Generalized Kuperberg invariants of $3$–manifolds

Abstract

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.