On the derived algebra of a centraliser

TL;DR

This paper characterizes when the centraliser of a nilpotent element in a classical Lie algebra is equal to its derived algebra, linking this property to the rigidity of the nilpotent element.

Contribution

It establishes a precise criterion connecting the equality of the centraliser and its derived algebra to the rigidity of the nilpotent element in classical Lie algebras.

Findings

$ abla_e = [ abla_e, abla_e] ext{ iff } e ext{ is rigid}$

If $e$ is in $[ abla_e, abla_e]$, then the nilpotent radical of $ abla_e$ equals $[ abla(1)_e, abla_e]$

Provides a characterization of the structure of centralisers of nilpotent elements in classical Lie algebras.

Abstract

Let $\g$ be a classical Lie algebra, $e$ a nilpotent of $\g$ element and $\gt g_e$ the centraliser of $e$ in $\g$. We prove that $\g_e=[\g_e,\g_e]$ if and only if $e$ is rigid. It is also shown that if $e$ is contained in $[\g_e,\g_e]$, then the nilpotent radical of $\g_e$ coincides with $[\g(1)_e,\g_e]$, where $\g(1)_e$ is an eigenspace of a characteristic of $e$ with the eigenvalue 1.