Additive primitive length in relatively free algebras

摘要

The additive primitive length of an element f of a relatively free algebra F-d(V) in a variety of algebras V is equal to the minimal number l such that f can be presented as a sum of l primitive elements. We give an upper bound for the additive primitive length of the elements in the d-generated polynomial algebra over a field of characteristic 0, d > 1. The bound depends on d and on the degree of the element. We show that if the field has more than two elements, then the additive primitive length in free d-generated nilpotent-by-abelian Lie algebras is bounded by 5 for d = 3 and by 6 for d > 3. If the field has two elements only, then our bounds are 6 for d = 3 and 7 for d > 3. This generalizes a recent result of Ela Aydin for two-generated free metabelian Lie algebras. In all cases considered in the paper, the presentation of the elements as sums of primitive elements can be found effectively in polynomial time.