# Chevalley Groups over an arbitrary ring.

My question is simply about the Chevalley groups over rings. In many books, including Carter's book on "Simple groups of Lie types", the groups are considered over fields. I have checked the computations and I noticed that the computations works over any commutative $\mathbb{Z}$-algebra. Why these groups are not introduced in the general format? Of course if these groups are defined over rings then they are not necessarily simple. Is it the reason why most people want to define them over fields?

Just to give a general definition of Chevalley groups over rings let me review the construction.

Let $L$ be a finite dimension complex simple Lie algebra with a Chevalley basis $$\{ e_r: r\in\Pi; e_r: r\in\Phi \},$$ where $\Pi$ is basis for the root system $\Phi$. Pick $\zeta\in\mathbb{C}$. Therefore $$\exp(\zeta ad_{e_r}),$$ is an Lie-algebra automorphism of $L$. One can observe that the entries of the matrix associated to $\exp(\zeta ad_{e_r})$, denoted by $A_r(\zeta)$, with respect to a Chevalley basis, are of the form $a\zeta^l$ where $a\in\mathbb{Z}$ and $l\in\mathbb{Z}_{\geq 0}$. Now let $B$ be a commutative $Z$ algebra with the structure map $\rho: \mathbb{Z}\to B$. Pick $b\in B$, then we consider the matrix $\overline{A_r(b)}$ which is obtained by transforming the entry of $a\zeta^l$ of the matrix $A_r(\zeta)$ into $\rho(a)b^l$. Then One can consider the linear transformation, denoted by $\overline{x_r(b)}$, obtained by $\overline{A_r(b)}$ on $L(B):=L(\mathbb{Z})\otimes_{\mathbb{Z}}B$. One can show that $\overline{x_r(b)}$ is indeed a Lie-algebra automorphism of $L(B)$. Then the Chevalley group $G_{ad}(B,\Phi)$, is defined by the subgroup of $GL(L(B))$ generated by $\overline{x_r(b)}$.

Is there a problem in this construction that I am not considering?

Many thanks for your answers.

• Volumes II and III of SGA3 are all about this story (giving an intrinsic definition of "Chevalley groups" over $\mathbf{Z}$ was Grothendieck's motivation for Demazure in his thesis which entailed much of SGA3). The reason "most people" define them only over fields is because the community of specialists in the representation and structure theory of algebraic (and arithmetic) groups and the community of experts in the level of scheme theory which pervades SGA3 has small overlap. Your construction is wrong even for ${\rm{PGL}}_2$ over $\mathbf{Q}$ (if meant at the level of $B$-valued points). Nov 14, 2013 at 0:35
• Marguax: I don't know much about the Scheme theoretical part of algebraic groups. You mentioned about the $PGL_2$ over $Q$. I don't see how to construct this by simple Lie algebra. Indeed the simple Lie algebra with the root system $A_1$ gives $PSL_2$. So I don't understand the mistake in my construction. Would you please explain more?
– M.B
Nov 14, 2013 at 0:51
• Dear M.B.: Sorry, I wasn't aware of your background, so my answer below will likely be useless to you. But in my answer I do address the distinction among ${\rm{SL}}_n(k)$, ${\rm{PGL}}_n(k)$, and ${\rm{SL}}_n(k)/\mu_n(k)$, and put your suggested construction into a broader framework. Nov 14, 2013 at 0:53
• M.B.: To construct the non-perfect group ${\rm{PGL}}_2(\mathbf{Q})$ as an abstract group, you have to manually fatten the diagonal by tossing in more generators on the diagonal. Or, at the cost of explicitness, you could take the algebraic $\mathbf{Q}$-group generated (in the sense of algebraic groups!) by the root groups in the adjoint representation of the Lie algebra; that is ${\rm{PGL}}_2$ as a group variety. Then form its group of rational points. Nov 14, 2013 at 4:59

## 2 Answers

The clean definition of adjoint Chevalley groups can be given in the spirit of what you are trying to do, but there is a hidden subtlety because it is only the torus in the simply connected case that is literally generated by the coroot groups (i.e., the simple positive coroots are a basis of its cocharacter group) whereas in the adjoint case it is the character group having a basis related to those simple positive coroots.

Since the root groups "come from" the simply connected central cover via isomorphisms, all you can hope to get using the root groups is at best the image of the points from the simply connected central cover in the adjoint form. And even that would only tend to work over fields, where we have Bruhat decomposition on rational points; already over a dvr there isn't quite a Bruhat decomposition, so it becomes rather subtle to try to get by with the group generated by points of root groups.

For example, in the case of ${\rm{PGL}}_n$ over a field $k$, what you get from the root groups is the image of ${\rm{SL}}_n(k)$ in ${\rm{PGL}}_n(k)$ (recall that ${\rm{SL}}_2(k)$ is generated by the standard unipotent points, so applied with the ${\rm{SL}}_2$ associated to each simple positive root this is "enough" to capture the $k$-points of the diagonal torus in ${\rm{SL}}_n$). Of course, the map ${\rm{SL}}_n(k) \rightarrow {\rm{PGL}}_n(k)$ has normal image with cokernel $k^{\times}/(k^{\times})^n$ that is typically huge (e.g., $k = \mathbf{Q}$).

