Cartan subalgebras of simple Lie algebras

Abstract

Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 7 p > 7 . Let H be a Cartan subalgebra of L, let L = H + Σ γ ∈ Γ L γ L = H + {\Sigma _{\gamma \in \Gamma }}{L_\gamma } be the Cartan decomposition of L with respect to H, and let H ¯ \bar H be the restricted subalgebra of Der L generated by ad H. Let T denote the maximal torus of H ¯ \bar H and I denote the nil radical of H ¯ \bar H . Then H ¯ = T + I \bar H = T + I . Consequently, each γ ∈ Γ \gamma \in \Gamma is a linear function on H.