1998/02/18 by Ulrike Tillmann, Tillmann, Ulrike · 4 citations
Mathematics · #16A16 #18D10 #57N10 #81E05 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.CT #math.GT #math.QA #math.RT #msc:16A16 #msc:18D10 #msc:57N10 #msc:81E05
paper · pdf · doi:10.48550/arxiv.math/9802089
Accepted for publication in the Journal of the LMS, April 1996
arxiv created 1998/02/18 · openalex publication_date 1998/02/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by ideas from string theory and quantum field theory new invariants of knots and 3-dimensional manifolds have been constructed from complex algebraic structures such as Hopf algebras (Reshetikhin and Turaev), monoidal categories with additional structure (Turaev and Yetter), and modular functors (Walker and Kontsevich). These constructions are very closely related. We take a unifying categorical approach based on a natural 2-dimensional generalization of a topological field theory in the sense of Atiyah and Segal, and show that the axioms defining these complex algebraic structures are a consequence of the underlying geometry of surfaces. In particular, we show that any linear category over a field with an action of the surface category is semi-simple and Artinian.