ring of functions on G (cf Jantzen's book "Representations of Algebraic Groups").


As for their uses, in positive characteristic the hyperalgebra of G captures the representation theory of G in the way that the enveloping algebra does in characteristic 0, i.e. the finite-dimensional representations of G are exactly the same as the finite-dimensional representations of the hyperalgebra. This fails completely for the enveloping algebra: instead, the enveloping algebra sees only the representations of the first Frobenius kernel of G.


