Abelian subalgebras and ideals of maximal dimension in Zinbiel algebras

Abstract

In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Zinbiel algebras. We study Zinbiel algebras containing maximal abelian subalgebras of codimension 1 and supersolvable Zinbiel algebras in which such subalgebras have codimension 2, and we also analyze the case of filiform Zinbiel algebras. We give examples to clarify some results, including listing the values for α and β for the low dimensional Zinbiel algebras over the complex field that have been classified.

Communicated by Alberto Elduque

Keywords:

• Abelian ideal
• abelian subalgebra
• nilpotent
• solvable
• supersolvable
• Zinbiel algebra

2020 Mathematics Subject Classification:

• 17A32
• 17B05
• 17B20
• 17B30
• 17B50

Acknowledgments

The authors are grateful to Professor Kaygorodov for spotting some typographical errors and for providing reference [3]. They are also grateful to the referee for their careful reading of the paper and helpful comments.

Additional information

Funding

The paper was partially supported by US-1262169, P20_01056, MTM2016-75024-P, and FEDER.