Remarks on the abelian ideals of a Borel subalgebra

2013/08/21 by Chao-Ping Dong, Dong, Chao-Ping
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1308.4464

openalex publication_date 2013/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \frb be a fixed Borel subalgebra of a finite-dimensional complex simple Lie algebra \frg. The Shi bijection associates to every ad-nilpotent ideal \fri of \frb a region V_\fri. In this paper, we show that \fri is abelian if and only if V_\fri∩ 2A is empty, if and only if the volume of V_\fri∩ 2A equals to that of A, where A is the fundamental alcove of the affine Weyl group. For certain flag of abelian ideals, we record an ascending property of their associated regions. We also determine the maximal eigenvalue m_r-1 of the Casimir operator on \wedge^r-1 \frg and the corresponding eigenspace M_r-1, where r is the number of positive roots.

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• Abelian ideals in a Borel subalgebra of a complex simple Lie algebra
• Abelian ideals and amazing roots
• Abelian ideals with given dimension in Borel subalgebras
• Abelian ideals of a Borel subalgebra and root systems
• Abelian ideals of a Borel subalgebra and long positive roots