• 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}
}
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… 

Figures from this paper

Triangular Prism Equations and Categorification
We introduce the triangular prism equations for fusion categories, which turn out to be equivalent to the pentagon equations in the spherical case (up to a change of basis), but provide insight to
Quantum Fourier analysis
TLDR
This paper introduces this mathematical subject, shows how it can solve some theoretical problems, and gives some applications to quantum physics with bounds on entropy and the analysis of quantum entanglement.
On Low Rank Fusion Rings
We present a method to generate all fusion rings of a specific rank and multiplicity. This method was used to generate exhaustive lists of fusion rings up to order 9 for several multiplicities. We
CLASSIFICATION OF GROTHENDIECK RINGS OF COMPLEX PIVOTAL FUSION CATEGORIES OF MULTIPLICITY ONE UP TO RANK SIX
Abstract. This paper classifies Grothendieck rings of complex pivotal fusion categories of multiplicity one up to rank six, as an application of a localization approach of the Pentagon Equation and
Interpolated family of non group-like simple integral fusion rings of Lie type
This paper computes the generic fusion rules of the Grothendieck ring of Rep(PSL(2, q)), q prime-power, by applying the Schur orthogonality relations on the generic character table. It then proves
Compact hypergroups from discrete subfactors
On odd-dimensional modular tensor categories
We study odd-dimensional modular tensor categories and maximally non-self dual (MNSD) modular tensor categories of low rank. We give lower bounds for the ranks of modular tensor categories in terms

References

SHOWING 1-10 OF 38 REFERENCES
On fusion categories
Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show
Classification of fusion categories of dimension pq
We provide a complete classification of fusion categories of Frobenius-Perron (FP) dimension pq over complex numbers, where p 1 and which have only one noninvertible object of dimension n.
Tensor Categories
  • American Mathematical Society
  • 2015
Quon Language: Surface Algebras and Fourier Duality
  • Zhengwei Liu
  • Mathematics
    Communications in Mathematical Physics
  • 2019
Quon language is a 3D picture language that we can apply to simulate mathematical concepts. We introduce the surface algebras as an extension of the notion of planar algebras to higher genus surface.
Yang-Baxter relation planar algebras
We give a new type of Schur-Weyl duality for the representations of a family of quantum subgroups and their centralizer algebra. We define and classify singly-generated, Yang-Baxter relation planar
Noncommutative Uncertainty Principles
Exchange relation planar algebras of small rank
The main purpose of this paper is to classify exchange relation planar algebras with 4 dimensional 2-boxes. Besides its skein theory, we emphasize the positivity of subfactor planar algebras based on
On a necessary condition for unitary categorification of fusion rings
In [LPW] Liu, Palcoux and Wu proved a remarkable necessary condition for a fusion ring to admit a unitary categorification, by constructing invariants of the fusion ring that have to be positive if
PIVOTAL FUSION CATEGORIES OF RANK 3
We classify all fusion categories of rank 3 that admit a pivotal structure over an algebraically closed field of characteristic zero. Also in the Appendix (joint with D. Nikshych) we give some
...
...