3-Leibniz bialgebras (3-Lie bialgebras)

TL;DR

This paper extends the concept of bialgebras to 3-Leibniz and 3-Lie algebras using cohomology, providing new theorems and examples to deepen understanding of their algebraic structures.

Contribution

It introduces the notion of 3-Leibniz bialgebras, extends existing theorems, and establishes a correspondence with Leibniz bialgebras, advancing the theory of higher-order algebraic structures.

Findings

Extended bialgebra notions to 3-Leibniz and 3-Lie algebras

Proved new theorems relating 3-Leibniz bialgebras to Leibniz bialgebras

Provided detailed examples illustrating the concepts

Abstract

The aim of this paper is to extend the notion of bialgebra for Leibniz algebras (and Lie algebras) to $3$-Leibniz algebras (and $3$-Lie algebras) by use of the cohomology complex of $3$-Leibniz algebras. Also, some theorems about Leibniz bialgebras are extended and proved in the case of $3$-Leibniz bialgebras ($3$-Lie bialgebras). Moreover, a new theorem on the correspondence between $3$-Leibniz bialgebra and its associated Leibniz bialgebra is proved. Finally, some examples are discussed in detail.