Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 5

Special issue on “Subfactors and Related Topics” in memory of Vaughan Jones

Guest Editors: Dietmar Bisch, Arthur Jaffe, Yasuyuki Kawahigashi, and Zhengwei Liu

Graded extensions of generalized Haagerup categories

Pages: 2335 – 2408

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n5.a3

Authors

Pinhas Grossman (School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, Australia)

Masaki Izumi (Department of Mathematics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto, Japan)

Noah Snyder (Department of Mathematics, Indiana University, Bloomington, In., U.S.A.)

Abstract

$\def\Z{\mathbb{Z}}$We classify certain $\Z_2$-graded extensions of generalized Haagerup categories in terms of numerical invariants satisfying polynomial equations. In particular, we construct a number of new examples of fusion categories, including: $\Z_2$-graded extensions of $\Z_{2n}$ generalized Haagerup categories for all $n \leq 5$; $\Z_2 \times \Z_2$-graded extensions of the Asaeda-Haagerup categories; and extensions of the $\Z_2 \times \Z_2$ generalized Haagerup category by its outer automorphism group $A_4$. The construction uses endomorphism categories of operator algebras, and in particular, free products of Cuntz algebras with free group $\mathrm{C}^\ast$-algebras.

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The authors’ work was supported in part by JSPS KAKENHI Grant Number JP20H01805; by ARC grants DP140100732, DP170103265, and DP200100067; and by NSF DMS grant number 2000093.

Received 27 January 2022

Received revised 20 December 2022

Accepted 12 February 2023

Published 30 January 2024