Generalizations of Frobenius-Schur indicators from Kuperberg invariants

Abstract

We introduce an approach to produce gauge invariants of any finite-dimensional Hopf algebras from the Kuperberg invariants of framed 3-manifolds. These invariants are generalizations of Frobenius-Schur indicators of Hopf algebras. The computation of Kuperberg invariants is based on a presentation of the framed 3-manifold in terms of Heegaard diagram with combings satisfying certain admissibility conditions. We provide framed Heegaard diagrams for two infinite families of small genus 3-manifolds, which include all the lens spaces, and some homology spheres. In particular, the invariants of the lens spaces $L(n,1)$ coincide with the higher Frobenius-Schur indicators of Hopf algebras. We compute the Kuperberg invariants of all these framed 3-manifolds, and prove that they are invariants of the tensor category of representations of the underlying Hopf algebra, or simply gauge invariants.