Some radicals, Frattini and Cartan subalgebras of Leibniz n-algebras

Abstract

In the present work, we introduce notions such as -solvability, - and -nilpotency and the corresponding radicals. We prove that these radicals are invariant under derivations of Leibniz -algebras. The Frattini and Cartan subalgebras of Leibniz -algebras are studied. In particular, we construct examples that show a classical result on conjugacy of Cartan subalgebras of Lie algebras, which also holds in Leibniz algebras and Lie -algebras, is not true for Leibniz -algebras.

Keywords:

• Lie n-algebras
• Leibniz n-algebras
• Frattini subalgebra
• Cartan subalgebra
• Jacobson radical

Classification:

• 17A32
• 17A42
• 17A60

Acknowledgments

The second author was supported by MICINN Grant MTM 2009-14464-C02 (European FEDER support included) and by Xunta de Galicia Grant Incite 09207215 PR. The third named author was partially supported by the Grants (RGA) No:11-018 RG/Math/AS_I–UNESCO FR: 3240262715 and NATO-Reintegration ref. CBP.EAP.RIG. 983169.