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May 22, 2020

Fusion Bialgebras and Fourier Analysis–New Analytic Obstructions of Categorification

In recent work joint with Jinsong Wu and Sebastien Palcoux, we introduce fusion bialgebras and their duals and systematically study their quantum Fourier analysis, inspired by quantum Fourier analysis on subfactors, such as quantum analogues of Hausdorff-Young inequality, Young’s inequality, sum-set estimates and uncertainty principles.  As an application, we discover new analytic obstructions on the unitary categorification of fusion rings. In particular, the Schur product property holds on the Grothendieck ring of a unitary fusion category, but not on a fusion ring, which turns out to be a surprisingly efficient analytic obstruction of unitary categorification.