On a type of commutative algebras

TL;DR

This paper introduces new concepts and solutions related to Jacobi-Jordan algebras, including classifications, extensions, and connections to the quantum Yang-Baxter equation, with concrete examples and automorphism group determinations.

Contribution

It develops a cohomological framework for classifying Jacobi-Jordan algebras and constructs new solutions to the quantum Yang-Baxter equation from nilpotent cases.

Findings

Constructed a new family of solutions for the quantum Yang-Baxter equation.

Classified Jacobi-Jordan algebras with specific surjective maps.

Determined automorphism groups of certain Jacobi-Jordan 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.