Fusion Bialgebras and Fourier Analysis

TL;DR

This paper develops the theory of fusion bialgebras, establishing Fourier analysis tools and inequalities, and applies these to classify certain fusion rings, revealing obstructions to categorification and subfactorization.

Contribution

It introduces fusion bialgebras and their Fourier analysis, proving key inequalities and classifying simple integral fusion rings up to rank 8.

Findings

Proves Hausdorff-Young inequality and uncertainty principles for fusion bialgebras.

Shows Schur product property and Young's inequality hold for fusion bialgebras but not always on their duals.

Classifies 34 simple integral fusion rings of Frobenius type up to rank 8.

Abstract

We introduce fusion bialgebras and their duals and systematically study their Fourier analysis. As an application, we discover new efficient analytic obstructions on the unitary categorification of fusion rings. We prove the Hausdorff-Young inequality, uncertainty principles for fusion bialgebras and their duals. We show that the Schur product property, Young's inequality and the sum-set estimate hold for fusion bialgebras, but not always on their duals. If the fusion ring is the Grothendieck ring of a unitary fusion category, then these inequalities hold on the duals. Therefore, these inequalities are analytic obstructions of categorification. We classify simple integral fusion rings of Frobenius type up to rank 8 and of Frobenius-Perron dimension less than 4080. We find 34 ones, 4 of which are group-like and 28 of which can be eliminated by applying the Schur product property on the dual. In general, these inequalities are obstructions to subfactorize fusion bialgebras.