Conjectures for the existence of an idempotent in \(\omega \)-polynomial algebras
type: Item
instance of: scholarly article
title: Conjectures for the existence of an idempotent in \(\omega \)-polynomial algebras
zbMATH Open document ID: 1252.17019
author: Michelle Nourigat
author: Richard Varro
published in: Discrete and Continuous Dynamical Systems. Series S
publication date: 2011-06-10T00:00:00Z
review text: A baric algebra is a commutative (but not necessarily associative) algebra \(A\) over a field \(F\) of characteristic not \(2\), endowed with a nonzero algebra homomorphism (the weight) \(\omega: A\rightarrow F\). Weight \(1\) idempotents play an important role in these algebras, as they allow a sort of Peirce decomposition, and because they represent situations of genetic equilibrium in the applications of these algebras to Genetics. The paper under review states two conjectures on the existence of weight \(1\) idempotents in baric algebras satisfying certain \(\omega\)-polynomial identities in one variable. These conjectures relate the existence of such idempotents to the multiplicity of \(1/2\) as a root of a certain (usual) polynomial attached to the \(\omega\)-identity. The conjectures are shown to hold for several classes of baric algebras: principal train algebras, \(\omega\)-PI algebras of degree \(4\) and plenary train algebras.
reviewed by: Alberto Elduque
Mathematics Subject Classification ID: 17D92
Mathematics Subject Classification ID: 17A30
zbMATH DE Number: 5906839
zbMATH Keywords: Baric algebra
zbMATH Keywords: idempotent
zbMATH Keywords: \(\omega\)-polynomial identity
zbMATH Keywords: train algebra
zbMATH Keywords: Bernstein algebra
zbMATH Keywords: mutation algebra
MaRDI profile type: Publication
full work available at URL: dcdss.2011.4.1543
OpenAlex ID: W2318042004
DOI: 10.3934/DCDSS.2011.4.1543
description: scientific article; zbMATH DE number 5906839