Tetrahedral symmetry of 6j-symbols in fusion categories

TL;DR

This paper proves tetrahedral symmetries of 6j-symbols in fusion categories using veined fusion categories, a new concept that simplifies calculations without requiring pivotal structures or unitarity.

Contribution

It introduces veined fusion categories and demonstrates their utility in establishing symmetries of 6j-symbols in general fusion categories.

Findings

Established tetrahedral symmetries for 6j-symbols in arbitrary fusion categories.

Introduced veined fusion categories as a new computational tool.

Connected algebraic properties of 6j-symbols to their geometric origins.

Abstract

We establish tetrahedral symmetries of 6j-symbols for arbitrary fusion categories under minimal assumptions. As a convenient tool for our calculations we introduce the notion of a veined fusion category, which is generated by a finite set of simple objects but is larger than its skeleton. Every fusion category C contains veined fusion subcategories that are monoidally equivalent to C and which suffice to compute many categorical properties for C. The notion of a veined fusion category does not assume the presence of a pivotal structure, and thus in particular does not assume unitarity. We also exhibit the geometric origin of the algebraic statements for the 6j-symbols.