
Journal of Lie Theory 26 (2016), No. 3, 659672 Copyright Heldermann Verlag 2016 Normalisers of Abelian Ideals of a Borel Subalgebra and ZGradings of a Simple Lie Algebra Dmitri I. Panyushev Institute for Information Transmission Problems of the R.A.S., Bolshoi Karetnyi per. 19, 127051 Moscow, Russia panyushev@iitp.ru Let g be a simple Lie algebra and Ab the poset of all abelian ideals of a fixed Borel subalgebra of g. If a is an element of Ab, then the normaliser of a is a standard parabolic subalgebra of g. We give an explicit description of the normaliser for a class of abelian ideals that includes all maximal abelian ideals. We also elaborate on a relationship between abelian ideals and Zgradings of g associated with their normalisers. Keywords: Root system, Borel subalgebra, minuscule element, abelian ideal. MSC: 17B20, 17B22, 20F55 [ Fulltextpdf (390 KB)] for subscribers only. 