Modular categories after Turaev

Daniel Steffen – 8 September 1999

These talks were an exposé of some definitions and results in V.G. Turaev, Quantum Invariants of Knots and 3-Manifolds, de Gruyter, 1994 (MR 95k:57014) on tortile tensor categories, modular categories and their connection to low-dimensional topology. For V a strict tortile tensor category, an isotopy invariant F on V-colored ribbon graphs with values in V is given. V-colored ribbon graphs are a generalization of framed tangles, whose components are equipped with objects and morphisms of V. A (quasi-)modular category is an additive strict tortile tensor category equipped with a finite family of simple objects (quasi-)dominating the category and respecting some axioms. Considering F on such a modular category W allows its extension to a topological invariant tau of closed connected oriented 3-manifolds with embedded W-colored ribbon graphs. For A a ribbon Hopf algebra, the category of A-modules of finite rank quasi-dominated by a finite family of simple A-modules and respecting the axioms is a quasi-modular category. A non-trivial geometrical example of a quasi-modular category is the refined skein category.