It is true that with the correct definition of $G^{\rm{ad}}(\Phi)$, the natural action on its Lie algebra $L$ over $\mathbf{Z}$ (the latter having a definition as you indicate, also in Carter's book) does make this a closed subgroup scheme of the automorphism scheme of $L$. So it does occur inside ${\rm{GL}}(L)$ "defined by some equations". But this is not a good way to understand its structure and properties over rings.

But let's come back to Carter's book. The point is that if $G$ is a split simply connected semisimple group over a finite field $k$ with $G^{\rm{ad}} = G/Z$ its adjoint central quotient (with $Z$ the schematic center of $G$), the isogeny $G \rightarrow G^{\rm{ad}}$ induces an injection of groups $G(k)/Z(k) \rightarrow G^{\rm{ad}}(k) \subset {\rm{GL}}(L_k)$. So voila, the group $G(k)/Z(k)$ is found inside ${\rm{GL}}(L_k)$, and as such it is literally generated by the $k$-points of the root groups. And away from some very low-rank cases over fields of size 2 or 3, this group is simple. But it is neither $G(k)$ nor $G^{\rm{ad}}(k)$ in general! That is, $G(k)/Z(k)$ is not "an algebraic group" in a reasonable sense in general.

So if those "simple finite groups of Lie type" $G(k)/Z(k)$ may be called "Chevalley groups", that is a misnomer in the sense that "Chevalley group" should be either an algebro-geometric structure or at least its group of points over some ring of interest, and $G(k)/Z(k)$ is typically neither of those. For example, if $G = {\rm{SL}}_n$ then $G(k)/Z(k)$ is the commutator subgroup of ${\rm{PGL}}_n(k)$ (sometimes called ${\rm{PSL}}_n(k)$, which is dangerous notation) but it is not an "algebraic group" in any reasonable sense in general. (Nonetheless, the theory of algebraic groups is very powerful to tell us things about $G(k)/Z(k)$!)

EDIT: In view of the OP's comments, probably I should explain why I say that the commonly seen notation "${\rm{PSL}}_n(k)$" for the group ${\rm{SL}}_n(k)/\mu_n(k)$ is quite dangerous. The notation suggests that if $K/k$ is a Galois extension of fields then the injection of groups ${\rm{SL}}_n(k)/\mu_n(k) \rightarrow {\rm{SL}}_n(K)/\mu_n(K)$ has image that is the ${\rm{Gal}}(K/k)$-invariants of the target (as is the case for the injection $X(k) \rightarrow X(K)$ of rational points of a variety $X$ over $k$). But this is totally false: there is an obstruction coming from the part of the Brauer group of $k$ killed by $K$.

So truly the construction $k \rightsquigarrow {\rm{SL}}_n(k)/\mu_n(k)$ is not at all like forming rational points of a variety (over a ground field that may be increased tomorrow).

In fact, the same phenomenon is even seen for ${\rm{PGL}}_n$ over rings more general than fields (or rather, more general than local rings and PID's). To be precise, if we consider the construction $R \rightsquigarrow {\rm{GL}}_n(R)/R^{\times}$ on rings (not just fields) then it is equally bad as above; e.g., one can make examples of number fields $K$ with nontrivial 2-torsion in the class group and finite Galois extensions $K'/K$ such that ${\rm{GL}}_2(O_K)/O_K^{\times}$ is strictly smaller than the subgroup of ${\rm{Gal}}(K'/K)$-invariants in ${\rm{GL}}_2(O_{K'})/O_{K'}^{\times}$. This may feel "weird" if you haven't seen it before, but it is entirely intuitive from the scheme-theoretic perspective. All of the same issues also arise in the more general Chevalley group situation (for $G(k)/Z(k)$ as above).

The only truly satisfactory way I have ever seen to come to grips with this kind of behavior for quotient constructions (over rings, and even over fields in tricky situations) in a systematic manner that "always works" (with good technique) is via group schemes. There are ways to grapple with it over fields without using schemes, but it tends to cause arguments to deviate a bit from group-theoretic intuition. And this includes even the case of Chevalley groups, over rings or fields. But it could just be a matter of taste; plenty of people have done very deep work in these matters without mastering the fancy algebraic geometry (e.g., Borel, Bruhat-Tits, et al.).

