Abstract

Frobenius-Schur indicators of spherical fusion categories are powerful tools in the study of algebraic properties of 3D-TFTs. Inspired by the close connections among (1) the Frobenius-Schur indicators of a semisimple Hopf algebra H, (2) the Kuperberg invariant of lens spaces associated to H, and (3) the Turaev-Viro invariants of lens spaces associated to the fusion category Rep(H), we investigate the Kuperberg invariants of framed 3-manifolds associated to general (not necessarily semisimple) Hopf algebras. We show that the Kuperberg invariants of framed lens spaces are categorical invariants, and they naturally generalize Frobenius-Schur indicators. The categorical invariance also holds for some other low-genus framed manifolds, suggesting the potential existence of a categorical construction of the Kuperberg invariant in the nonsemisimple setting. This talk is based on joint work with Liang Chang and Siu-Hung Ng