Pre-modular fusion categories of small global dimensions

TL;DR

This paper extends the understanding of fusion categories by establishing a Lagrange-like theorem for their global dimensions and classifying all pre-modular fusion categories with small integer global dimensions up to 10.

Contribution

It introduces a Lagrange theorem analogue for fusion categories and provides a complete classification of pre-modular categories with small global dimensions.

Findings

Proved a Lagrange theorem analogue for fusion categories.

Classified pre-modular fusion categories with global dimensions ≤ 10.

Established foundational results for small-dimensional fusion categories.

Abstract

We first prove an analogue of Lagrange theorem for global dimensions of fusion categories, then we give a complete classifications of pre-modular fusion categories of integer global dimensions less than or equal to $10$.