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Recall that we are assuming f is algebraically closed

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Unformatted text preview: by definition of tβ we have: ,t β (hα ) = κ(tβ , hα ) = 2κ(tβ , tα ) 2(β , α) = ∈Z κ(tα , tα ) (α, α) where we used the definition of the inner product on H ∗ : (α, β ) := κ(tα , tβ ). Proposition 8.5.1. The set of roots Φ spans H ∗ . Proof. Suppose not. Then there is some h ∈ H so that α(h) = 0 for all α ∈ Φ. But then [hx] = α(h)x = 0 for all x ∈ Lα and [hx] = 0 for all x ∈ H since H is abelian. So, h commutes with all the generators of L and is therefore in Z (L). But Z (L) = 0 since L is semisimple. ￿ This means we can choose a basis for H ∗ consisting of roots: α1 , · · · , αn . If β ∈ Φ then ￿ β= ci αi where ci ∈ F . Recall that we are assuming F is algebraically closed with characteristic 0. Thus F contains Q, the rational numbers. Claim 1: ci ∈ Q. 34 MATH 223A NOTES 2011 LIE ALGEBRAS Pf: (β , αj ) = ￿ ci (αi , αj ). Therefore, 2(β , αj ) ￿ 2(αi , αj ) = ci (αj , αj ) (αj , αj ) These fractions are Cartan integers. Also the matrix (αi , αj ) is nonsingular since the form is nondegenerate. Therefore, ci is the ratio of two determinants of integer matrices and therefore a rational number. This implies that the roots lie in the Q-span of α1 , · · · , αn . Let EQ denote this rational vector space. Claim 2: The inner product (·, ·) on EQ has rational values and is positive definite. Pf: Take any λ ∈ H ∗ . Then (λ, λ) = κ(tλ , tλ ) = Tr(ad tλ ad tλ ) =(1) ￿ β ∈Φ β (tλ )2 =(2) ...
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.

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