On a type of commutative algebras

Abstract: We introduce some basic concepts for Jacobi-Jordan algebras such as: representations, crossed products or Frobenius/metabelian/co-flag objects. A new family of solutions for the quantum Yang-Baxter equation is constructed arising from any $3$-step nilpotent Jacobi-Jordan algebra. Crossed products are used to construct the classifying object for the extension problem in its global form. For a given Jacobi-Jordan algebra $A$ and a given vector space $V$ of dimension $\mathfrak{c}$, a global non-abelian cohomological object ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$ is constructed: it classifies, from the view point of the extension problem, all Jacobi-Jordan algebras that have a surjective algebra map on $A$ with kernel of dimension $\mathfrak{c}$. The object ${\mathbb G} {\mathbb H}^{2} \, (A, \, k)$ responsible for the classification of co-flag algebras is computed, all $1 + {\rm dim} (A)$ dimensional Jacobi-Jordan algebras that have an algebra surjective map on $A$ are classified and the automorphism groups of these algebras is determined. Several examples involving special sets of matrices and symmetric bilinear forms as well as equivalence relations between them (generalizing the isometry relation) are provided.