From Trigroups To Leibniz 3-Algebras

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.