In [FSS12, Proposition 2.10], for a finite-dimensional ribbon factorizable Hopf algebra $H$, the dual $H^*$ admits a Frobenius algebra structure in $\mathrm{Bimod}(H)$. 

In [EGNO15, Example 7.20.6], if $ H $ is a unimodular finite-dimensional Hopf algebra, then $ H^* $ serves as the canonical Frobenius algebra object in the category ${\rm Rep}(H \otimes H^{\rm cop})$. This result should generalize [FSS12, Proposition 2.10] mentioned above.

In [LW23, Section 6] various rigid Frobenius algebras within non-semisimple modular tensor categories are discussed. 

*Regarding the examples mentioned above, it is still necessary to verify when they are connected.*

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*References*

[EGNO15] Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. *Tensor Categories*. Mathematical Surveys and Monographs, 205. American Mathematical Society, Providence, RI, 2015. xvi+343 pp.

[FSS12] Fuchs, Jürgen; Schweigert, Christoph; Stigner, Carl. Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms. J. Algebra 363 (2012), 29--72. 

[LW23] Laugwitz, Robert; Walton, Chelsea. *Constructing non-semisimple modular categories with local modules.* Comm. Math. Phys. 403 (2023), no. 3, 1363--1409.