Cohomology and deformations of Jacobi–Jordan algebras

Abstract The purpose of this paper is to introduce a (full) cohomology of Jacobi–Jordan algebras, which form a subclass of Jordan algebras, and to develop extensions and deformation theories of Jacobi–Jordan algebras based on this cohomology. 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.