On Generalizations of Jacobi-Jordan Algebras

In this paper, we present some generalizations of Jacobi-Jordan algebras. More concretely, we will focus on noncommutative Jacobi-Jordan algebras, Malcev-Jordan algebras, and general Jacobi-Jordan algebras. We adapt a method, used to classify Poisson algebras, in order to classify all general Jacobi-Jordan algebras up to dimension 4, and, in particular, all noncommutative Jacobi-Jordan algebras up to dimension 4. We present the classification of Malcev-Jordan algebras up to dimension 5. As the class of Jacobi-Jordan algebras (commutative algebras that satisfy the Jacobi identity), we find that Malcev-Jordan algebras are Jordan algebras but not necessarily nilpotent. However, we show that the classification of nilpotent Malcev-Jordan algebras is sufficient to obtain the classification of the whole class.