A universal plane of diagrammatic categories

TL;DR

This paper introduces quantum skein relations for a family of ribbon categories parametrized by the projective plane over the rationals, revealing special points related to the Freudenthal magic square and exceptional series.

Contribution

It establishes a universal framework linking diagrammatic categories with quantum algebra structures via the projective plane parametrization.

Findings

Thirteen special points admit ribbon functors to invariant tensor categories.

These points form three projective lines, including the Freudenthal magic square and exceptional series.

Provides a new geometric perspective on quantum algebra classifications.

Abstract

We introduce quantum skein relations for a family of ribbon categories parametrised by the projective plane over $\bQ$. There are thirteen points for which the ribbon category admits a ribbon functor to a category of invariant tensors for a quantised enveloping algebra. These thirteen points lie on three projective lines. One line gives the first row of the Freudenthal magic square and another gives the fourth row which is the exceptional series.