2020/07/01 by Tzu-Chen Huang, Huang, Tzu-Chen, Ying-Hsuan Lin +1
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA) #Quantum Algebra (math.QA) #Strongly Correlated Electrons (cond-mat.str-el)
paper · pdf · doi:10.48550/arxiv.2007.00670
openalex publication_date 2020/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A fusion category is called transparent if the associator involving any invertible object is the identity map. For the Haagerup-Izumi fusion rings with G = ℤ2n+1 (the ℤ3 case is the Haagerup fusion ring with six simple objects), the transparent ansatz reduces the number of independent F-symbols from order O(n6) to O(n2), rendering the pentagon identity practically solvable. Transparent Haagerup-Izumi fusion categories are thereby constructively classified up to G = ℤ9, recovering all known Haagerup-Izumi fusion categories to this order, and producing new ones. Transparent Haagerup-Izumi fusion categories additionally satisfying S4 tetrahedral invariance are further classified up to G = ℤ15, and the explicit F-symbols for the unitary ones, including the Haagerup H3 fusion category, are compactly presented. The F-symbols for the Haagerup H2 fusion category are also presented. Going beyond, the transparent ansatz offers a viable course towards constructing novel fusion categories for new fusion rings.