Ternary algebras and groups

TL;DR

This paper constructs explicit groups linked to ternary algebras, extending Lie (super)algebras, using variables from the three-exterior algebra, and provides matrix representations for these groups.

Contribution

It introduces a new construction of groups associated with ternary algebras and extends Lie algebra concepts using three-exterior algebra variables.

Findings

Explicit group constructions for ternary algebras

Matrix representations involving three-exterior algebra

Extension of Lie (super)algebras to order three

Abstract

We construct explicitly groups associated to specific ternary algebras which extend the Lie (super)algebras (called Lie algebras of order three). It turns out that the natural variables which appear in this construction are variables which generate the three-exterior algebra. An explicit matrix representation of a group associated to a peculiar Lie algebra of order three is constructed considering matrices with entry which belong to the three exterior algebra.