Conjectures for the existence of an idempotent in \(\omega \)-polynomial algebras
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.
scientific article; zbMATH DE number 5906839