Graded extensions of generalized Haagerup categories

TL;DR

This paper classifies certain graded extensions of generalized Haagerup categories using polynomial invariants, leading to new examples of fusion categories with complex algebraic structures.

Contribution

It introduces a classification method for graded extensions of generalized Haagerup categories and constructs several new fusion categories with specific grading groups.

Findings

Classified $bZ_2$-graded extensions via polynomial invariants.

Constructed new fusion categories including extensions of $bZ_{2n}$ and Asaeda-Haagerup categories.

Used operator algebra endomorphisms and free products of Cuntz and free group C$^*$-algebras.

Abstract

We classify certain $\mathbb{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: $\mathbb{Z}_2 $-graded extensions of $\mathbb{Z}_{2n} $ generalized Haagerup categories for all $n \leq 5 $; $\mathbb{Z}_2 \times \mathbb{Z}_2 $-graded extensions of the Asaeda-Haagerup categories; and extensions of the $\mathbb{Z}_2 \times \mathbb{Z}_2 $ generalized Haagerup category by its outer automorphism group $A_4 $. The construction uses endomorphism categories of operator algebras, and in particular, free products of Cuntz algebras with free group C$^*$-algebras.