ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic group G over k is a variety over k together with maps of varieties m : G G G and i : G G giving G the structure of a group. A morphism of algebraic groups ϕ : G H is both a group homomorphism and a morphism of varieties. Notice that by the group structure we have i 2 = id G so that i is in fact an isomorphism of varieties. The comorphisms m : k[g] k[g] k[g] and i : k[g] k[g] on the coordinate algebra of G give k[g] the structure of a Hopf algebra. Thus we have an equivalence of categories between finitely generated commutative Hopf algebras over k and algebraic groups over k. To first see the interaction of the group and variety structures, consider the following definition. Definition 1.2. For any g G, define λ g : k[g] k[g] by λ g f(x) = f(g 1 x) = f +m g 1 x k[g]/m g 1 x, an automorphism of the k-algbera k[g]. λ g is called left translation of functions by g. I claim that the comorphism of λ g is l g 1 : G G, where l g (x) = gx. To see this, consider λ 1 g (m x ): λ 1 g (m x ) = {f k[g] f(g 1 x) = 0} = {f k[g] f m g 1 x} = m g 1 x so that λ g(x) = g 1 x. From this discussion we see that l g is an isomorphism of varieties, whose inverse is l g 1. It s just as natural to define right translation of functions by g as the map ρ g f(x) = f(xg). The comorphism of ρ g can be seen, as above, to be r g : G G where r g (x) = xg. The map ρ : G GL(k[G]) given by g ρ g is an injective morphism of algebraic groups. Injectivity follows from the fact that f(g) = f(h) for all f k[g] if and only if m g = m h. 2. Subgroups Lemma 2.1. If A G is any subset of G, not necessarily a subgroup, then ga = ga for all g G. Date: Spring 2013. 1
Proof. Since l g : G G is a homeomorphism, any closed set containing ga is of the form ge for some closed set E containing A. Homeomorphisms split across intersections so we have ga = g E = ge = ga A E A E Corollary 2.2. If H G, then H G. Proof. Since i : G G is a homeomorphism, a similar argument to the one in the lemma shows that i(h) = i(h) = H so that H is closed under inverses. Similarly, for any x H we have xh = xh = H. Now take any x H. Corollary 2.3. If H G then H G. Proof. Clearly the result of the lemma also holds in the case of multiplication on the right, so that we have ghg 1 = ghg 1 = H. From abstract group theory, we know that the product of two subgroups is a subgroup when one normalizes the other, and the product is normal when both are normal. What can we say about the product of two closed subgroups? We begin with some preliminary work. Lemma 2.4. Let U G be any dense open set, and V be any nonempty open set. Then UV = G. Proof. Choose any g G. Since i and l g are both homeomorphisms, we have gv 1 is open in G. It follows that gv 1 U, so that g = uv for some u U, v V. Definition 2.5. Let X be any topological space. A subset A X is locally closed if it is an intersection of an open set and a closed set. A subset A X is constructible if it is the union of finitely many locally closed sets. We will need the following two facts about constructible sets which we state without proof. (1) Constructible sets contain a dense open set in their closure (2) The image of a constructible set under a morphism of varieties is again constructible. Constuctible subgroups are closed. To see this, suppose H is a constructible subgroup and let U be a dense open set of H, which is a subgroup by corollary 2.2. Then by lemma 2.4 we have H = UU HH = H One result of this discussion is that if ϕ : G G is a map of algebraic groups, then ϕ(g) is closed in G (G is closed, so constructible, so ϕ(g) is a constructible subgroup, so closed). We can finally state the following result. Proposition 2.6. Suppose A and B are closed subgroups of an algebraic group G, and B normalizes A. Then AB is a closed subgroup of G. Proof. The only new content here is that AB is closed. This follows from AB being the image of the morphism of varieties m : A B G, and the second fact listed above (A B is constructible because it is closed in G G). 2
3. The Identity Component Proposition 3.1. Let G = G i be the decomposition of G into its irreducible components. There is a unique irreducible component containing e G. Proof. Renumbering if necessary, let G 1,..., G n be the irreducible components containing e. Irreducible components are necessarily closed, and the product of closed, irreducible sets is again closed and irreducible, because the tensor product of finitely generated domains over an algebraically closed field k is again a domain (true, but not clear). It follows that G 1... G n is irreducible as it is the image of G 1... G n under the continuous map G... G G. Since e G 1... G n, and the G i are maximal, it follows that G 1... G n G i for some i. It is also true that G j G 1... G n for all j. It follows that n = 1. Denote by G the unique irreducible component of G containing the identity. We call G connected if G = G, i.e., if there is only one irreducible component. 4. Embedding in GL(n, k) The following theorem allows us to always keep in mind matrices when working with any affine algebraic group. Before stating the theorem, we state a finiteness result that is used in the proof of the theorem. Lemma 4.1. k[g] is a union of finite-dimensional G-stable subspaces, where G acts on k[g] via G GL(k[G]) which maps g to ρ g. Proof. Let f k[g]. If is the coproduct of the Hopf algebra k[g], write (f) = f i g i. Then, for any g, x G we have: ρ g f(x) = f(g 1 x) = (f)(g 1, x) = f i (g 1 )g i (x) so that ρ g f = f i (g 1 )g i. It follows that the span of the g i is a finite dimensional vector space which contains all ρ g f for all g G. It then follows that the span of all ρ g f is finite dimensional, and G-stable. Theorem 4.2. Let G be an algebraic group over k. subgroup of some GL(n, k). 