The index of a Lie algebra, the centraliser of a nilpotent element, and the normaliser of the centraliser

Details The index of a Lie algebra, the centraliser of a nilpotent element, and the normaliser of the centraliser The index of a Lie algebra, the centraliser of a nilpotent element, and the normaliser of the centraliser

2001-07-04

arxiv.org/abs/math/0107031v1

Mathematics

Algebraic Geometry

22 pages, LaTeX, to appear in Math Proc. Camb. Phil. Soc

Scientific paper

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.

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