On bicrossed product of fusion categories and exact factorizations

TL;DR

This paper introduces a new framework for constructing and analyzing fusion categories through bicrossed products and exact factorizations, extending concepts from group theory to fusion categories.

Contribution

It generalizes the bicrossed product concept to fusion categories, characterizes all exact factorizations as bicrossed products, and provides explicit examples involving Tambara-Yamagami and pointed fusion categories.

Findings

Every exact factorization can be realized as a bicrossed product.

Explicit fusion rules and associativity constraints are provided for new examples.

The framework extends the classical bicrossed product from groups to fusion categories.

Abstract

We introduce the notion of a matched pair of fusion rings and fusion categories, generalizing the one for groups. Using this concept, we define the bicrossed product of fusion rings and fusion categories and we construct exact factorizations for them. This concept generalizes the bicrossed product, also known as external Zappa-Sz\'ep product, of groups. We also show that every exact factorization of fusion rings can be presented as a bicrossed product. With this characterization, we describe the adjoint subcategory and universal grading group of an exact factorization of fusion categories. We give explicit fusion rules and associativity constraints for examples of fusion categories arising as a bicrossed product of combinations of Tambara-Yamagami categories and pointed fusion categories. These examples are new to the best of the knowledge of the authors.