Cohomology and deformations of Jacobi-Jordan algebras

TL;DR

This paper introduces a new cohomology and deformation theory for Jacobi-Jordan algebras, using zigzag cohomology, and explores their properties, extensions, and applications.

Contribution

It develops a novel zigzag cohomology framework for Jacobi-Jordan algebras and links cohomology groups to algebraic extensions and deformations.

Findings

Defined zigzag cohomology for Jacobi-Jordan algebras

Connected first and second cohomology groups to extensions and deformations

Provided examples and applications of the theory

Abstract

In this paper, we define and develop a cohomology and deformation theories of Jacobi-Jordan algebras. We construct a cohomology based on two operators, called zigzag cohomology, and detail the low degree cohomology spaces. We describe the relationships between first and second cohomology groups with extensions and deformations. Moreover, we consider cohomology properties of pseudo-euclidean Jacobi-Jordan algebras and provide a deformation theory that fits with our zigzag cohomology of Jacobi-Jordan algebras. Furthermore, the paper includes several examples and applications.