CLASSIFICATION OF SEMISIMPLE ALGEBRAIC GROUPS J. WARNER 1. Definitions Let k be an algebraically closed field, and let G be an algebraic group defined over k. Lemma 1.1. If A and B are normal, solvable subgroups of G, so is AB. Lemma 1.2. If A and B are closed subgroups of G, and if B normalizes A, then AB is a closed subgroup of G The previous two lemmas show that any algebraic group has a unique largest, closed, normal, solvable subgroup. The identity component of this subgroup is thus the unique largest, closed, normal, solvable, connected subgroup of G. Definition 1.3. Let G be an algebraic group. The radical of G, denoted R(G), is the largest closed, normal, solvable, connected subgroup of G. Example 1.4. Let G = GLn(k). Example 1.5. Let G = SLn(k). Then R(G) = e. Definition 1.6. If G 6= e is connected, and if R(G) = e, then we call G semisimple. Example 1.7. By example 1.5, we have that SLn(k) is a semisimple algebraic group. The connectedness of SLn(k) follows from Corollary 7:5 in [H] ∼ ⊕n Definition 1.8. A torus is any algebraic group isomorphic to Dn(k) = (Gm) , the set of invertible diagonal n × n matrices. An algebraic group is diagonalizable if it is isomorphic to a subgroup of a torus. A maximal torus of an algebraic group G is a subgroup T that is a torus not properly contained in any other subgroup of G that is also a torus. Example 1.9. Dn(k) is a maximal torus in GLn(k). Theorem 1.10. Let G be a connected algebraic group. Then the maximal tori in G are all conjugate. Definition 1.11. For a connected algebraic group G, define the rank of G to be the dimension of a maximal torus inside of G. To any affine algebraic group G we may associate the lie algebra L(G) = g which can be viewed as the space of all left-invariant derivations of G. There is a vector space isomorphism g ! T (G)e, and we can translate the bracket in g to T (G)e via this isomorphism. Given a map of algebraic groups ' : G ! G0, there is an induced map on the tangent spaces, which gives us a map d' : L(G) ! L(G0), called the differential of '. Date: Fall 2013. 1 2 J. WARNER Definition 1.12. For any algebraic group G, a character of G is any morphism of algebraic groups G ! Gm. Pointwise multiplication gives the set of all characters of G, X(G), the structure of an abelian group. Lemma 1.13. If G is diagonalizable, then X(G) is finitely generated. Lemma 1.14. If G is connected, then X(G) is torsion free. The two lemmas tell us that X(T ) is a free abelian group of finite rank for any torus T . ◦ ◦ Lemma 1.15. If H is a diagonalizable subgroup of G, then NG(H) = CG(H) In particular, the lemma implies that the group W (G; S) = NG(S)=CG(S) is ◦ finite for any torus S ⊂ G. This is because the group NG(H)=NG(H) is finite, with order equal to the number of connected components of NG(H). We call the group W (G; S) the Weyl group of G relative to S. If S is maximal, then we have the Weyl group of G. 2. Root Systems Let E be a real vector space. Exercise 2.1. If α 2 E is a nonzero vector, and Ψ is a finite spanning set of E containing α, then there is a unique linear transformation τα which maps Ψ to itself, maps α to −α, and fixes pointwise a subspace of codimension 1. Definition 2.2. An abstract root system of E is a subset Ψ of E satisfying (R1) Ψ is a finite spanning set of E, not containing 0. (R2) If α 2 Ψ, the only multiples of α in Ψ are ±α. (R3) If α 2 Ψ, there exists a reflection τα which leaves Ψ stable. (R4) If α; β 2 Ψ, then τα(β) − β is an integral multiple of α. The elements of Ψ are called roots. The rank of Ψ is the dimension of E. The finite group W (Ψ) ⊂ GL(E) generated by the τα is called the Weyl group of Ψ. There is an inner product (α; β) on E such that the elements of W = W (Ψ) are all orthogonal transformations. With this inner product we have the formula: τα(β) = β − hβ; αiα where 2(β; α) hβ; αi = (α; α) Definition 2.3. A base of Ψ is a subset ∆ = fα1; : : : ; αrg is a basis of E, such that in the unique expression X α = ciαi the ci are all integers with common sign. The elements of a base are called simple roots or fundamental roots. Denote by Ψ+ the collection of all roots in Ψ with pos- itive integral coefficients, called positive roots, and Ψ− those with negative integral coefficients, called negative roots. Exercise 2.4. Show that bases exist, that W permutes the collection of bases freely and transitively, and every root lies in at least one base. CLASSIFICATION OF SEMISIMPLE ALGEBRAIC GROUPS 3 Definition 2.5. A root system Ψ is said to be irreducible if it is not the disjoint union of two mutually orthogonal proper subsets. Exercise 2.6. Every root system is the disjoint union of uniquely determined irreducible root systems in certain subspaces of E. Given an irreducible root system Ψ and a base ∆, we obtain a directed, connected graph, D = D(Ψ; ∆) as follows. The vertices of D are the the simple roots, and the number of edges connecting simple roots α to β is hβ; αihα; βi. Place an arrow on the edges pointing towards the shorter root if such a shorter root exists. Theorem 2.7. The correspondence Ψ ! D(Ψ) is one-to-one between irreducible root systems and connected Dynkin diagrams. 3. Abstract Root System of (G; T ) Fix a maximal torus T for an algebraic group G. T acts on G via conjugation, that is we have a map T ! Aut(G) which sends g to Int g, where Int g(h) = ghg−1. Let Ad g : g ! g be the differential of Int g, so that Ad : T ! Aut g is a map of algebraic groups. The image of T is a diagonalizable subgroup, say H. We can thus write H M a g = g gα α2Φ(G;T ) where mathfrakgα = fx 2 g j h · x = α(h) · x; 8h 2 Hg: Here Φ(G; T ) ⊂ X(H) consists of all α 2 X(H) with gα 6= 0. Definition 3.1. The elements of Φ(G; T ) are called the roots of G relative to T . Theorem 3.2. Let G be semisimple, and T ⊂ G a maximal torus. Consider the real vector space E = R ⊗Z X(T ), a vector space of finite dimension equal to the rank of X(T ) as a free abelian group. We can embed Φ(G; T ) in E via the map α ! (1 ⊗ α). Under this inclusion, Φ(G; T ) is an abstract root system in E, whose Weyl group is isomorphic to the Weyl group of G. The root system of G is not necessarily irreducible. Theorem 3.3. Let G be semisimple, and let fGi j i 2 Ig be the minimal closed connected normal subgroups of positive dimension (as varieties). Then I is finite, say I = f1; : : : ; ng, G = G1 :::Gn, and the root systems Ψ(Gi) correspond to the irreducible root systems of Ψ(G). The Gi are noncommutative, with no nontrivial, proper, closed connected nor- mal subgroups. Such groups are called simple, and have irreducible roots systems. It follows that the simple algebraic groups are classified by connected Dynkin dia- grams. References [H] J.E. Humphreys, Linear Algebraic Groups, Springer-Verlag, 1975..