Itô’s theorem and metabelian Leibniz algebras

Abstract

We prove that the celebrated Itô’s theorem for groups remains valid at the level of Leibniz algebras: if is a Leibniz algebra such that , for two abelian subalgebras and , then is metabelian, i.e. . A structure-type theorem for metabelian Leibniz/Lie algebras is proved. All metabelian Leibniz algebras having the derived algebra of dimension are described, classified and their automorphisms groups are explicitly determined as subgroups of a semidirect product of groups associated to any vector space .

Keywords:

• metabelian Leibniz (Lie) algebras

AMS Subject Classifications:

• 17A32
• 17A60
• 17B30

Acknowledgements

The authors are grateful to Otto H. Kegel for his comments on a previous version of the paper.

Notes

A.L. Agore is Postdoctoral Fellow of the Fund for Scientific Research Flanders (Belgium) (F.W.O. Vlaanderen). This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, grant number 88/05.10.2011.