5. Quotients Then G is isomorphic to a closed Given an abstract group G and a normal subgroup N, we know that the quotient G/N is also a group. In the case of algebraic groups, we would like to make sense of what G/N means when N is closed. Here is a result towards this end. For any vector space V, call a morphism of algebraic groups ϕ : G GL(V ) a rational representation. Theorem 5.1. Let G be an algebraic group, and N a closed, normal subgroup of G. Then there exists a rational representation ϕ : G GL(V ) such that ker ϕ = N. As groups, we know that G/N = ϕ(g), and ϕ(g) is a closed subgroup in GL(V ), so it is an affine algebraic group. Thus, we think of G/N as the image of the rational representation given in the theorem. It can be shown that the quotient algebraic groups obtained in this way from two different rational representations are isomorphic. 3
6. Jordan Decomposition For any algebraic group G, and any element g G, we seek a canonical decomposition of g into the product of a semisimple and unipotent element. First we work in GL(n, k). An element g GL(n, k) is called semisimple if its minimal polynomial has distinct roots. Since k is algebraically closed, its minimal polynomial factors into distinct linear factors, and thus g is semisimple if and only if g is diagonalizable. We say that g is unipotent if g I n is nilpotent, or equivalently, if the only eigenvalue of g is 1. Theorem 6.1. For any g GL(n, k), there are unique, commuting elements s, u GL(n, k) such that g = su, s is semisimple, and u is unipotent. This unique decomposition is called the Jordan decomposition. Earlier we considered the faithful representation of an algebraic group G via right translation of functions ρ : G GL(k[G]). We would like to see that this representation in some sense preserves the Jordan decomposition. First, we need to make some modifications to the idea of semisimple and unipotent for infinite dimensional vector spaces. For an infinite dimensional vector space V, the strict definition of semisimple and unipotent in GL(V ) is too restrictive for our purposes, so we will need to work with local conditions. If V is an infinite dimensional vector space that can be written as the union of finite dimensional subspaces stable under x GL(V ), then we say x is locally unipotent (semisimple) if its restriction to the each stable subspace is unipotent (semisimple). Now we can state that right translation of functions preserves the Jordan decomposition. Theorem 6.2. Let g GL(n, k) have decomposition given by g = su. Then ρ g GL(k[GL(n, k)]) has Jordan decomposition given by ρ s ρ u. This gives us some idea concerning how to extend our decomposition to the case of arbitrary algebraic groups, which is done by the following theorem. Theorem 6.3. For an arbitrary algebraic group G, and any g G, there exists unique commuting elements s, u G such that g = su, with ρ s semisimple and ρ u unipotent. Also, if ϕ : G G is a map of algebraic groups, then ϕ(s)ϕ(u) is the Jordan decomposition of ϕ(g) in G This allows us to define unipotence and semisimplicity in an arbitrary algebraic group. That is, we say g is unipotent (semisimple) if ρ g is unipotent (semisimple). The theorem just stated shows that unipotence and semisimplicity are preserved under morphisms of algebraic groups. In positive characteristic, we have the following equivalent definition of the unipotent elements of G. Proposition 6.4. Let the characteristic of k be p > 0. Then U(G) = {g G g pt = e, some t N} Proof. We work under some fixed embedding into GL(n, k). If g I is nilpotent, then (g I) pt = g pt I = 0 for large enough t. Conversely, if g pt = I, then (g I) pt = g pt I = 0 is nilpotent. 4
7. The Radical of G The following result must be shown before our definition of the radical subgroup of G. Proposition 7.1. Let G be an algebraic group. Then G contains a unique maximal, normal, solvable subgroup H, which is necessarily closed. Proof. That H must be closed follows from corollary 2.3 and??. To see the result, suppose that H is a maximal, normal, solvable subgroup, and let K be any other such subgroup. Then by proposition??, it follows that HK is normal and solvable. Since H HK, we know by maximality that H = HK so that K H. But K is also maximal, so K = H. Definition 7.2. For any algebraic group G, denote by R(G) the identity component of the unique maximal, normal, solvable subgroup of G. R(G), called the radical of G, is the unique maximal, normal, solvable, connected subgroup of G. A connected algebraic group G e is semisimple if R(G) = e. Since the last nonzero term in the derived series of a solvable group is commutative and normal ([H], Proposition 17.2), it follows that G is semisimple if and only if G is connected and contains no nontrivial commutative, normal, connected subgroups. Example 7.3. Consider SL(n, k), matrices with determinant 1. Inside of SL(n, k) we have the group µ n (k) of nth roots of unity of k which embeds as diagonal matrices, one matrix for each root having that root in each of its diagonal entries. µ n (k) is actually the center of SL(n, k), and is thus normal and solvable. However, it is not connected, as each matrix represents its own irreducible component. It is know that the quotient P SL(n, k) = SL(n, k)/µ n (k) is a simple group References [H] J.E. Humphreys, Linear Algebraic Groups, Springer-Verlag, 1975. 5