A Cuntz algebra approach to the classification of near-group categories

Abstract

We classify C${}^{\ast }$ near-group categories by using Vaughan Jones theory of subfactors and the Cuntz algebra endomorphisms. Our results show that there is a sharp contrast between two essentially different cases, integral and irrational cases. When the dimension of the unique non-invertible object is an integer, we obtain a complete classification list, and it turns out that such categories are always group theoretical. When it is irrational, we obtain explicit polynomial equations whose solutions completely classify the C${}^{\ast }$ near-group categories in this class.

135 pages.Theorem 2.2 and Theorem 13.3 are renewed and typographical errors are corrected

References in corpus (2)

• The classification of subfactors of index at most 5
• Cyclic extensions of fusion categories via the Brauer-Picard groupoid

Cited by in corpus (4)

• Generalized Orbifold Construction for Conformal Nets
• Uniqueness of unitary structure for unitarizable fusion categories
• The classification of $3^n$ subfactors and related fusion categories
• Classification of $\mathbb{Z}/2\mathbb{Z}$-quadratic unitary fusion categories