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Fusion bialgebras and Fourier analysis: Analytic obstructions for unitary categorification

Zhengwei Liu Yau Mathematical Sciences Center and Department of Mathematics, Tsinghua University, Beijing, 100084, China; Beijing Institute of Mathematical Sciences and Applications, Huairou District, Beijing, 101408, China Sebastien Palcoux Beijing Institute of Mathematical Sciences and Applications, Huairou District, Beijing, 101408, China Jinsong Wu Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin, 150001, China

Category Theory Functional Analysis Quantum Algebra Rings and Algebras Spectral Theory and Operator Algebra mathscidoc:2207.04003

Advances in Mathematics, 390, 107905, 2021.10
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.
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@inproceedings{zhengwei2021fusion,
  title={Fusion bialgebras and Fourier analysis: Analytic obstructions for unitary categorification},
  author={Zhengwei Liu, Sebastien Palcoux, and Jinsong Wu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707155839565300575},
  booktitle={Advances in Mathematics},
  volume={390},
  pages={107905},
  year={2021},
}
Zhengwei Liu, Sebastien Palcoux, and Jinsong Wu. Fusion bialgebras and Fourier analysis: Analytic obstructions for unitary categorification. 2021. Vol. 390. In Advances in Mathematics. pp.107905. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707155839565300575.
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