1993/10/31 by John W. Barrett, Bruce W. Westbury · 175 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Geometric and Algebraic Topology #Mathematics #Pure mathematics #hep-th #math.QA
paper · pdf · doi:10.1006/aima.1998.1800
published in Advances in Mathematics 143(2), 357-375 (Elsevier BV) · 16 pages. Minor corrections
arxiv created 1998/07/22 · openalex publication_date 1999/05/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras and the motivating application is the definition of 6j-symbols as used in topological field theories. We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a monoidal category with duals following MacLane (1963). In the second section we give the definition of a spherical category, and construct a natural quotient which is also spherical. In the third section we define spherical Hopf algebras so that the category of representations is spherical. Examples of spherical Hopf algebras are involutory Hopf algebras and ribbon Hopf algebras. Finally we study the natural quotient in these cases and show it is semisimple.