Complements of Intervals and Prefrattini Subalgebras of Solvable Lie Algebras

TL;DR

This paper explores the structure of solvable Lie algebras by examining complements of intervals and prefrattini subalgebras, extending concepts from group theory to Lie algebra context.

Contribution

It introduces a Lie-theoretic analogue of prefrattini subgroups and studies their conjugacy properties within solvable Lie algebras.

Findings

Conjugacy of certain subalgebras is established for a broad class of solvable Lie algebras.

A new framework for understanding complements in subalgebra intervals is developed.

The work generalizes group-theoretic concepts to Lie algebra theory.

Abstract

In this paper we study a Lie-theoretic analogue of a generalisation of the prefrattini subgroups introduced by W. Gasch\"utz. The approach follows that of P. Hauck and H. Kurtzweil for groups, by first considering complements in subalgebra intervals. Conjugacy of these subalgebras is established for a large class of solvable lie algebras.