The inner ideals of the classical Lie algebras, related gradings and Jordan pairs

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 a classical Lie 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 An, Bn, Cn and Dn 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 G2, F4, E6, E7 and E8) 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 three last-named authors with E. Neher. This reduces the question to deal with root systems.

(This paper has now appeared under the title "The inner ideals of the simple finite dimensional Lie algebras" in J. Lie Theory 22 (2012), No. 4, 907--929)


C. Draper < cdf@uma.es >

A. Fernández López < emalfer@agt.cie.uma.es >

E. García < esther.garcia@urjc.es >

M. Gómez Lozano < magomez@agt.cie.uma.es >