• Dear Marguax. I am so thankful for answering to my question But unfortunately since I don't know much about the theory of algebraic groups I can not understand many points that you have made in your answer. It would be very good if you would please give me a reference with more details for these.
– M.B
Nov 14, 2013 at 3:05
• Moreover I have seen the following paper by Borel on "Properties and linear representations of Chevalley groups". In this paper he also has defined the Chevalley groups over field with the way that I presented in my question. So I just want to know if there is any mistake if one define it over rings. Perhaps what you are trying to mention is that this construction is not general enough. Do I see it correctly?
– M.B
Nov 14, 2013 at 3:06
• @M.B.: There is a typo in 3.3(1): $G_{\pi,k}$ at the start should be $G_{\pi,K}$, with $K$ an algebraically closed field containing a given field $k$. In 3.3(5) he says (as I did) that for the "Chevalley group" $G_{\pi}$ as a $\mathbf{Z}$-group, the "group generated by $x_a(t)$'s" is the image in $G_{\pi}(k)$ of the $k$-points of the simply connected central cover of $G_{\pi}$. In section 4 he abandons pure group theory to make $G_{\pi}$ as a $\mathbf{Z}$-group scheme. Upshot: his abstract group $G_{\pi,k}$ is not $G_{\pi}(k)$. He calls $G_{\pi,k}$ a "Chevalley group", but this is dangerous! Nov 14, 2013 at 4:24
• In the end, the main issue is probably want you want to do with these things. There may be entrenched incompatible traditions in the group theory and group scheme communities for the meaning of "Chevalley group", and the arbiter of one's preference is what one wants to do with it. The algebraic geometer's viewpoint is very robust for varying rings and fields, and recovers the other viewpoint in a clean way. I am skeptical that "group generated by root groups" is a useful notion except over fields (where Bruhat decomposition is available), but I am open to hearing to the contrary. Nov 14, 2013 at 4:36
• The 1996 paper of Vavilov and Plotkin mentioned in my comment to Jim's answer addresses the role of "group generated by root groups" beyond the setting of fields. (It seems to be of more interest than I had expected, but nonetheless they also discuss the fact that with general rings these are smaller than the groups of ring-valued points of the Chevalley group schemes, and it is the latter groups of points which they call "Chevalley groups".) Nov 14, 2013 at 15:17

A couple of further clarifications, to supplement the extensive answer by marguax and the many comments:

1. Chevalley's influential 1955 paper was mainly concerned with finding a uniform approach to most of the known simple finite groups of Lie type (supplemented soon afterward by the introduction of twisted groups as well as the groups of Suszuki and Ree). The original "Chevalley group" was defined as a group of automorphisms of a certain Lie algebra, generated by unipotent elements; this group is simple over almost all fields (except a few very small ones). The Lie algebra here is obtained by first reducing mod $$p$$ a Chevalley $$\mathbb{Z}$$-basis of a simple Lie algebra over $$\mathbb{C}$$, then tensoring with a field of prime characteristic $$p$$.

A subtle point here is that the "Chevalley Lie algebra" obtained over an algebraically closed field is actually the Lie algebra of the corresponding simply connected algebraic group, whereas the Chevalley groups themselves are closer to the adjoint groups.

Anyway, Steinberg in his 1967-68 Yale lectures broadened the notion of "Chevalley group" by using other faithful Lie algebra representations, to include special linear groups and the like.

1. In fact, there is a huge amount of literature about Chevalley groups over (commutative) rings. Here the ideal structure of the ring contributes to normal subgroup structure in the group, so the groups are typically non-simple. But they do come up naturally in algebraic K-theory, including the study of the congruence subgroup problem. N.A. Vavilov and many others have written extensively about such groups. There is less textbook literature along these lines, except for a few books on algebraic K-theory including one by Hahn and O'Meara. And the entire subject becomes quite technical.

ADDED: Concerning the more sophisticated viewpoint of Chevalley-Demazure group schemes, there is an important recent paper by Lusztig (not yet freely available online) Study of a $$\mathbf{Z}$$-form of the coordinate ring of a reductive group. The arXiv preprint is here.

• Steinberg's Yale notes build only simply connected forms, yes? (Section 4 of Borel's article which the OP looked at goes further.) For your #2, the "Chevalley groups" in Vavilov et al. are the ring-valued points of Chevalley-Deamzure group schemes, whereas what the OP and Borel call a "Chevalley group" is called there an "elementary subgroup". The link between them and "rare" cases of equality (and role of Bruhat decomposition in that) are addressed in the first 2 pages of the 1996 Acta Applicandae paper of Vavilov and Plotkin; also see the last paragraph of section 1. Nov 14, 2013 at 15:11
• Dear Jim: Many thanks for the answer. You have mentioned that the Chevalley group, constructed by the Chevalley basis, has been considered for commutative rings. So I would like to know if the construction is the same as I presented in my question. This is the reason I asked the question.
– M.B
Nov 14, 2013 at 16:02
• @marguax: No, Steinberg works with an arbitrary lattice between the root lattice and the weight lattice. His "Chevalley groups" are generated by elementary (unipotent) matrices, but in his lectures and related papers he deals with related covering groups as well. The notes are online at www.math.ucla.edu/~rst/ Your further comments are helpful. Nov 14, 2013 at 17:27
• @M.B.: The definition of "Chevalley group" varies in different sources, so it's always important to pin it down. In his many papers, Vavilov works with the points of a Chevalley-Demazure group scheme but also with the usually smaller "elementary Chevalley group"; the difference is an important part of studying the groups over rings. See his long survey: Structure of Chevalley groups over commutative rings.* Nonassociative algebras and related topics (Hiroshima, 1990), 219–335, World Sci. Publ., River Edge, NJ, 1991. Your construction is much narrower, similar to Chevalley's. Nov 14, 2013 at 17:35