Algebraic Groups Curtis Heberle MTH 189 Final Paper 12/14/2010 The primary objects of study in algebraic geometry are varieties. Having become acquainted with these objects, it is interesting to consider when a set has the structure of both a variety and some other mathematical object. We consider in particular the case where a set has the structure of both a variety and an abstract group. To that end, we define an algebraic group to be a variety X together with regular maps m: X X X and i: X X satisfying the typical rules for multiplication and inverse in a group. Morphisms of algebraic groups are also defined in a very natural way -- as maps which are both regular maps between varieties (i.e., morphisms of varieties) and homomorphisms of groups. As usual, we define an isomorphism to be a morphism for which an inverse morphism exists. Algebraic groups have certain nice properties. For example, algebraic groups have nonsingular varieties. We omit a formal proof, but the idea is as follows: left-multiplying by a fixed element g of the group gives an automorphism of the group. Since the multiplication map is regular, the left-multiplication also gives a regular map between the variety and itself, so that left-multiplication gives an algebraic group automorphism. Under such a mapping the geometric properties of the points are preserved; that is, nonsingular points map to nonsingular points. However, by choosing g appropriately, we can map any point to any other point; thus either all points are singular or no points are. This fact explains the requirement of nonsingularity in the familiar example we now describe. We have seen, in section 2.8 of Reid's Undergraduate Algebraic Geometry, that given an irreducible, nonsingular cubic C in the projective plane, we can define multiplication and inversion on the points of C as follows: fix a point O C. Then for A, B C, let A 1 be the
3rd point of intersection of C with the line OA, and let A B = R 1, where R is the 3rd point of intersection of C with the line AB. Reid provides proof that these operations do indeed impose a group structure on C, and that the maps given by multiplication and inverse are regular; thus we can realize such curves as algebraic groups. In particular, they are simple examples of abelian varieties, which we now define. An abelian variety is an algebraic group the underlying variety of which is complete. We say a variety X is complete when, given any variety Y, the projection X Y Y maps closed sets to closed sets. While there exist non-projective complete varieties, examples of such are beyond our present scope. For our purposes we note without proof that projective varieties are complete, while affine varieties are not. Abelian varieties have certain nice properties; in particular, the group multiplication on an abelian variety is (as the name might be taken to suggest) always commutative. We give no further examples of abelian varieties. Moving on from the projective case, we next consider algebraic groups where the underlying variety is affine. We will refer to such structures as affine algebraic groups. As our first examples of an affine algebraic group, we consider the general linear group, GL n (K), i.e. the set of non-singular n n matrices with entries from the field K. To show that this is an algebraic group we must establish two things: that it is a variety, and that its multiplication and inverse (in this case, ordinary matrix multiplication/inversion) define regular maps. To see that GL n (K) is a variety, we consider the criteria for a membership. An n n matrix A is nonsingular (and thus an element of GL n (K)) if det(a) 0. However, det(a) is just a polynomial in the n 2 entries of A, so this condition naturally corresponds to a standard open subset of A n 2. We recall from Reid that standard open subsets in affine space are in fact varieties. In particular, we see that GL n (K) is the variety in A n 2 +1 given by V(det A Y 1). That the multiplication in GL n (K) defines a regular map is easy to see: it is just ordinary matrix multiplication, and when two matrices are multiplied the coefficients of the product can be written as polynomial functions (that is, regular functions) of the
coefficients of the factors. It is less obvious that the inverse map is regular, but recalling from linear algebra the formula A 1 = adj A det A, where adj(a) is the adjugate of A, a matrix whose coefficients depend on determinants of submatrices of A (and thus depend polynomially on the coefficients of A). Thus the coefficients of A -1 are given by polynomial functions of the coefficients of A, divided by the determinant of A, which we know to be nonzero. The classical subgroups of GL n K can easily be seen to be subvarieties as well. For example, the special linear group of matrices with determinant 1 is a subvariety, as its members must be in GL n K as satisfying det A 1 = 0, which is simply a polynomial condition on the entries in A. As another example, the orthogonal group (matrices A for which A 1 = A t ) can be characterized as the matrices whose column vectors v 1, v n are orthonormal, i.e. for which v i v j = 0 v i v i 1 = 0. for i j, These are again simply polynomial conditions on the elements of A, and so define the orthogonal group as a subvariety (as well as a subgroup) of GL n (K). It is no coincidence that the examples we have given of affine algebraic groups are all subgroups of the general linear group. Algebraic groups which are subgroups of the general linear group we call linear algebraic groups, and we state the following theorem: every affine algebraic group is a linear algebraic group. A detailed proof of this theorem is given in Meinolf Geck's An Introduction to Algebraic Geometry and Algebraic Groups; we conclude our discussion with a partial overview of the proof. Given an affine algebraic group G, we wish to show that G is isomorphic to a subgroup of the general linear group. Since GL n (K) can thought of as the automorphism group of an n-dimensional K-vector space, we can restate our goal as embedding G in the automorphism group of some finite-dimensional vector space. To begin, we show that there exists a homomorphism (of groups) between G and the automorphism group of its own coordinate ring, k[g]. Given x G, define the translation
ρ x : G G, y yx. This is an automorphism of G, since left-multiplying a group by one of its elements returns the whole group. Thus the induced map ρ x : k G k[g] is an automorphism of k G. We note that ρ x ρ y = ρ xy, so the map ρ: G Aut k G, x ρ x is a (group) homomorphism between G and Aut k G. The result above is similar to what we want, but we have the problem that k[g] need not be finite-dimensional. We want to find a finite-dimensional vector space the automorphism group of which contains a subgroup isomorphic to G. We construct the space as follows: let F be a finite subset of k[g]. Then define the subspace E span ρ x f f F, x G}. By a series of calculations involving the multiplication map on G along with the ρ x maps, it can be shown that E is in fact the vector space we're looking for. That is, it is finite dimensional, and G is isomorphic to a Zariski-closed subgroup of its automorphism group. Again, we omit the details, but we note that the isomorphism is given by the map ρ E : G Aut E, y ρ y. where here we view ρ y as acting only on E, not on the entire coordinate ring. To summarize, we can show that G is isomorphic (via the coordinate ring map induced by simple translations) to a subgroup the automorphism group of a finite-dimensional subspace of its coordinate ring, and thus to a subgroup of the general linear group. References Geck, Meinolf. An introduction to algebraic geometry and algebraic groups. Oxford University Press, USA, 2003. Harris, Joe. Algebraic geometry: a first course. Springer, 1992. Mumford, David. Abelian Varieties. Oxford University Press, 1970. Reid, Miles. Undergraduate algebraic geometry. Cambridge University Press, 1988.