Categories generated by a trivalent vertex View Full Text


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Article Info

DATE

2017-04

AUTHORS

Scott Morrison, Emily Peters, Noah Snyder

ABSTRACT

This is the first paper in a general program to automate skein theoretic arguments. In this paper, we study skein theoretic invariants of planar trivalent graphs. Equivalently, we classify trivalent categories, which are nondegenerate pivotal tensor categories over generated by a symmetric self-dual simple object X and a rotationally invariant morphism 1→X⊗X⊗X. Our main result is that the only trivalent categories with dimHom(1→X⊗n) bounded by 1, 0, 1, 1, 4, 11, 40 for 0≤n≤6 are quantum SO(3), quantum G2, a one-parameter family of free products of certain Temperley-Lieb categories (which we call ABA categories), and the H3 Haagerup fusion category. We also prove similar results where the map 1→X⊗3 is not rotationally invariant, and we give a complete classification of nondegenerate braided trivalent categories with dimensions of invariant spaces bounded by the sequence 1, 0, 1, 1, 4. Our main techniques are a new approach to finding skein relations which can be easily automated using Gröbner bases, and evaluation algorithms which use the discharging method developed in the proof of the 4-color theorem. More... »

PAGES

817-868

References to SciGraph publications

  • 2011-10. The Exoticness and Realisability of Twisted Haagerup–Izumi Modular Data in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1999-04. Exotic Subfactors of Finite Depth with Jones Indices \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} in COMMUNICATIONS IN MATHEMATICAL PHYSICS
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  • 2010-04. On braided fusion categories I in SELECTA MATHEMATICA
  • 2013-02. Canonical Basis for Quantum in LETTERS IN MATHEMATICAL PHYSICS
  • 1979. Probabilistic algorithms for sparse polynomials in SYMBOLIC AND ALGEBRAIC COMPUTATION
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  • 1993-03. Representation theory of osp(1,2)q in ZEITSCHRIFT FÜR PHYSIK C PARTICLES AND FIELDS
  • 2012-05. Quantum Subgroups of the Haagerup Fusion Categories in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • Identifiers

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    http://scigraph.springernature.com/pub.10.1007/s00029-016-0240-3

    DOI

    http://dx.doi.org/10.1007/s00029-016-0240-3

    DIMENSIONS

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