Mathematics Colloquia and Seminars

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I outline the proof of the following theorem: Let k be an algebraically closed field and C a semisimple spherical k-linear tensor category with simple unit, finitely many isoclasses of simple objects and non-zero dimension. Then the center/quantum double D(C) is semisimple and modular (in the sense of Turaev) and dim D(C) = (dim C)^2. The machinery developed for the proof employs 2-categories and Frobenius algebras in (abstract) tensor categories and is interesting in itself since it captures all the algebraic aspects of subfactor theory. We also mention some results concerning invariants of 3-manifolds.