Structure Theory of Semisimple Lie Algebras II
نویسندگان
چکیده
3. The restriction K|gα×g−α , i.e. K(a, b) with a ∈ gα and b ∈ g−α, is non-degenerate, so it induces a pairing between gα and g−α. In particular, we have dim gα = dim g−α. 4. The restriction K|h×h is non-degenerate, hence we have an isomorphism ν : h → h∗ given by ν(h)(h′) = K(h, h′) for all h, h′ ∈ h. The map ν induces a bilinear form on h∗ by K(α, β) = β ( ν−1(α) ) = α ( ν−1(β) ) for all α, β ∈ h∗. We proved that K(α, α) 6= 0 if α ∈ ∆. 5. For all α ∈ ∆, e ∈ gα and f ∈ g−α, we have: [e, f ] = K(e, f)ν−1(α).
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