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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.07409 |
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| _version_ | 1866915333222170624 |
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| author | Chang, Liang Ng, Siu-Hung Wang, Yilong |
| author_facet | Chang, Liang Ng, Siu-Hung Wang, Yilong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07409 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalizations of Frobenius-Schur indicators from Kuperberg invariants Chang, Liang Ng, Siu-Hung Wang, Yilong Quantum Algebra Geometric Topology Rings and Algebras 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. |
| title | Generalizations of Frobenius-Schur indicators from Kuperberg invariants |
| topic | Quantum Algebra Geometric Topology Rings and Algebras |
| url | https://arxiv.org/abs/2506.07409 |