Surprising properties of centralisers in classical Lie algebras

TL;DR

This paper investigates the algebraic and geometric properties of centralisers of nilpotent elements in classical Lie algebras, revealing new insights into their centers, commuting varieties, and Poisson structures.

Contribution

It provides new results on the structure of centralisers, their centers, and Poisson geometry in classical Lie algebras, extending understanding of their invariants and commuting pairs.

Findings

Characterization of the center of $g_e$

Description of commuting varieties associated with $g_e$

Analysis of Poisson structures on $g_e^*$

Abstract

Let $g$ be a classical Lie algebra, i.e., either $gl_n$, $sp_n$, or $so_n$ and let $e\in g$ be a nilpotent element. We study various properties of centralisers $g_e$. The first four sections deal with rather elementary questions, like the centre of $g_e$, commuting varieties associated with $g_e$, or centralisers of commuting pairs. The second half of the paper addresses problems related to different Poisson structures on $g_e^*$ and symmetric invariants of $g_e$.