Trivalent Categories for Adjoint Representations of Exceptional Lie Algebras

TL;DR

This paper explores a universal categorical framework that, through specific quotients, models either vector product algebras or the representation categories of exceptional Lie algebras, revealing deep structural connections.

Contribution

It introduces a universal pivotal symmetric monoidal category generated by a Schurian object with skew-symmetric multiplication, linking it to exceptional Lie algebra representations.

Findings

Quotients yield vector product algebras

Quotients realize categories of exceptional Lie algebra representations

Establishes a categorical foundation for exceptional Lie structures

Abstract

We consider the universal pivotal, symmetric, monoidal, $\Bbbk$-linear category, generated by a Schurian object with a skew-symmetric multiplication, and study some of its quotients. We show that these quotients give rise to either vector product algebras or representation categories of exceptional Lie algebras.