Volume 19, issue 5 (2019)

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Generalized Kuperberg invariants of $3$–manifolds

Rinat Kashaev and Alexis Virelizier

Algebraic & Geometric Topology 19 (2019) 2575–2624

In the 1990s, based on presentations of 3–manifolds by Heegaard diagrams, Kuperberg associated a scalar invariant of 3–manifolds to each finite-dimensional involutory Hopf algebra over a field. We generalize this construction to the case of involutory Hopf algebras in arbitrary symmetric monoidal categories admitting certain pairs of morphisms called good pairs. We construct examples of such good pairs for involutory Hopf algebras whose distinguished grouplike elements are central. The generalized construction is illustrated by an example of an involutory super-Hopf algebra.

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invariants of $3$–manifolds, Hopf algebras
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 16T05
Received: 18 June 2018
Revised: 5 November 2018
Accepted: 17 November 2018
Published: 20 October 2019
Rinat Kashaev
Section de Mathématiques
Université de Genève
Alexis Virelizier
Départment de Mathématiques
Université de Lille