2001/07/04 by Dmitri I. Panyushev, Panyushev, Dmitri I. · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT
paper · pdf · doi:10.48550/arxiv.math/0107031
22 pages, LaTeX, to appear in Math Proc. Camb. Phil. Soc
arxiv created 2001/07/04 · openalex publication_date 2001/07/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the index for several natural classes of non-reductive subalgebras of semisimple Lie algebras. Namely, we look at parabolic subalgebras, centralisers of nilpotent elements, and the normalisers of the centralisers. We discuss a conjecture of Elashvili to the effect that the index of any centraliser is equal to the rank of the semisimple algebra in question. It is shown that Elashvili's conjecture is true for `small' and `large' orbits. Some properties of the index for the normaliser of the centraliser are proved. In particular, we prove that, for a regular nilpotent element, the normaliser of the centraliser is a Frobenius Lie algebra.