Finitely semisimple spherical categories and modular categories are self-dual

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.

42 pages; LaTeX with xypic macros; v2: typos corrected

References in corpus (13)

• Spherical Categories
• Four dimensional topological quantum field theory, Hopf categories, and the canonical bases
• Axioms for Weak Bialgebras
• State sum construction of two-dimensional open-closed Topological Quantum Field Theories
• Spin Foam Diagrammatics and Topological Invariance
• Generalized Lattice Gauge Theory, Spin Foams and State Sum Invariants
• The equality of 3-manifold invariants
• Tannaka-Krein reconstruction and a characterization of modular tensor categories
• Adjointable monoidal functors and quantum groupoids
• A canonical Tannaka duality for finite seimisimple tensor categories
• From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence of tensor categories
• From Subfactors to Categories and Topology II. The quantum double of tensor categories and subfactors
• Hopf Algebroid Symmetry of Abstract Frobenius Extensions of Depth 2

Cited by in corpus (6)

• Cardy algebras and sewing constraints, I
• Cardy algebras and sewing constraints, II
• Dichromatic state sum models for four-manifolds from pivotal functors
• Tannaka-Krein reconstruction for coactions of finite quantum groupoids
• Yetter-Drinfel'd algebras and coideals of Weak Hopf $C^{\ast }$-Algebras
• Weak Hopf Algebras unify the Hennings-Kauffman-Radford and the Reshetikhin-Turaev invariant