Jacobi–Jordan-admissible algebras and pre-Jacobi–Jordan algebras

The main purpose of this paper is to study Jacobi–Jordan-admissible algebras and some particular classes like pre-Jacobi–Jordan algebras. First, we provide characterizations of Jacobi–Jordan algebras by means of oxidation process. Then, we discuss Jacobi–Jordan-admissible algebras for which we obtain a surprising characterization as 3-power anti-associative algebras. Moreover, we introduce a concept of [Formula: see text]-anti-associative algebra, generalizing anti-associative algebras and prove that they are equivalent to Jacobi–Jordan-admissible algebras. Furthermore, we study the structure of pre-Jordan–Jacobi algebras which form a particular class of Jacobi–Jordan-admissible algebras. Moreover, an inductive description by a process of oxidation is provided for an interesting particular class of pre-Jacobi–Jordan algebras, called strong pre-Jacobi–Jordan algebras.