Jacobi-Jordan conformal algebras: Basics, Constructions and related structures

The main purpose of this paper is to introduce and investigate the notion ofJacobi-Jordan conformal algebra. They are a generalization of Jacobi-Jordanalgebras which correspond to the case in which the formal parameter lambdaequals 0. We consider some related structures such as conformal modules,corresponding representations and O-operators. Therefore, conformal derivationsfrom Jacobi-Jordan conformal algebras to their conformal modules are used todescribe conformal derivations of Jacobi-Jordan conformal algebras ofsemidirect product type. Moreover, we study a class of Jacobi-Jordan conformalalgebras called quadratic Jacobi-Jordan conformal algebras, which arecharacterized by mock-Gel'fand Dorfman bialgebras. Finally, the C[delta]-splitextending structures problem for Jacobi-Jordan conformal algebras is studied.Furthermore, we introduce an unified product of a given Jacobi-Jordan conformalalgebra $J$ and a given C[delta]-module K. This product includes some otherinteresting products of Jacobi-Jordan conformal algebras such as twistedproduct or crossed product. Using this product, a cohomological type object isconstructed to provide a theoretical answer to the C[delta]-split extendingstructures problem.