Fusion bialgebras and Fourier analysis

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