9 - The classification of semisimple algebraic groups

Summary

The aim here is to achieve a classification of semisimple algebraic groups in terms of combinatorial data. It is clear from the previous section that the set of roots plays an essential role in the structure of reductive groups. We now formalize this concept.

Root systems

Let G be a connected reductive group and T ≤ G a maximal torus. Then associated to this we have a finite set of roots Φ ⊂ X ≔ X(T) with the finite Weyl group W acting faithfully on X, preserving Φ (see Proposition 8.4). Recall the group Y = Y(T) of cocharacters of T and the pairing 〈, 〉 : X × Y → ℤ defined in Section 3.2. We identify X and Y with subgroups of E ≔ X ⊗_ℤ ℝ and E^∨ ≔ Y ⊗_ℤ ℝ, respectively, and denote the induced pairing on E × E^∨ also by 〈, 〉. The actions of W on X and on Y may be extended to actions on E and E^∨. Recall the reflections s_α ∈ W introduced in Section 8.4.

We first axiomatize the combinatorial properties satisfied by these data.

Definition 9.1 A subset Φ of a finite-dimensional real vector space E is called an (abstract) root system in E if the following properties are satisfied:

1. (R1) Φ is finite, 0 ∉ Φ, 〈Φ〉 = E;

2. (R2) if c ∈ ℝ is such that α, cα ∈ Φ, then c = ±1;

3. (R3) […]