A class of nonassociative algebras including flexible and alternative algebras, operads and deformations

TL;DR

This paper explores a specific class of nonassociative algebras characterized by symmetric associator relations, focusing on third power associative algebras and their connections to operads and deformations.

Contribution

It introduces a new class of nonassociative algebras with symmetric associator relations, extending the understanding of flexible, alternative, and third power associative algebras.

Findings

Identified two types of nonassociative algebras with symmetric associator relations.

Connected third power associative algebras to operads and deformation theory.

Extended previous work on Lie-admissible algebras to a new algebraic class.

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 on three elements. The first one corresponds to the Lie-admissible algebras that we studied in a previous paper. Here we are interested by the second one corresponding to the third power associative algebras.