Tame automorphisms of a free dual Leibniz algebra of rank two

Abstract

The article gives a definition of -variety of algebras and proves that the variety of dual Leibniz algebras is a -variety. In [Alimbaev A.A., Naurazbekova A.S., Umirbaev U. Linearization of automorphisms and triangulation of derivations of a free algebras of rank 2 // Siberian Electronic Mathematical Reports. – 2019. – Vol. 16. – P. 1133-1146.], the results concerning the linearization of automorphisms and the triangulation of derivations of free algebras of rank two -varieties are obtained. As a consequence of the results of this work, we have shown the following: the group of tame automorphisms of a free dual Leibniz algebra in two variables admits the structure of an amalgamated free product, any reductive group of tame automorphisms of a free dual Leibniz algebra in two variables over a field of characteristic zero is linearizable, and any locally nilpotent derivation of this algebra is triangulable.