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 $\mathfrak{g}$ 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 $P^{\ast }⋉(k^{\ast }\times {\mathrm{Aut}}_k(P))$ associated to any vector space $P$.

Final version; to appear in Linear Multilinear Algebra

References in corpus (1)

• Derived length and nildecomposable Lie algebras

Cited by in corpus (1)

• Metabelian associative algebras