Richard Varro

Les Algèbres de Mutation

The definition of ω M-algebra is introduced and we study a special class : the basic algebras with nilpotent kernels of index 2, named mutation algebras.

Étude des ω-Pi Algèbres Commutatives de Degré 4: III. Algèbres Barycentriques Invariantes par Gamétisation

We continue the study of the algebras satisfying an identity of the shape aX2X2 + bX4 − λX3 − μX2 − νX, we introduce here the case a + b = 1, ab ≠ 0, λ = 2, and μ + ν = −1. By studying a 26 × 27 linear system, we show that this class admits a partition into six parts among which four correspond to algebras not verifying other identity of degree ≤4. In this article we study these four algebras. We show they satisfy principal and plenary train identities of rank > 4, we determine the algebras admitting an idempotent and we give their Peirce decompositions in the two cases: with and without idempotent.

Sur la Classification des Nilalgèbres Commutatives de Nilindice 3

ABSTRACT We give some structure results and recursive-like methods for constructions and classifications of commutative nilalgebras of nilindex 3.

Train Algèbres Alternatives: Relation Entre Nilindice et Degré

We determine train polynomials for power associative algebras and for alternative train algebras. We show bonds between polynomials and nilindices of some factors of the Peirce decomposition.

Sur Les Train Algèbres De Degré Quatre: Structures Et Classifications

We study train algebras of fourth degree, give some genetic examples and explicit the plenary train identities associated. These algebras fall ultimately into four classes; the first two of them do not have 1/2 as train root and thus have idempotents. For these types we provide structure theorems, which are then used for classification purposes in small dimensions.

Critère d’existence d’idempotent basé sur les algèbres de Rétrocroisement

ABSTRACT We study the relationship of backcrossing algebras with mutation algebras and algebras satisfying ω-polynomial identities: we show that in a backcrossing algebra every element of weight 1 generates a mutation algebra and that for any polynomial identity f there is a backcrossing algebra satisfying f. We give a criterion for the existence of idempotent in the case of baric algebras satisfying a nonhomogeneous polynomial identity and containing a backcrossing subalgebra. We give numerous genetic interpretations of the algebraic results.

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.

Identités Multilinéaires de Degré 4 pour les Algèbres de Bernstein et de Mutation Non Commutatives

We give the identities generating the spaces of degree 4 multilinear identities satisfied by the non commutative 0th-order Bernstein algebras and the non commutative mutation algebras. For it, we use a method reducing the search for multilinear identities to the resolution of linear systems.

Dupliquée d'une train algèbre et d'une algèbre de Bernstein périodique

We show that if A2 is a train algebra of rank r, then the duplicate of A is train of rank r + 1. Also, if A2 is a Bernstein algebra of order n and period p, the commutative duplicate of A is Bernstein of order n + 1 and period p.