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Mapping class group actions on quantum doubles. (English) Zbl 0833.16039
The author gives a short survey of counterparting the heuristic quantum field theory by mathematically rigorous constructions by applying quasitriangular Hopf algebras, or quantum groups, to replace the field theoretical machinery. An extended topological quantum field theory (TQFT) assigns to every surface $$S$$ a category $${\mathcal C}_S$$ and a functor $$\Phi_S:\text{Cob}_S\to{\mathcal C}_S$$ [see M. Atiyah, Publ. Math., Inst. Hautes Etud. Sci. 68, 175-186 (1988; Zbl 0692.53053)]. The author defines and studies the action of operators generating the mapping class group $${\mathcal D}:=\pi_0(\text{Diff}(T,D))$$ on the double $$D({\mathcal A})$$ of a finite-dimensional Hopf algebra. He uses the results for a rigorous proof of the modular relationships and determines the protective phase of the universal TQFT as $$\nu^{-3}$$. The structure of the representation of $$\text{SL}(2,\mathbb{Z})$$ on the center of $$D(B_q)$$ [see V. Turaev: Quantum Invariants of 3-Manifolds. (Publ. Inst. Rech. Math. Avancée, Strasbourg, Preprint: ISSN 0755-3390)] is considered by restricting the action on mapping class groups. The author expects to find higher dimensional algebraic representations of $$\text{SL}(2,\mathbb{Z})$$ while starting from higher-rank quantum groups for which the orders of nilpotencies of central elements will be higher.

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 17B37 Quantum groups (quantized enveloping algebras) and related deformations 57T05 Hopf algebras (aspects of homology and homotopy of topological groups) 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 81R05 Finite-dimensional groups and algebras motivated by physics and their representations 81T05 Axiomatic quantum field theory; operator algebras
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