Journal of Lie Theory 33 (2023), No. 3, 799--830
Copyright Heldermann Verlag 2023
On the Classification of 2-Solvable Frobenius Lie Algebras
André Diatta
Aix-Marseille Université, Institut Fresnel, Marseille, France
andre.diatta@fresnel.fr
Bakary Manga
Dép. de Mathématiques et Informatique, Université Cheikh Anta, Diop de Dakar, Sénégal
bakary.manga@ucad.edu.sn
Ameth Mbaye
Dép. de Mathématiques et Informatique, Université Cheikh Anta, Diop de Dakar, Sénégal
ameth3.mbaye@ucad.edu.sn
[Abstract-pdf]
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 $\mathfrak{sl}(4,\mathbb{R})$ 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 $\mathfrak{sl}(n,\mathbb{R})$.
Keywords: Frobenius Lie algebra, 2-step solvable exact symplectic Lie algebra, symplectic Lie group, maximal Abelian subalgebra, nonderogatory endomorphism, cyclic matrix, companion matrix, Kravchuk signature, Cartan subalgebra, Jordan form.
MSC: 17B05, 17B08, 15A27, 53A15, 53D15, 22E60, 17B60, 70G45, 16W25, 13B25.
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