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arXiv · 2506.07409

Generalizations of Frobenius-Schur indicators from Kuperberg invariants

Liang Chang·Siu-Hung Ng·Yilong Wang
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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.

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BibTeXRIS

Liang Chang, Siu-Hung Ng, Yilong Wang. 2025-06-09. Generalizations of Frobenius-Schur indicators from Kuperberg invariants. https://arxiv.org/abs/2506.07409

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