On Properties of Principal Elements of Frobenius Lie Algebras
| Autor: | Diatta, A., Bakary Manga |
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| Přispěvatelé: | Diatta, Andre |
| Předmět: |
Mathematics - Differential Geometry
17B05 17B08 22E60 seaweed Lie algebra FOS: Physical sciences Frobenius Lie algebra Mathematical Physics (math-ph) [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph] Commutative Algebra (math.AC) Mathematics - Commutative Algebra [MATH.MATH-AC] Mathematics [math]/Commutative Algebra [math.AC] [MATH.MATH-SG] Mathematics [math]/Symplectic Geometry [math.SG] Differential Geometry (math.DG) Mathematics - Symplectic Geometry affine Lie algebra Mathematics::Quantum Algebra FOS: Mathematics symplectic Lie algebra Symplectic Geometry (math.SG) affi ne motion [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph] [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG] Mathematical Physics |
| Zdroj: | Web of Science BASE-Bielefeld Academic Search Engine Scopus-Elsevier |
| Popis: | 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. Latex, 16 pages. The last version appeared at Journal of Lie Theory. Keywords and phrases: Frobenius Lie algebra, affine Lie algebra, Left symmetric Lie algebra, affine motion, symplectic Lie algebra, seaweed Lie algebra, symplectic Lie group, invariant symplectic structure, invariant affine structure |
| Databáze: | OpenAIRE |
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