Nonassociative algebras of anti-biderivation-type

TL;DR

This paper introduces and characterizes a new class of nonassociative algebras called D-algebras, explores their relationships with known algebraic structures, and classifies complex 3-dimensional instances.

Contribution

It defines D-algebras, studies their properties, relationships with other algebraic structures, and provides a classification of 3-dimensional complex D-algebras.

Findings

D-algebras include Jacobi-Jordan, symmetric anti-Leibniz, and anti--LR algebras.

Anti--LR algebras under commutator produce -sf4 algebras.

Every Jacobi-Jordan algebra admits a non-trivial post-Jacobi-Jordan structure.

Abstract

The main purpose of this paper is to study the class of Jacobi-Jordan-admissible algebras, such that its product is an anti-biderivation of the related Jacobi-Jordan algebra. We called it as $\mathcal A{\rm BD}$-algebras. First, we provide characterizations of algebras in this class. Furthermore, we show that this class of nonassociative algebras includes Jacobi-Jordan algebras, symmetric anti-Leibniz algebras, and anti-${\rm LR}$-algebras. In particular, we proved that anti-${\rm LR}$-algebras under the commutator product give $\mathfrak{s}_4$-algebras, which were recently introduced by Filippov and Dzhumadildaev. In addition, we then study $\mathcal A$flexible ${\mathcal A}{\rm BD}$-algebras. Then, we introduce the post-Jacobi-Jordan structures on Jacobi-Jordan algebras and establish results that each Jacobi-Jordan algebra admits a non-trivial post-Jacobi-Jordan structure. At the end of the paper, we give the algebraic classification of complex $3$-dimensional $\mathcal A{\rm BD}$-algebras.