Generalizations of Frobenius-Schur indicators from Kuperberg invariants

TL;DR

This paper develops a method to derive gauge invariants from Kuperberg invariants for finite-dimensional Hopf algebras, generalizing Frobenius-Schur indicators, and computes these invariants for various 3-manifolds.

Contribution

It introduces a new approach linking Kuperberg invariants to Frobenius-Schur indicators and provides explicit computations for specific 3-manifolds including lens spaces.

Findings

Kuperberg invariants generalize Frobenius-Schur indicators

Invariants are computed for lens spaces and homology spheres

Invariants depend only on the tensor category of representations

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.