Hopf algebras for ternary algebras

TL;DR

This paper develops a universal enveloping algebra for ternary Lie algebras of order three, proves a Poincaré-Birkhoff-Witt theorem, and shows it has a Hopf algebra structure, linking it to three-exterior algebra variables.

Contribution

It introduces a universal enveloping algebra for Lie algebras of order three and establishes its Hopf algebra structure, extending classical Lie theory.

Findings

Proved a Poincaré-Birkhoff-Witt theorem in this context

Constructed a Hopf algebra structure on the universal enveloping algebra

Linked the algebra's dual to variables generating the three-exterior algebra

Abstract

We construct an universal enveloping algebra associated to the ternary extension of Lie (super)algebras called Lie algebra of order three. A Poincar\'e-Birkhoff-Witt theorem is proven is this context. It this then shown that this universal enveloping algebra can be endowed with a structure of Hopf algebra. The study of the dual of the universal enveloping algebra enables to define the parameters of the transformation of a Lie algebra of order three. It turns out that these variables are the variables which generate the three-exterior algebra.