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- W66975369 abstract "Abstract . In this note we prove that for any two restricted rootsα, β of a real semisimple Lie algebra g , such that α + β =0, the corre-sponding root subspaces satisfy [g α , g β ]=g α + β . Key words: real semisimple Lie algebra, root systems, reducedroots, root subspaces AMS subject classifications: 17B20Received February 26, 2004 Accepted March 31, 2004Let g be a real semisimple Lie algebra, a a Cartan subspace of g and R the(restricted) root system of the pair (g , a) in the dual space a ∗ of a . For α ∈ R denote by g α the corresponding root subspace of g :g α = {x ∈ g;[ h,x ]= α ( h ) x ∀h ∈ a }. The aim of this note is to prove the following theorem:Theorem. Let α,β ∈ R be such that α + β =0. Then either [ x, g α ]=g α + β ∀x ∈ g β { 0 } or [ x, g β ]=g α + β ∀x ∈ g α { 0 }. Although the proof is very simple and elementary, the assertion does not seemto appear anywhere in the literature. The argument for the proof is from [2], whereit is used to prove [g α , g α ]=g 2 α (a fact which is also proved in [3], 8.10.12), as wellas that the nilpotent constituent in an Iwasawa decomposition is generated by theroot subspaces corresponding to simple roots.Let" @default.
- W66975369 created "2016-06-24" @default.
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- W66975369 date "2004-06-26" @default.
- W66975369 modified "2023-09-24" @default.
- W66975369 title "A note on the root subspaces of real semisimple Lie algebras" @default.
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