2013/08/27 by Masaki Izumi, Izumi, Masaki, Scott Morrison +3 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1308.5723
openalex publication_date 2013/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a pair of fusion categories C and D, we may form the free product C * D and the tensor product C \boxtimes D. It is natural to think of the tensor product as a quotient of the free product. What other quotients are possible? When C=D=A2, there is an infinite family of quotients interpolating between the free product and the tensor product (closely related to the A2n-1(1) and Dn+2(1) subfactors at index 4). Bisch and Haagerup discovered one example of such an intermediate quotient when C=A2 and D=T2, and suggested that there might be another family here. We show that such quotients are characterized by parameters n ≥ 1 and ω with ω2n=1. For n=1,2,3, we show ω must be 1, and construct the corresponding quotient (n=1 is the tensor product, n=2 is the example discovered by Bisch and Haagerup, and n=3 is new). We further show that there are no such quotients for 4 ≤ n ≤ 10. Our methods also apply to the case when C=D=T2, and we prove similar results there. During the preparation of this manuscript we learnt of an independent result of Liu's on subfactors. With the translation between the subfactor and fusion category settings provided here, it follows there are no such quotients for any n ≥ 4.