On Normal Tensor Functors and Coset Decompositions for Fusion Categories

We consider two approaches to isotopy invariants of oriented links: one from ribbon categories and the other from generalized Yang–Baxter (gYB) operators with appropriate enhancements. The gYB-operators we consider are obtained from so-called gYBE objects following a procedure of Kitaev and Wang. We show that the enhancement of these gYB-operators is canonically related to the twist structure in ribbon categories from which the operators are produced. If a gYB-operator is obtained from a ribbon category, it is reasonable to expect that two approaches would result in the same invariant. We prove that indeed the two link invariants are the same after normalizations. As examples, we study a new family of gYB-operators which is obtained from the ribbon fusion categories SO (N)2, where N is an odd integer. These operators are given by 8 × 8 matrices with the parameter N and the link invariants are specializations of the two-variable Kauffman polynomial invariant F.

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The center of the extended Haagerup subfactor has 22 simple objects

International Journal of Mathematics ◽

10.1142/s0129167x17500094 ◽

2017 ◽

Vol 28 (01) ◽

pp. 1750009 ◽

Cited By ~ 1

Author(s):

Scott Morrison ◽

Kevin Walker

Keyword(s):

Fusion Category ◽

Dimension Function ◽

Fusion Ring ◽

Fusion Categories ◽

Modular Data ◽

Grothendieck Rings ◽

Combinatorial Data

We explain a technique for discovering the number of simple objects in [Formula: see text], the center of a fusion category [Formula: see text], as well as the combinatorial data of the induction and restriction functors at the level of Grothendieck rings. The only input is the fusion ring [Formula: see text] and the dimension function [Formula: see text]. In particular, we apply this to deduce that the center of the extended Haagerup subfactor has 22 simple objects, along with their decompositions as objects in either of the fusion categories associated to the subfactor. This information has been used subsequently in [T. Gannon and S. Morrison, Modular data for the extended Haagerup subfactor (2016), arXiv:1606.07165 .] to compute the full modular data. This is the published version of arXiv:1404.3955 .

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Solvability of a Class of Braided Fusion Categories

Applied Categorical Structures ◽

10.1007/s10485-012-9299-y ◽

2013 ◽

Vol 22 (1) ◽

pp. 229-240 ◽

Cited By ~ 5

Author(s):

Sonia Natale ◽

Julia Yael Plavnik

Keyword(s):

Fusion Categories

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Boundary Topological Entanglement Entropy in Two and Three Dimensions

Communications in Mathematical Physics ◽

10.1007/s00220-021-04191-y ◽

2021 ◽

Author(s):

Jacob C. Bridgeman ◽

Benjamin J. Brown ◽

Samuel J. Elman

Keyword(s):

General Property ◽

Entanglement Entropy ◽

Quantum Correlations ◽

Three Dimensions ◽

Topological Phases ◽

Fusion Categories ◽

Special Cases ◽

Constructive Proofs ◽

And Algebra ◽

Topological Entanglement Entropy

AbstractThe topological entanglement entropy is used to measure long-range quantum correlations in the ground space of topological phases. Here we obtain closed form expressions for the topological entropy of (2+1)- and (3+1)-dimensional loop gas models, both in the bulk and at their boundaries, in terms of the data of their input fusion categories and algebra objects. Central to the formulation of our results are generalized $${\mathcal {S}}$$ S -matrices. We conjecture a general property of these $${\mathcal {S}}$$ S -matrices, with proofs provided in many special cases. This includes constructive proofs for categories up to rank 5.

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