On properties of principal elements of Frobenius Lie algebras

TL;DR

This paper studies principal elements in Frobenius Lie algebras, establishing their properties, embedding structures, and classifications, with implications for solutions to the Classical Yang-Baxter Equation and Lie algebra representations.

Contribution

It demonstrates embeddings of Lie algebras with left symmetric structures into sl(m,K), and characterizes principal elements' semisimplicity under certain conditions, advancing classification of Frobenius Lie algebras.

Findings

Any Lie algebra with a left symmetric algebra structure can be embedded into some sl(m,K).

If a Frobenius Lie algebra has only inner derivations, its principal elements are semisimple over C.

Certain Frobenius Lie algebras have nonsemisimple principal elements and nonrational eigenvalues.

Abstract

We investigate the properties of principal elements of Frobenius Lie algebras, following the work of M. Gerstenhaber and A. Giaquinto. We prove that any Lie algebra with a left symmetric algebra structure can be embedded, in a natural way, as a subalgebra of some sl(m,K), for K= R or C. Hence, the work of Belavin and Drinfeld on solutions of the Classical Yang-Baxter Equation on simple Lie algebras, applied to the particular case of sl(m, K) alone, paves the way to the complete classification of Frobenius and more generally quasi-Frobenius Lie algebras. We prove that, if a Frobenius Lie algebra has the property that every derivation is an inner derivation, then every principal element is semisimple, at least for K=C. As an important case, we prove that in the Lie algebra of the group of affine motions of the Euclidean space of finite dimension, every derivation is inner. We also bring a class of examples of Frobenius Lie algebras, that hence are subalgebras of sl(m, K), but yet have nonsemisimple principal elements as well as some with semisimple principal elements having nonrational eigenvalues, where K=R or C.