A class of prime fusion categories of dimension $2^N$

Abstract

We study a class of strictly weakly integral fusion categories ${\mathfrak{I}}_{N,\zeta }$, where $N\ge 1$ is a natural number and $\zeta$ is a $2^N$th root of unity, that we call $N$-Ising fusion categories. An $N$-Ising fusion category has Frobenius-Perron dimension $2^{N+1}$ and is a graded extension of a pointed fusion category of rank 2 by the cyclic group of order ${\mathbb{Z}}_{2^N}$. We show that every braided $N$-Ising fusion category is prime and also that there exists a slightly degenerate $N$-Ising braided fusion category for all $N>2$. We also prove a structure result for braided extensions of a rank 2 pointed fusion category in terms of braided $N$-Ising fusion categories.

amslatex, 22 pages. Added assumption in Theorem 5.2 and details in proof of Theorem 5.3

References in corpus (1)

• Generalized Tambara-Yamagami categories