Algebraic Groups in Characteristic p

Question

It is well-known that Lie groups are, under nice conditions, essentially determined by their Lie-algebras. What's the corresponding statement for algebraic groups over fields of finite characteristic?

Answer 1

Since the question is somewhat open-ended, it may be useful to add further comments to what Dietrich and Marguax have said. [Also, it's important to correct Dietrich's first sentence: While any finite dimensional Lie algebra over $\mathbb{C}$ is the Lie algebra of some Lie group, it need not be the Lie algebra of an algebraic group. Some references are given in my answer to an earlier question here.]

1) In the setting of (real or complex) Lie groups, it's always essential to focus on connected groups, since by definition the Lie algebra only depends on the identity component of the group. Moreover, the classical correspondence between Lie groups and their Lie algebras isn't quite bijective: semisimple groups with the same root system have isomorphic Lie algebras even though the groups may vary through an isogeny class from simply connected to adjoint type. Apart from such qualifications, the exponential and logarithm maps allow one to pass back and forth between the analytic and the algebraic theories pretty effectively.

2) In the 1950s Chevalley extended the work on linear algebraic groups begun by Kolchin and Borel, imitating the Lie algebra correspondence to some extent. Here the Lie algebra is geometrically the tangent space to the group at the identity, so again one wants to consider mainly the connected groups (equivalent to being irreducible in the Zariski topology). Over an algebraically closed field of characteristic 0, there is even a partial substitute for analytic methods in his use of formal exponential power series. But obviously this has its limits, and in prime characteristic it breaks down badly. Even in characteristic 0, the Jordan-Chevalley decomposition in the group allows one to distinguish the additive and multiplicative groups (as algebraic groups) but not their Lie algebras.

3) For the crucial study (including classification) of semisimple groups, Chevalley abandoned the Lie algebra but was still able to imitate almost exactly the Cartan-Killing classification by root data. Here again it's essential to focus on simply connected groups, since isogenous groups often have isomorphic Lie algebras (though in characteristic $p$ that's not always true).

4) For semisimple (connected and simply connected) $G$, the question asked amounts to this: If the Lie algebras of two such groups are isomorphic over an algebraically closed field of prime characteristic, are the groups isomorphic as algebraic groups? To this the answer turns out to be YES, but it seems to require case-by-case study using the Chevalley classification. For some $p$, the Lie algebra of a simple $G$ is not simple, which complicates matters. Probably the most thorough study was done by G.M.D. Hogeweij in his Utrecht thesis: see his papers in Indag. Math. 44 (1982).

5) The books by Chevalley, Demazure-Gabriel, and myself all contain various details about the characteristic 0 correspondence for algebraic groups and their Lie algebras, which concerns mainly centers, centralizers, etc. But examples show clearly how much breaks down in characteristic $p$. So one always has to be cautious.

P.S. Besides the further insights into algebraic groups and their Lie algebras given by scheme theory and by formal groups, the hyperalgebra approach is quite useful. It's developed thoroughly in the book Representations of Algebraic Groups by J.C. Jantzen (second ed., AMS, 2003), where the complications in characteristic $p$ representation theory are explored.

Answer 2

Over the complex numbers, connected linear algebraic groups correspond to Lie algebras in the usual way. This Lie correspondence breaks down over number fields, and breaks down even more over fields of prime characteristic. This was first shown by Chevalley in the early 1950s, and it started a search for a good substitute for the Lie algebra of an algebraic group. This involved formal groups, hyperalgebras, Lazard correspondence etc. For this see for example the book "Formal Groups and Applications" by Michiel Hazewinkel.

Answer 3

If you focus on the "good" Lie groups, namely the connected semisimple ones, then the classification in the compact case or the complex-analytic case is achieved by means of the Lie algebra just up to isogeny (the precise ambiguity implicit in the "essentially" in your question), and the Lie algebras that arise are the semisimple ones over $\mathbf{R}$ with negative-definite Killing form in the compact case and the semisimple complex Lie algebras in the complex-analytic case. Both cases are classified by root systems, thanks to Cartan, Killing, et al.

But one can do so much better: the root datum, an integral refinement of the root system introduced by Demazure in SGA3 for algebraic-group purposes, classifies such analytic groups in both cases up to isomorphism (no isogeny ambiguity). And building on ideas of Chevalley, Borel, Tits, et al., this works algebraically over any field: the "split" connected semisimple linear algebraic groups over any field whatsoever (or even any commutative ring, with appropriate definitions) are classified up to isomorphism by root data which are "semisimple" (i.e., roots span the rationalization of the character lattice). This tells one much much more than can be achieved by formal groups (which cannot detect the underlying algebraicity, though are certainly useful tools, albeit more often in the commutative case where Lie theory and root systems are largely useless).