Chevalley Groups over an arbitrary ring.

Question

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.

Answer 1

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 $G(k)/Z(k)$!)

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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.).

Answer 2

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.

2. 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.