From Trigroups To Leibniz 3-Algebras

TL;DR

This paper explores the mathematical structures of trigroups, their connection to 3-racks, and how smooth trigroups lead to Leibniz 3-algebras, expanding the understanding of higher algebraic systems.

Contribution

It introduces the category of trigroups, analyzes their relationship with 3-racks, and shows how smooth trigroups generate Leibniz 3-algebras.

Findings

3-racks can be constructed by conjugating trigroups

Trigroups with smooth structures produce Leibniz 3-algebras

Establishes connections between trigroups, 3-racks, and Leibniz 3-algebras

Abstract

In this paper, we study the category of trigroups as a generalization of the notion of digroup [4] and analyze their relationship with 3-racks [1] and Leibniz 3-algebras [6]. Trigroups are essentially associative trioids in which there are bar-units and bar-inverses. We prove that 3-racks can be constructed by conjugating trigroups. We also prove that trigroups equipped with a smooth manifold structure produce Leibniz 3-algebras via their associated Lie 3-racks.