On Generalizations of Jacobi–Jordan Algebras

Abstract

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.

Keywords

• Jacobi–Jordan algebras
• Jordan algebras
• noncommutative Jordan nilalgebras
• algebraic classification