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WHATIS... ? a Linear Algebraic ? Skip Garibaldi

From a marketing perspective, algebraic groups group over F, a , or a is are poorly named.They are not the groups you met a “” in the of aﬃne varieties as a student in abstract , which I will call over F, smooth , or topological , concrete groups for clarity. Rather, an algebraic respectively. For algebraic groups, this implies group is the analogue in algebra of a topological that the G(K) is a concrete group for each ﬁeld group (fromtopology) or a Liegroup (from K containing F. and ). Algebraic groups provide a unifying language Examples for apparently diﬀerent results in algebra and The basic example is the general . 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 , Steenrod operations in Chow theory, n-by-n over K—to the 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 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 over F, we view them as collections of matrices that is also a group and a Lie group is a smooth over every K of F. 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 —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 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 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 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 , 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