Finitely semisimple spherical categories and modular categories are self-dual

2008/06/30 by Hendryk Pfeiffer
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math-ph #math.MP #math.QA #msc:16W30 #msc:18D10

paper · pdf · doi:10.1016/j.aim.2009.03.002

published as Advances in Mathematics 221 No. 5 (2009) 1608-1652 · 42 pages; LaTeX with xypic macros; v2: typos corrected

arxiv created 2009/03/03 · openalex publication_date 2009/03/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

We show that every essentially small finitely semisimple k-linear additive spherical category in which k=End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category with respect to the long forgetful functor is self-dual as a Weak Hopf Algebra.

Citations

• Axioms for Weak Bialgebras
• A canonical Tannaka duality for finite seimisimple tensor categories
• Tannaka–Kreıˇn reconstruction and a characterization of modular tensor categories
• From subfactors to categories and topology II: The quantum double of tensor categories and subfactors