Anti-Leibniz algebras: A non-commutative version of mock-Lie algebra

TL;DR

This paper introduces anti-Leibniz algebras as a non-commutative extension of mock-Lie algebras, providing classification and exploring new algebraic structures via averaging operators and embedding tensors.

Contribution

It defines anti-Leibniz algebras, classifies low-dimensional cases, and constructs new algebraic structures using averaging operators and embedding tensors.

Findings

Classification of low-dimensional anti-Leibniz algebras

Construction of anti-associative dialgebras and trialgebras

Introduction of embedding tensors in algebraic structure building

Abstract

Leibniz algebras are non skew-symmetric generalization of Lie algebras. In this paper we introduce the notion of anti-Leibniz algebras as a "non commutative version" of mock-Lie algebras. Low dimensional classification of such algebras is given. Then we investigate the notion of averaging operators and more general embedding tensors to build some new algebraic structures, namely anti-associative dialgebras, anti-associative trialgebras and anti-Leibniz trialgebras.