On a pre-Jacobi-Jordan algebra: relevant properties and double construction

TL;DR

This paper introduces pre-Jacobi-Jordan algebras, explores their properties, and constructs new algebraic structures through double constructions, including classification in low dimensions.

Contribution

It defines pre-Jacobi-Jordan algebras, studies their properties, and provides a classification in dimension two along with new double construction methods.

Findings

Defined pre-Jacobi-Jordan algebras and studied bimodules and matched pairs.

Constructed a double algebra from a pre-Jacobi-Jordan algebra and its dual.

Classified pre-Jacobi-Jordan algebras in dimension two.

Abstract

We introduce a pre-Jacobi-Jordan algebras and study some relevant properties such as bimodules, matched pairs. Besides, we established a pre-Jacobi-Jordan algebra built as a direct sum of a given pre-Jacobi-Jordan algebra $(\A, \cdot)$ and its dual $(\A^{\ast}, \circ),$ endowed with a non-degenerate symmetric bilinear form $B,$ where $\cdot$ and $\circ$ are the products defined on $\A$ and $\A^{\ast},$ respectively. Finally, after pre-Jacobi-Jordan algebras classification in dimension two, we thoroughly give some double constructions of pre-Jacobi-Jordan algebraic structures.