Simple Lie Algebras of Toral Rank One

Abstract

Abstract. Let L be a finite-dimensional simple lie algebra over an algebraically closed field of characteristic p > 7. Let L have Carian decomposition L -H + '2yeTL1-If T generates a cyclic group then L is isomorphic to sl(2, F) or to one of the simple Lie algebras of generalized Cartan type W<\ : n) or H(l : n : «)(2>.The object of this paper is to classify the simple Lie algebras with "small" Cartan subalgebras. If H is a Cartan subalgebra of L there are two natural ways to measure its size: the dimension of H or the dimension of a maximal torus in a restricted subalgebra generated by H.