Ideals in Parabolic Subalgebras of Simple Lie Algebras

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Chari, Vyjayanthi
Dolbin, R. J.
Ridenour, T.
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Abstract
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We study ad-nilpotent ideals of a parabolic subalgebra of a simple Lie algebra. Any such ideal determines an antichain in a set of positive roots of the simple Lie algebra. We give a necessary and sufficient condition for an antichain to determine an ad-nilpotent ideal of the parabolic. We write down all such antichains for the classical simple Lie algebras and in particular recover the results of D. Peterson. In section 2 of the paper we study the unique ideal in a parabolic which is irreducible as a module for the reductive part and give several equivalent statements that are satisfied by the corresponding subset of roots.
Comment: 14 pages
Keywords
Mathematics - Representation Theory, 17B20
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