Categories generated by a trivalent vertex

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 C 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 dim Hom(1 → X ⊗n ) bounded by 1, 0, 1, 1, 4, 11, 40 for 0 ≤ n ≤ 6 are quantum SO(3), quantum G 2 , 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. arXiv:1501.06869v3 [math.QA]

The Brauer–Picard groups of fusion categories coming from the ADE subfactors

“…Both of these planar algebras contain sub-planar algebras generated by the trivalent vertex. By considering the fusion rules for Ad(A 7 ) we can see that both these sub-planar algebras have box space dimensions (1, 0, 1, 1, 3, ....), thus by the main theorem of [28] must be SO(3) q for q either e iπ 4 or e 3iπ 4 . As the categorical dimension of both f (2) and f (4) in Ad(A 7 ) is 1 + √ 2, we must have that both sub-planar algebras are SO(3) e iπ 4 .…”

“…Considering dimensions there are two possible simple objects that f (0) ⊠ f (2) may be sent to. These are f (0) ⊠ f (2) , and f (28) ⊠ f (2) . Aiming for a contradiction, suppose there was an auto-equivalence of A 29 ⊠ Ad(A 4 ) bop sending f (0) ⊠ f (2) to f (28) ⊠ f (2) .…”

“…These are f (0) ⊠ f (2) , and f (28) ⊠ f (2) . Aiming for a contradiction, suppose there was an auto-equivalence of A 29 ⊠ Ad(A 4 ) bop sending f (0) ⊠ f (2) to f (28) ⊠ f (2) . In the category Ad(A 4 ) bop there exists a non-zero morphism f (2) ⊗ f (2) → f (2) .…”

Classifying fusion categories $$\otimes $$-generated by an object of small Frobenius–Perron dimension

“…As a complete classification of fusion categories is hopelessly out of reach with current techniques, current research into the classification of fusion categories focuses on classifying "small" fusion categories, where small can have a variety of different meanings. Examples of such partial classifications can be found in [28] where a classification of pivotal fusion categories with exactly three simple objects is given, or in [25] where a classification of pivotal fusion categories with restrictions on the size of certain hom spaces is found.…”

Locality and the uniqueness of quantum invariants

“…The uniqueness of a local invariant given the data from the representation theory of a Lie algebra was used implicitly in [9,12] and [14] to determine link polynomials associated with SU (n), A 2 and G 2 . The idea is further explored in [13], along with an exhaustive analysis of categories of diagrams with a single trivalent vertex.…”