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Additive primitive length in relatively free algebras
- Publication Year :
- 2018
- Publisher :
- arXiv, 2018.
-
Abstract
- 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 Ayd��n 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.<br />LATEX, 9 pages
- Subjects :
- Pure mathematics
Computer Science::Information Retrieval
General Mathematics
010102 general mathematics
Astrophysics::Instrumentation and Methods for Astrophysics
Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing)
Mathematics - Rings and Algebras
Automorphism
Mathematics - Commutative Algebra
Commutative Algebra (math.AC)
01 natural sciences
TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES
Rings and Algebras (math.RA)
Free algebra
FOS: Mathematics
Computer Science::General Literature
0101 mathematics
Element (category theory)
13F20, 13P05, 14E07, 14R10, 17B30, 17B40
ComputingMilieux_MISCELLANEOUS
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....c48e5aebfec3a386d74e75d68fbb6ff0
- Full Text :
- https://doi.org/10.48550/arxiv.1812.04585