Abstract

There exists two types of nonassociative algebras whose associator satisfies a symmetric relation associated with a 1-dimensional invariant vector space with respect to the natural action of the symmetric group Σ3. The first one corresponds to the Lie-admissible algebras and this class has been studied in a previous paper of Remm and Goze. Here we are interested by the second one corresponding to the third power associative algebras.

Highlights

  • We have classified for binary algebras, Cf. [1], relations of nonassociativity which are invariant with respect to an action of the symmetric group on three elements Σ3 on the associator

  • In particular we have investigated two classes of nonassociative algebras, the first one corresponds to algebras whose associator Aμ satisfies

  • In studies of Remm [1], we have studied the operadic and deformations aspects of the first one: the class of Lie-admissible algebras

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Summary

Introduction

We have classified for binary algebras, Cf. [1], relations of nonassociativity which are invariant with respect to an action of the symmetric group on three elements Σ3 on the associator. In particular we have investigated two classes of nonassociative algebras, the first one corresponds to algebras whose associator Aμ satisfies. Where τij denotes the transposition exchanging i and j, c is the 3-cycle (1,2,3) These relations are in correspondence with the only two irreducible one-dimensional subspaces of [Σ3] with respect to the action of Σ3, where [Σ3] is the group algebra of Σ3. Recall that a third power associative algebra is an algebra ( ,μ) whose associator satisfies Aμ (x, x, x) = 0 Recall that a quadratic operad is Koszul if the free -algebra based on a -vector space V is Koszul, for any vector space V This property is conserved by duality and can be studied using generating functions of and of !

We also have
We deduce that
Cohomology and Deformations
Recall that an alternative algebra is given by the relation
From literature of

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