Simple Lie algebras of toral rank one

Abstract

Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 7 p > 7 . Let L have Cartan decomposition L = H + ∑ γ ∈ Γ L γ L = H + {\sum _{\gamma \in \Gamma }}{L_\gamma } . If Γ \Gamma generates a cyclic group then L is isomorphic to sl ( 2 , F ) {\text {sl}}(2,F) or to one of the simple Lie algebras of generalized Cartan type W ( 1 : n ) W(1:{\mathbf {n}}) or H ( 2 : n : Φ ) ( 2 ) H{(2:{\mathbf {n}}:\Phi )^{(2)}} .