On some “minimal” Leibniz algebras

Abstract

The aim of this paper is to describe some “minimal” Leibniz algebras, that are the Leibniz algebras whose proper subalgebras are Lie algebras, and the Leibniz algebras whose proper subalgebras are abelian.

On ideals and contraideals in Leibniz algebras

“…In the theory of Leibniz algebras, it begins to be used only now. So, in [8], the Leibniz algebras, whose subalgebras are Lie algebras, and Leibniz algebras, whose subalgebras are Abelian were studied. In paper [9], the Leibniz algebras, whose subalgebras are ideals, have been investigated.…”

On the role played by anticommutativity in Leibniz algebras

“…In this, we convince ourselves by looking at cyclic subgroups in groups and cyclic subalgebras in Leibniz algebras. Cyclic subgroups in an arbitrary group are commutative, while cyclic subalgebras in Leibniz algebras do not generally possess anticommutativity, as can be seen from their description given in [1].…”

On analogs of some group-theoretic concepts and results for Leibniz algebras

“…Thus, a cyclic Leibniz algebra 〈 a 〉, where an element a has infinite depth, is hypocentral and has infinite dimension (see [9]). …”