Sur les identités polynomiales vérifiées par les algèbres de rétrocroisement

ABSTRACT

We study the ideal of polynomial identities of a single indeterminate satisfied by all backcrossing algebras. For this we distinguish two categories according to whether or not these algebras satisfy an identity for the plenary powers. For each category, we give the generators for the vector space of identities, a condition for any object belonging to one of these two categories verify a given identity, a necessary and sufficient condition that a polynomial is an identity and we study the existence of an idempotent element. We give a method which brings the search of identities satified by the backcrossing algebras to the solution of linear systems and we illustrate this method by constructing generators of homogeneous and non homogeneous identities of degrees less than 8.

KEYWORDS:

• Altitude of a polynomial
• backcrossing algebras
• Baric algebras
• idempotent element
• mutation algebras
• ω-polynomial identities
• rooted trees
• Specht’s question
• T-ideal

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 17D92
• Secondary: 17A30

Acknowledgment

This paper was completed while Cristián Mallol was visiting the University of Montpellier. This author gratefully acknowledges the Universidad de la Frontera in Temuco, Chile for making this visit possible and the Institut de Mathématiques et de Modélisation de Montpellier for their warm hospitality.

Notes

¹Ici et dans la suite nous avons utilisé le logiciel MAPLE 9 pour la résolution des systèmes linéaires ayant plus de 6 inconnues.