It\^o's theorem and metabelian Leibniz algebras

TL;DR

This paper extends Itô's theorem from group theory to Leibniz algebras, proving that certain decompositions imply the algebra is metabelian, and classifies all such algebras with one-dimensional derived algebra.

Contribution

It introduces a Leibniz algebra analogue of Itô's theorem, provides a structure theorem for metabelian Leibniz algebras, and classifies those with a one-dimensional derived algebra.

Findings

Leibniz version of Itô's theorem proven

Structure theorem for metabelian Leibniz algebras established

Classification and automorphism groups of certain metabelian Leibniz algebras provided

Abstract

We prove that the celebrated It\^{o}'s theorem for groups remains valid at the level of Leibniz algebras: if $\mathfrak{g}$ is a Leibniz algebra such that $\mathfrak{g} = A + B$, for two abelian subalgebras $A$ and $B$, then $\mathfrak{g}$ is metabelian, i.e. $[ \, [\mathfrak{g}, \, \mathfrak{g}], \, [ \mathfrak{g}, \, \mathfrak{g} ] \, ] = 0$. A structure type theorem for metabelian Leibniz/Lie algebras is proved. All metabelian Leibniz algebras having the derived algebra of dimension $1$ are described, classified and their automorphisms groups are explicitly determined as subgroups of a semidirect product of groups $P^* \ltimes \bigl(k^* \times {\rm Aut}_{k} (P) \bigl)$ associated to any vector space $P$.