相对自由代数中的可加本原长度
Additive primitive length in relatively free algebras
摘要：
相对自由代数$F_d$的元素$f$在各种代数中的加性本原长度等于使$f$可表示为$\ell$本原元之和的最小数$\ell$。给出了特征为0的域上$d$生成的多项式代数$d>1$中元素的加法本原长度的一个上界。边界取决于$d$和元素的阶数。我们证明了，如果域有两个以上的元素，那么在自由$d$生成的幂零阿贝尔李代数的加法本原长度是有界的5为$d=3$和6为$d>3$。如果字段只有两个元素，那么我们的界限是6为$d=3$和7为$d>3$。这推广了Ela Aydın关于二元生成自由亚阿贝尔李代数的一个结果。在所有的情况下，本文中考虑的元素作为原始的总和的介绍可以有效地在多项式时间内找到。
英文摘要：
The additive primitive length of an element $f$ of a relatively free algebra $F_d$ in a variety of algebras is equal to the minimal number $\ell$ such that $f$ can be presented as a sum of $\ell$ 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 bound are 6 for $d=3$ and 7 for $d>3$. This generalizes a recent result of Ela Aydın for two-generated free metabelian Lie algebras. In all cases considered in the paper the presentation of the elements as sums of primitive can be found effectively in polynomial time.
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