# Does the classification of reductive groups follow from that of semisimple groups?

I had a question for anyone familiar with the proofs of the classification of reductive groups. I skipped most of the details of classification when I originally learned linear algebraic groups, and now I'm trying to go back and fill the gaps in my knowledge.

Let $G$ be a connected algebraic group over an algebraically closed field. If $G$ is reductive, then $G$ is classified up to isomorphism by its root datum, and conversely every root datum belongs to some connected, reductive group.

If $G$ is semisimple, then $G$ is classified up to isomorphism by its root system and its fundamental group, and conversely every pair of a root system and a subgroup of the weight lattice modulo the root lattice belongs to some semisimple group.

Taking the classification theorem for semisimple groups for granted (say existence and uniqueness), is it possible to deduce the classification theorem for connected, reductive groups? Or does the classification theorem for reductive groups need to be done again from scratch.

• Yes. For any field $k$ and connected reductive $k$-group $G$ there is the canonical isomorphism $(Z \times G')/\mu \simeq G$ where $G'$ is the connected semisimple derived group, $Z$ is the maximal central torus, and $\mu = Z \cap G'$ is a central $k$-subgroup scheme of $G'$ (and if $H$ is a connected semisimple $k$-group, $\mu \subset H$ is a central $k$-subgroup scheme, and $T$ is a $k$-torus equipped with an inclusion of $\mu$ then $G :=(T \times G')/\mu$ is connected reductive with derived group $G'$, maximal central torus $T$, etc.) – nfdc23 Dec 13 '16 at 1:56
• @D_S: As nfdc23 indicates, the answer to your basic question is 'yes', but already over an algebraically closed field it's a bit tricky to state a comprehensive theorem. For example, even when $G$ is connected and simple (as an algebraic group), it isn't always determined up to isomorphism by its root system and fundamental group (as you assert and as I loosely formulated it in the first printing of my 1975 book). I'll try to write a detailed answer if I have time later today, including reductive groups in the mix. – Jim Humphreys Dec 14 '16 at 15:41
• @JimHumphreys, your answer is very nice, but it's not clear to me how it indicates that a connected simple group over an algebraically closed field "isn't always determined up to isomorphism by its root system and fundamental group". (Of course isogenies are relevant, but change the fundamental group, at least infinitesimally.) Could you say more? – LSpice Dec 23 '19 at 23:32
• @LSpice: Sorry for the delay in responding, but I had to search awhile to find older printings of my book GTM 21. In Theorem 32.1, not having yet embraced the idea of "root datum", I gave too superficial an account of isomorphism between "simple" algebraic groups. In the revised printing I added the obvious exception, the group of type D$_\ell$ with even rank $\ell\geq 6$. (What is actually proved here is the equivalent Theorem' following Chevalley which involves extending a map. A complicated but concrete method compared to Takeuchi, later Steinberg.) – Jim Humphreys Dec 28 '19 at 20:36
• Oh, right! I missed that we are considering the fundamental group abstractly, not with a specified embedding in the weight-modulo-root group, so that we can't distinguish the three copies of $\operatorname C_2$ inside $\operatorname W(\mathsf D_{2n})$, even though two of them really do give non-isomorphic groups in the non-triality case. Thanks! – LSpice Dec 28 '19 at 20:44

As indicated in the comments, there is no need to redo the entire classification argument when passing from "semisimple" to "reductive". But it's useful to recall some of the history. The emphasis in the Chevalley seminar 1956-58 was on achieving a uniform classification of semisimple algebraic groups over an algebraically closed field of arbitrary characteristic. For this a slightly more general treatment of isogenies between such groups is more natural. (Chevalley discovered for example that the simple groups of types $B_\ell$ and $C_\ell$ are isogenous in characteristic 2 while their groups of rational points are isomorphic even though the underlying algebraic groups are not).

A little later, Demazure and Grothendieck translated most of this into the more flexible language of group schemes in SGA3, while Borel and Tits by 1965 expanded the framework by defining reductive algebraic groups over an arbitrary field. (For such groups Tits achieved a classification, modulo some later reformulation of his theorem stated in the proceedings of the 1965 Boulder AMS Institute. The main unsolved problem is to classify the $k$-anisotropic groups, which vary a lot for different fields of definition $k$.)

The Chevalley seminar and the other sources in French are available online from numdam, but note that SGA3 has been re-edited in recent years while a corrected typeset version of the Chevalley seminar was published in 2005 by Springer. (In my 1975 textbook, I mostly followed the Chevalley seminar; but when the characteristic is not 2 or 3, my 1966 thesis showed that it's also possible to rely more on the Lie algebra as in the classical characteristic 0 case.)

Variants were found by M. Takeuchi and T.A. Springer, using for example ideas of Serre and Steinberg about generators and relations for the groups. But in spite of the differences in the published approaches, all require a lot of detail about the internal structure of simple algebraic groups (those with no proper closed normal subgroups): Bruhat decomposition, generation by tori along with root subgroups. A key conclusion is that two such algebraic groups are isomorphic precisely when they have the same root system (or Dynkin diagram) and the same fundamental group, except in type $D_\ell$ with $\ell >2$ even, when you have to distinguish the half-spin and special orthogonal groups. (Note too that the work of Tits gave a more precise picture of the internal structure: when $G$ is a simple algebraic group, its only proper normal subgroups are those contained in the finite center.)

Using the methods of Chevalley (1955), one further shows the existence of all possible types of simple algebraic groups. The classification of possible semisimple groups is then a routine but slightly messy exercise: start with a product of simply connected simple groups, then factor out a subgroup of the (finite) center. There may be a great many possibilities.

Translating all of this into the language of reductive groups is then a matter of reading between the lines in Springer's textbook: given an isomorphism of root data, one gets an isomorphism of root systems along with a comparison of fundamental groups, etc. Unfortunately, there is no single source in the literature for a truly unified treatment of all these matters, including isogenies. But the core of it all is the study of the Borel/Chevalley structure theory. The transition to reductive groups is needed mainly because these are more natural for induction purposes than the semisimple ones: Levi subgroups of parabolics are reductive but seldom semisimple. However, central tori over an algebraically closed field are fairly innocuous.

• Minor detail: The definition of simple should include "connected" (i.e. no proper normal, closed, connected subgroup). – Tobias Kildetoft Dec 19 '16 at 9:27
• @Tobias: Thanks for adding the essential word "connected" here. The standard example is a special linear group, which usually has a nontrivial finite center (not connected) but is considered "simple" as an algebraic group. This is the definition I used in my 1975 book, though such conventions about the term "simple" always need to be stated explicitly. – Jim Humphreys Dec 19 '16 at 14:47