Lie algebras with a finite number of ideals

Abstract

In this paper, we focus on the structure of the variety of Lie algebras with a finite number of ideals and their graph representations using Hasse diagrams. The large number of necessary conditions on the algebraic structure of this type of algebras leads to the explicit description of those algebras in the variety with trivial Frattini subalgebra. To illustrate our results, we have included and discussed lots of examples throughout the paper.

Keywords:

• Lie algebra
• ideal
• Frattini subalgebra
• lattice
• Hasse diagram

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors are partially funded by grant MTM2017-83506-C2-1-P of Ministerio de Economía, Industria y Competitividad, Gobierno de España (Spain). J. Roldán-López is supported by a predoctoral research grant FPI-2018 of ‘Universidad dela Rioja’.