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Tetrahedral forms in monoidal categories and 3-manifold invariants

Geer, Nathan ; Kashaev, Rinat ; Turaev, Vladimir

In: Journal für die reine und angewandte Mathematik (Crelles Journal), 2012, vol. 2012, no. 673, p. 69-123

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    Title
    • Tetrahedral forms in monoidal categories and 3-manifold invariants
    Author
    • Geer, Nathan. Mathematics and Statistics, Utah State University, Logan, Utah 84322, USA
    • Kashaev, Rinat. Section de Mathématiques, Université de Genève, 2-4, rue du Lièvre, Case postale 64, 1211 Genève 4, Switzerland
    • Turaev, Vladimir. Department of Mathematics, Indiana University, Rawles Hall, 831 East 3rd St, Bloomington, IN 47405, USA
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Journal für die reine und angewandte Mathematik (Crelles Journal), 2012, vol. 2012, no. 673, p. 69-123. De Gruyter
    Other Version
    Classification
    • Health
    OAI-PMH Identifier
    • oai:doc.rero.ch:299620
    Summary
    We introduce systems of objects and operators in linear monoidal categories called Ψ̂-systems. A Ψ̂-system satisfying several additional assumptions gives rise to a topological invariant of triples (a closed oriented 3-manifold M, a principal bundle over M, a link in M). This construction generalizes the quantum dilogarithmic invariant of links appearing in the original formulation of the volume conjecture. We conjecture that all quantum groups at odd roots of unity give rise to Ψ̂-systems and we verify this conjecture in the case of the Borel subalgebra of quantum 𝔰𝔩2