• Corpus ID: 204904803

Fusion Bialgebras and Fourier Analysis

@article{Liu2019FusionBA,
  title={Fusion Bialgebras and Fourier Analysis},
  author={Zhengwei Liu and Sebastien Palcoux and Jinsong Wu},
  journal={arXiv: Quantum Algebra},
  year={2019}
}

• Zhengwei Liu, S. Palcoux, Jinsong Wu
• Published 26 October 2019
• Mathematics
• arXiv: Quantum Algebra

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 3780. 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.