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If i < j, then it is easy to check that sei −ej (a1 , . . , an+1 ) = (a1 , . . , aj , . . , ai , . . , an+1 ) (ai and aj swapped) For all α, β ∈ An , one has sα (β) − β = − β, α α, and β, α ∈ {0, ±1, ±2}, so An is indeed a root system. It is clearly reduced. ” Thus we can identify W(An ) with the set of permutation matrices in GL(V ). Note that the standard inner product ·, · on V is W -invariant. The Dynkin diagram of An is • • ··· • • (n vertices). 2 (type A1 × A1 ). This root system lives inside V = R2 and R = {±(α, 0), ±(0, β)}.

The Dynkin diagram is • • ··· • • • (n vertices). 7 (type Dn , n 4). The ambient vector space is V = Rn , and the set of roots is {ei } ∪ {±ei ± ej : i < j}. The Dynkin diagram is • (n vertices). 8 (type E6 ). The space V is the subspace of R8 consisting of vectors x such that x6 = x7 = −x8 . The roots are {±ei ± ej : i < j 5}, together with all ± where 5 1 2 (−1)ν(i) ei e8 − e7 − e6 + , i=1 ν(i) is even. 9 (type E7 ). Here V is the hyperplane in R8 orthogonal to e7 + e8 . The set of roots is {±ei ± ej : i < j 6} ∪ {±(e7 − e8 )}, together with all ± where 1 2 6 (−1)ν(i) ei e7 − e8 + , i=1 ν(i) is odd.

If u, v : X → G are two morphisms of schemes over k, then TranspG (u, v) is represented by a closed subscheme of G. Proof. 5]. 2. Let k be a field, G/k a group scheme of finite type, and H ⊂ G a closed subgroup scheme. Then CG (H) and NG (H) are represented by closed subgroup schemes of G Clearly CG (H) ⊂ NG (H). 3 Borel subgroups For this section, k is an algebraically closed field of characteristic zero. 1. Let G/k be a linear algebraic group. A Borel subgroup of G is a connected solvable subgroup B ⊂ G that is maximal with respect to those properties.

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