WHATIS... ? a Linear Algebraic Group? Skip Garibaldi
From a marketing perspective, algebraic groups group over F, a Lie group, or a topological group is are poorly named.They are not the groups you met a “group object” in the category of aﬃne varieties as a student in abstract algebra, which I will call over F, smooth manifolds, or topological spaces, concrete groups for clarity. Rather, an algebraic respectively. For algebraic groups, this implies group is the analogue in algebra of a topological that the set G(K) is a concrete group for each ﬁeld group (fromtopology) or a Liegroup (from analysis K containing F. and geometry). Algebraic groups provide a unifying language Examples for apparently diﬀerent results in algebra and The basic example is the general linear group number theory. This uniﬁcation can not only sim- GLn for which GLn(K) is the concrete group of plify proofs, it can also suggest generalizations invertible n-by-n matrices with entries in K. It is the and bring new tools to bear, such as Galois co- collection of solutions (t, X)—with t ∈ K and X an homology, Steenrod operations in Chow theory, n-by-n matrix over K—to the polynomial equation etc. t ·det X = 1. Similar reasoning shows that familiar matrix groups such as SLn, orthogonal groups, Deﬁnitions and symplectic groups can be viewed as linear A linear algebraic group over a ﬁeld F is a smooth algebraic groups. The main diﬀerence here is that aﬃne variety over F that is also a group, much instead of viewing them as collections of matrices like a topological group is a topological space over F, we view them as collections of matrices that is also a group and a Lie group is a smooth over every extension K of F. manifold that is also a group. (For nonexperts: Roughly, the theory of linear algebraic groups it is useful to think of an aﬃne variety G as a generalizes that of linear Lie groups over the real natural assignment—i.e., a functor—that takes any or complex numbers to give something that makes ﬁeld extension K of F and gives the set G(K) of sense over an arbitrary ﬁeld. The category of linear common solutions over K of some ﬁxed family of algebraic groups over R contains a full subcategory polynomials with coeﬃcients in F.) equivalent to the compact Lie groups; see [3, Properly speaking, for each of the three types §5]. And the parameterization of the irreducible of groups in the previous paragraph, one needs to ﬁnite-dimensional representations of a complex require that the group operations are morphisms reductive Lie group in terms of dominant weights in the appropriate category, so, e.g., for a topo- holds more generally for so-called “split reductive” logical group G, the multiplication G × G → G and groups over any ﬁeld. inversion G → G are required to be continuous. Algebraic groups allow one to deal system- In more categorical language, a linear algebraic atically with familiar matrix groups and their generalizations in a way that works over arbitrary Skip Garibaldi is Winship Professor of Mathematics at ﬁelds, whether they are the rationals for number Emory University. His email address is [email protected] theory, ﬁnite ﬁelds for ﬁnite group theory, or the ams.org. real or complex numbers for geometry. This is not The author thanks B. Conrad, G. McNinch, R. Parimala, just a language. There is enough theory available G. Prasad, and J.-P. Serre for their valuable suggestions. that one can often avoid computing with actual
October 2010 Notices of the AMS 1125 matrices, or at least with matrices larger than Group Theory 2-by-2! In the list of ﬁnite simple (concrete) groups, most are of Lie type.That is, take a linear algebraicgroup Local-Global Principles G that is absolutely simple, simply connected, and Quadratic forms and division algebras over a deﬁned over a ﬁnite ﬁeld F that is not very small. global ﬁeld1 F are determined by their proper- Then the concrete group G(F) modulo its center is ﬁnite and simple. This is helpful because one ties at the completions Fv of F; this is roughly the content of the Hasse-Minkowski theorem and can use the general framework of algebraic groups the Albert-Brauer-Hasse-Noether theorem, respec- to prove theorems about these ﬁnite groups. One tively. These theorems can be viewed as saying