Jacobi–Jordan Conformal Algebras: Basics, Constructions and Related Structures

The main purpose of this paper is to introduce and investigate the notion of Jacobi–Jordan conformal algebras. They are a generalization of Jacobi–Jordan algebras which correspond to the case in which the formal parameter $\lambda$ equals 0. We consider some related structures such as conformal modules, corresponding representations and $\mathcal{O}$-operators. Therefore, conformal derivations from Jacobi–Jordan conformal algebras to their conformal modules are used to describe conformal derivations of Jacobi–Jordan conformal algebras of the semidirect product type. Moreover, we study a class of Jacobi–Jordan conformal algebras called quadratic Jacobi–Jordan conformal algebras, which are characterized by mock-Gel’fand–Dorfman bialgebras. Finally, the $\mathbb{C}[\partial ]$-split extending structures problem for Jacobi–Jordan conformal algebras is studied. Furthermore, we introduce an unified product of a given Jacobi–Jordan conformal algebra J and a given $\mathbb{C}[\partial ]$-module K. This product includes some other interesting products of Jacobi–Jordan conformal algebras such as the twisted product and crossed product. Using this product, a cohomological type object is constructed to provide a theoretical answer to the $\mathbb{C}[\partial ]$-split extending structures problem.