Hopf algebras with triality

TL;DR

This paper extends the concept of triality from groups and loops to Hopf algebras, showing that universal enveloping algebras of Lie algebras with triality also possess this property, leading to new constructions for Malcev algebras.

Contribution

It demonstrates that universal enveloping algebras of Lie algebras with triality are Hopf algebras with triality, providing a novel approach to Malcev algebra enveloping algebras.

Findings

Universal enveloping algebra of Lie algebra with triality has triality.

New construction method for universal enveloping algebras of Malcev algebras.

Extension of triality concepts to Hopf algebras.

Abstract

In this paper we revisit and extend the constructions of Glauberman and Doro on groups with triality and Moufang loops to Hopf algebras. We prove that the universal enveloping algebra of any Lie algebra with triality is a Hopf algebra with triality. This allows us to give a new construction of the universal enveloping algebras of Malcev algebras. Our work relies on the approach of Grishkov and Zavarnitsine to groups with triality.