example of this is Deligne-Lusztig theory, which that the natural map is the most eﬀective approach to the complex representations of the ﬁnite groups of Lie type. 1 1 (∗) H (F, G) → Y H (Fv ,G) The construction of simple concrete groups in places v of F the previous paragraph works for many algebraic groups G and many ﬁelds F, not just for ﬁnite in Galois cohomology is injective, where G is an ﬁelds. But for precisely which G does it work? orthogonal group or PGL . This reformulation of n The Kneser-Tits problem (1964) asks: Let G be a the problem in terms of algebraic groups has two linear algebraic group that is simply connected, is advantages. First, we see that the two theorems absolutely simple, and contains GL1. Is G(F) mod- are two faces of the same phenomenon, and ulo its center simple? Much like the Hasse principle second, we get a natural general question: For discussed above, Kneser-Tits is a generalization ∗ which linear algebraic groups G is ( ) injective for in terms of algebraic groups of earlier problems— every global ﬁeld F? The Hasse principle (proved such as the Tannaka-Artin problem—regarding by Kneser,Harder,and Chernousov) saysthat (∗) is more classical algebraic structures. injective if G is connected and absolutely simple. (A The answer to Kneser-Tits seems to depend linear algebraic group is absolutely simple if, when on the arithmetic complexity of the ﬁeld F. The viewed as an algebraic group over an algebraic answer can be “no” for ﬁelds of dimension at least closure of F, it is not commutative and does 4 (Platonov, 1975). not contain any proper, connected, and normal In contrast, the answer is “yes” for global ﬁelds, algebraic subgroups besides the identity.) This which “are 2-dimensional”. This was an open broadly generalizes the two results mentioned at question for some time, until Philippe Gille settled the start of the paragraph. the last remaining case in 2007 by discovering If we weaken our request that (∗) be injective, the following interesting criterion that works over we arrive at a substantial result due to Borel and every ﬁeld F: for a group G as in Kneser-Tits, the Serre (early 1960s) that (∗) has ﬁnite kernel for concrete group G(F) modulo its center is simple (a every linear algebraic group G in the case in which purely algebraic criterion) if and only if the variety F is a number ﬁeld. This result has recently been G is, roughly speaking, “path connected” (a purely extended to global ﬁelds of prime characteristic geometric criterion). See  for details. by Brian Conrad. Because these latter ﬁelds are We still don’t know the answer to Kneser-Tits not perfect, this extension relies on the classiﬁca- for other ﬁelds of dimension 2 and don’t even tion of “pseudo-reductive” linear algebraic groups have strong indications of what the answer should recently completed by Conrad, Ofer Gabber, and be for dimension 3. Gopal Prasad. In closing, I urge you: Please do not be misled Another way to generalize the local-global ques- by the short list of topics in this article. There are many other areas of mathematics in which tion is to weakenthe hypotheses on F, for example, algebraic groups play an essential role, such as to assume that F has cohomological dimension at the Langlands program, geometric invariant theory most 2, which holds for totally imaginary number and Schubert varieties in algebraic geometry, Tits’s ﬁelds and C(x, y). For such ﬁelds and G simply theory of buildings... Algebraic groups deserve connected, Serre’s “Conjecture II” (1962) asserts more attention. that H1(F, G) is zero. This is known to hold if F is the function ﬁeld of a complex surface (de Jong, References He, and Starr, 2008) or G is a classical group (Bayer  P. Gille, Le problème de Kneser-Tits, Astérisque 326 and Parimala, 1995). There are many other results (2009), 39–82. on this conjecture and generalizations such as the  V. P. Platonov and A. Rapinchuk, Algebraic Groups Hasse principle conjecture II; Google can provide and Number Theory, Academic Press, Boston, 1994. details.  J.-P. Serre, Gèbres, L’Enseignement Math. 39 (1993), 33–85. (Reprinted in Oeuvres, vol. IV, pp. 272–324.)
1A global ﬁeld is a ﬁnite extension of Q or k(t) for k a ﬁnite ﬁeld.
1126 Notices of the AMS Volume 57, Number 9