The generalised nilradical of a Lie algebra

TL;DR

This paper explores generalizations of the nilradical in Lie algebras, showing that for any Lie algebra, a certain quotient relates to derivations of an ideal linked to the socle of a semisimple algebra.

Contribution

It introduces new generalizations of the nilradical that preserve key properties and establishes a representation of Lie algebras via derivations of these generalized structures.

Findings

For every Lie algebra L, L/Z(N) embeds into Der(N*?)

Defines a class of ideals N*? related to the socle of semisimple Lie algebras

Extends properties of the classical nilradical to broader contexts

Abstract

A solvable Lie algebra L has the property that its nilradical N contains its own centraliser. This is interesting because gives a representation of L as a subalgebra of the derivation algebra of its nilradical with kernel equal to the centre of N. Here we consider several possible generalisations of the nilradical for which this property holds in any Lie algebra. Our main result states that for every Lie algebra L, L/Z(N), where Z(N) is the centre of the nilradical of L, is isomorphic to a subalgebra of Der(N?*) where N*? is an ideal of L such that N?*/N is the socle of a semisimple Lie algebra.