On locally defined formations of soluble Lie and Leibniz algebras

TL;DR

This paper investigates the local definability of formations in soluble Lie and Leibniz algebras, revealing unique and multiple local definitions depending on the algebra type and field characteristic.

Contribution

It characterizes which formations are locally defined in Lie and Leibniz algebras over different fields, highlighting differences from finite soluble group formations.

Findings

Formations of nilpotent and soluble Lie algebras are the only locally defined formations over characteristic 0.

The formation of all soluble Lie algebras has many local definitions over characteristic 0.

Over non-zero characteristic, saturated formations of soluble Lie algebras have at most one local definition.

Abstract

It is well-known that all saturated formations of finite soluble groups are locally defined and, except for the trivial formation, have many different local definitions. I show that for Lie and Leibniz algebras over a field of characteristic 0, the formations of all nilpotent algebras and of all soluble algebras are the only locally defined formations and that the latter has many local definitions. Over a field of non-zero characteristic, a saturated formation of soluble Lie algebras has at most one local definition but a locally defined saturated formation of soluble Leibniz algebras other than that of nilpotent algebras has more than one local definition.