On slightly degenerate fusion 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 $2p^{n}d$ and $4p^{n}d$ 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.

23 pages; typos are corrected and some references are added