Tannaka–Kreıˇn reconstruction and a characterization of modular tensor categories

2007/11/30 by Hendryk Pfeiffer · 1 citation
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.jalgebra.2009.02.026

published as Journal of Algebra 321 No. 12 (2009) 3714-3763 · 52 pages; LaTeX2e; xypic and pstricks macros; v2: typos corrected

arxiv created 2009/03/25 · openalex publication_date 2009/04/05 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We show that every modular category is equivalent as an additive ribbon category to the category of finite-dimensional comodules of a Weak Hopf Algebra. This Weak Hopf Algebra is finite-dimensional, split cosemisimple, weakly cofactorizable, coribbon and has trivially intersecting base algebras. In order to arrive at this characterization of modular categories, we develop a generalization of Tannaka-Krein reconstruction to the long version of the canonical forgetful functor which is lax and oplax monoidal, but not in general strong monoidal, thereby avoiding all the difficulties related to non-integral Frobenius-Perron dimensions.

Citations

• Duality for actions of weak Kac algebras and crossed product inclusions of II₁ factors
• Axioms for Weak Bialgebras
• Frobenius algebras and planar open string topological field theories
• A canonical Tannaka duality for finite seimisimple tensor categories
• Invariants of 3-manifolds via link polynomials and quantum groups
• Classical and quantum conformal field theory

Cited by

• Finitely semisimple spherical categories and modular categories are self-dual