Journal of Lie Theory 22 (2012), No. 4, 907--929
Copyright Heldermann Verlag 2012
The Inner Ideals of the Simple Finite Dimensional Lie Algebras
Cristina Draper
Dep. de Matemática Aplicada, Universidad de Málaga, 29071 Málaga, Spain
cdf@uma.es
Antonio Fernández López
Dep. de Matemática Aplicada, Universidad de Málaga, 29071 Málaga, Spain
emalfer@uma.es
Esther García
Dep. de Matemática Aplicada, Universidad Rey Juan Carlos, 28933 Móstoles -- Madrid, Spain
esther.garcia@urjc.es
Miguel A. Goméz Lozano
Dep. Álgebra, Geometría y Topología, Universidad de Málaga, 29071 Málaga, Spain
magomez@agt.cie.uma.es
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 Al, Bl, Cl and Dl 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 last three named authors with E. Neher ["A construction of gradings of Lie algebras", Int. Math. Res. Not. IMRN 16, Art. ID mm051, 34 (2007)]. This reduces the question to deal with root systems.
Keywords: Lie algebra, Jordan pairs, inner ideal, subquotient, grading.
MSC: 17B30
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