The Inner Ideals of the Simple Finite Dimensional Lie Algebras

The inner ideals of the simple finite dimensional Lie algebras over an algebraically closed field of characteristic 0 are classified up to conjugation by automorphisms of the Lie algebra, and up to Jordan isomorphisms of their corresponding subquotients (any proper inner ideal of such an algebra is abelian and therefore it has a subquotient which is a simple Jordan pair).While the description of the inner ideals of the Lie algebras of types A l , B l , C l and D l can be obtained from the Lie inner ideal structure of the simple Artinian rings and simple Artinian rings with involution, the description of the inner ideals of the exceptional Lie algebras (types G 2 , F 4 , E 6 , E 7 and E 8 ) remained open.The method we use here to classify inner ideals is based on the relationship between abelian inner ideals and Z -gradings, obtained in a recent paper of the last three named authors with E. Neher.This reduces the question to deal with root systems.