Primitive elements of free metabelian algebras of rank two

Abstract

Let F(x, y) be a relatively free algebra of rank 2 in some variety of algebras over a field K of characteristic 0. In this paper we consider the problem whether p(x, y) ∈ F(x, y) is a primitive element (i.e. an automorphic image of x): (i) If F(x, y)/(p(x, y)) ≅ F(z), the relatively free algebra of rank 1 (ii) If p(f, g) is primitive for some injective endomorphism (f, 9) of F(x, y) (iii) If p(x, y) is primitive in a relatively free algebra of larger rank. These problems have positive solutions for polynomial algebras in two variables. We give the complete answer for the free metabelian associative and Lie algebras and some partial results for free associative algebras.