On the classification of 2-solvable Frobenius Lie algebras

TL;DR

This paper classifies 2-solvable Frobenius Lie algebras, providing a complete structure description, classifications in low dimensions, and corrections to existing MASA classifications, with implications for endomorphism and Cartan subalgebra characterizations.

Contribution

It offers a comprehensive classification of 2-solvable Frobenius Lie algebras, including new results on their structure, classifications in low dimensions, and corrections to prior MASA classifications.

Findings

Classified 2-solvable Frobenius Lie algebras as semidirect sums of vector spaces and MASAs.

Provided a complete classification for nonderogatory endomorphisms and MANS of class 2.

Corrected and extended the list of MASAs in sl(4, R).

Abstract

We discuss the classification of 2-solvable Frobenius Lie algebras. We prove that every 2-solvable Frobenius Lie algebra splits as a semidirect sum of an n-dimensional vector space V and an n-dimensional maximal Abelian subalgebra (MASA) of the full space of endomorphisms of V. We supply a complete classification of 2-solvable Frobenius Lie algebras corresponding to nonderogatory endomorphisms, as well as those given by maximal Abelian nilpotent subalgebras (MANS) of class 2, hence of Kravchuk signature (n-1,0,1). In low dimensions, we classify all 2-solvable Frobenius Lie algebras in general up to dimension 8. We correct and complete the classification list of MASAs of sl(4, R) by Winternitz and Zassenhaus. As a biproduct, we give a simple proof that every nonderogatory endormorphism of a real vector space admits a Jordan form and also provide a new characterization of Cartan subalgebras of sl(n, R).