The centralisers of nilpotent elements in classical Lie algebras

2004/07/05 by Oksana Yakimova, O. S. Yakimova, Yakimova, O. S. · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT) #math.RT

paper · pdf · doi:10.48550/arxiv.math/0407065

Replaced with english translation

openalex publication_date 2004/07/05 · arxiv created 2004/09/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The index of a finite-dimensional Lie algebra g is the minimum of dimensions of stabilisers g_α of elements α∈ g^*. Let g be a reductive Lie algebra and z(x) a centraliser of a nilpotent element x∈ g. Elashvili has conjectured that the index of the centraliser z(x) equals the index of g, i.e., the rank of g. Here Elashvili's conjecture is proved for reductive Lie algebras of classical type. It is shown that in cases g=gl_n and g=sp_2n the coadjoint action of z(x) has a generic stabiliser. Also, we give an example of a nilpotent element x∈ so₈ such that the coadjoint action of z(x) has no generic stabiliser.

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