Jacobi-Jordan conformal algebras: Basics, Constructions and related structures
Taoufik Chtioui, Sami Mabrouk, Abdenacer Makhlouf
TL;DR
This work defines Jacobi-Jordan conformal algebras as conformal analogues of Jacobi-Jordan algebras and develops their module theory, $\mathcal{O}$-operators, and symplectic structures, linking them to left anti-symmetric conformal algebras. It then characterizes quadratic Jacobi-Jordan conformal algebras via mock-Gel'fand Dorfman bialgebras and outlines how such structures emerge from anti-Novikov algebras. A central theme is the $\mathbb{C}[\partial]$-split extending structures problem, for which the authors introduce unified products and a cohomological-type classification (via $\text{CExtd}$ and $\text{CExtd}'$), providing a general framework to classify all extensions of a Jacobi-Jordan conformal algebra by a module. The framework specializes to twisted and crossed products, offering concrete constructions to realize and analyze extensions where the base algebra sits inside larger JJ conformal algebras.
Abstract
The main purpose of this paper is to introduce and investigate the notion of Jacobi-Jordan conformal algebra. 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 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 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 C[delta]-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 C[delta]-module K. This product includes some other interesting products of Jacobi-Jordan conformal algebras such as twisted product or crossed product. Using this product, a cohomological type object is constructed to provide a theoretical answer to the C[delta]-split extending structures problem.