2022/03/12 by Zhengwei Liu, Liu, Zhengwei, Sébastien Palcoux +4 · 1 citation
Mathematics · #16T20 (Secondary) #18M05 #18M20 #18M30 (Primary) 57K16 #20G42 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2203.06522
openalex publication_date 2022/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the triangular prism equations (TPE) for fusion categories, obtained by evaluating triangular prisms in terms of tetrahedra. Using an oriented graphical calculus, we show that the geometric symmetries of the regular tetrahedron are preserved. In the spherical case, we prove that the TPE are equivalent to the pentagon equations after a suitable change of basis. These equations provide new insight for managing complexity via localization. As a consequence, and using the Fuchs-Runkel-Schweigert theorem on the second Frobenius-Schur indicator, we obtain new categorification criteria. As an application, we solve all remaining open cases to complete the classification of unitary 1-Frobenius simple integral fusion categories up to rank 8 and up to Frobenius-Perron dimension 20000.