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Journal of Lie Theory 26 (2016), No. 3, 659--672
Copyright Heldermann Verlag 2016

Normalisers of Abelian Ideals of a Borel Subalgebra and Z-Gradings 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

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 Z-gradings of g associated with their normalisers.

Keywords: Root system, Borel subalgebra, minuscule element, abelian ideal.

MSC: 17B20, 17B22, 20F55

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