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Abelian ideals of Borel subalgebras and affine Weyl groups. (English) Zbl 1112.17011

From the introduction: In this paper we investigate in detail certain remarkable posets \({\mathcal I}_{\text{ab}}(\tau,\alpha)\) of abelian ideals of a Borel subalgebra of a simple Lie algebra. These posets depend upon a long root \(\alpha\) and an element \(\tau\) of the coroot lattice (subject to certain natural restrictions), and form a partition of all abelian ideals of a Borel subalgebra. As a consequence of our analysis, we may deduce uniform formulas for the dimensions of maximal and minimal ideals in \({\mathcal I}_{\text{ab}}(\tau,\alpha)\). Our work generalizes and refines work of Panyushev and Suter on similar topics, using as a key tool the combinatorics of affine Weyl groups and root systems. Our formulas also provide a uniform explanation of classical results of Mal’tsev on the maximal dimension of an abelian subalgebra of a simple Lie algebra.
We also obtain results on ad-nilpotent ideals which complete the analysis started in [P. Cellini and P. Papi, J. Algebra 225, No. 1, 130–141 (2000; Zbl 0951.17003) and ibid. 258, No. 1, 112–121 (2002; Zbl 1033.17008)].

MSC:

17B20 Simple, semisimple, reductive (super)algebras
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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