On slightly degenerate fusion categories

TL;DR

This paper investigates properties of slightly degenerate fusion categories, establishing divisibility conditions, nilpotency results for certain dimensions, and classifying specific types of these categories.

Contribution

It provides new divisibility results, nilpotency classifications, and a classification of certain slightly degenerate fusion categories, advancing understanding in fusion category theory.

Findings

Squares of simple object dimensions divide half of the total dimension.

Fusion categories with FP-dimensions 2p^nd and 4p^nd are nilpotent.

Classification of slightly degenerate generalized Tambara-Yamagami categories.

Abstract

In this paper, we first show for a slightly degenerate pre-modular fusion category $\mathcal{C}$ that squares of dimensions of simple objects divide half of the dimension of $\mathcal{C}$, and that slightly degenerate fusion categories of FP-dimensions $2p^nd$ and $4p^nd$ are nilpotent, where $p$ is an odd prime and $d$ is an odd square-free integer. Then we classify slightly degenerate generalized Tambara-Yamagami fusion categories and weakly integral slightly degenerate fusion categories of particular dimensions.