# Is there a reasonable way to define “reductive Lie algebra” in prime characteristic?

Among the finite dimensional Lie algebras over a field of characteristic 0, there is a sensible definition of "reductive Lie algebra" going back at least to the 1960 first chapter of N. Bourbaki's treatise Groupes et algebres de Lie ($\S6$, no. 4 in the first printing, which I have at hand). Here the concise definition just requires that $\mathfrak{g}$ have a completely reducible ("semisimple") adjoint representation. This is immediately shown to be equivalent to a number of other conditions, e.g., that the derived algebra is semisimple, or that the nilpotent radical is zero, or that the radical of $\mathbb{g}$ is the center.

Is there a reasonable definition of "reductive Lie algebra" over a field of characteristic $p>0$?

For example, the undefined term occurs in the title of a 2000 paper by J.C. Jantzen here, though the text suggests right away that he is referring to the Lie algebra of a (say connected) reductive algebraic group $G$.

The current widespread use of the term "reductive" is sometimes problematic though well-motivated by the inductive role of Levi subgroups of parabolic subgroups in a given semisimple $G$. It reflects the influence of the 1965 paper by Borel-Tits here. For affine algebraic groups (or group schemes) the notion "reductive" just means that the unipotent radical of the group is trivial. The structure theory then shows that such a (connected) algebraic group is an almost-direct product of a semisimple group and an algebraic torus. This of course relies on the intrinsic nature of the Jordan-Chevalley decomposition, which is missing from the classical theory of Lie groups. There one often focuses on semisimple groups, having found (thanks to Levi and Mal'cev) that an arbitrary connected Lie group is a semidirect product of a semisimple Lie group (unique up to conjugacy) and a solvable Lie group (the radical).

The term "reductive" does have the drawback of suggesting complete reducibility of finite dimensional representations ("linear reductivity"). In characteristic 0, Mostow had already studied carefully the Lie groups which satisfy such complete reducibility for relevant finite dimensional representations. This property carries over easily to semisimple (hence also reductive) algebraic groups thanks to Chevalley's classification combined with Weyl's classical theorem for Lie algebras. But linear reductivity fails badly for reductive algebraic groups in prime characteristic. Here there is a weaker substitute called "geometric reductivity" conjectured by Mumford to hold for all reductive groups and eventually proved by Haboush. In any case, the Bourbaki definition of "reductive Lie algebra" isn't helpful in characteristic $p$. And the Lie algebra of a reductive group sometimes has a quirky structure, as seen already for $\mathfrak{gl}_n$ or $\mathfrak{sl}_n$ when $p | n$. It's definitely not easy to characterize such Lie algebras within the much larger class of restricted Lie algebras.

• When you write that the Bourbaki definition of "reductive Lie algebra" isn't helpful in characteristic $p$ (of course, Prop. 5 of section 6.4 of Ch. I very much illuminates that definition!), it focuses attention on an important point: helpful for what purpose? Any sense of "reasonable" should be determined by what one wants to do with the notion. Already for "semisimple" (Def. 1 in 6.1 of Ch. I, illuminated by Thm. 1 there) it may be disorienting that when $p|n$, $\mathfrak{sl}_n$ isn't semisimple but $\mathfrak{pgl}_n$ is. So some clarification on the motivating context would help. – nfdc23 Oct 29 '17 at 16:36
• There is no "Bourbaki definition of reductive Lie algebra in positive characteristic", because the chapter (I,§6: Semisimple Lie algebras) in which this definition can be found, starts with an explicit assumption that the ground field has characteristic zero. – YCor Oct 29 '17 at 16:51
• @YCor: That is certainly a good point. It is perhaps also worth noting that some of the equivalent conditions to semisimplicity proved by Bourbaki in char. 0 do still hold in positive characteristic, as shown in Seligman's book Modular Lie Algebras (which I had in mind when writing my comment above, as I should have made clearer there). In lieu of a motivating purpose, it's hard to know what is a "reasonable" definition. – nfdc23 Oct 29 '17 at 17:55
• @nfdc23: I realize that "reasonable" is vague, but I'm wondering if the default definition "Lie algebra of a reductive group" (tossing all problems back into algebraic groups) can be replaced by a purely Lie algebraic definition here. Lie algebras of algebraic groups in characteristic $p>0$ being restricted (= Lie $p$-algebras), the restricted Lie algebras form a natural starting point. Since basic notions about (for instance) support varieties apply to this family, it would be useful to bypass the problematic question of what "semisimple" means for a restricted Lie algebra. – Jim Humphreys Oct 29 '17 at 22:31
• I'm not sure how you are making sense of the nilpotent radical in char 0. One has $Lie(\mathbb G_a) = Lie (\mathbb G_m) = k$, where k is the field one is working over. To decide whether this Lie algebra consists of nilpotent or semisimple elements you need more information, like a faithful representation. Of course in char p one can use the p-map to define nilpotence intrinsically. So I'm not sure how to get the equivalence of 'the nilpotent radical is zero, or that the radical of 𝕘 is the center.' – David Stewart Oct 30 '17 at 9:02