Let $H$ be a finite-dimensional Hopf algebra over the complex field.
Question: Does $ H $ have a finite number of Hopf subalgebras?
In the case where $ H $ is semisimple, the answer is yes. According to [EW14, Theorem 3.6], a semisimple Hopf algebra over an algebraically closed field of characteristic zero has a finite number of right coideal subalgebras. Furthermore, [EW14, Example 3.5] presents a non-semisimple example—Sweedler's 4-dimensional Hopf algebra—that has infinitely many right coideal subalgebras, though it does have a finite number of Hopf subalgebras. Please note that [EW14, Theorem 3.6] has been generalized to apply to any algebraically closed field, as stated in [S17, Theorem 1].
A colleague suggested another perspective: Based on [T79, Theorems 1,2], there is a one-to-one correspondence between the Hopf subalgebras of $ H $ and the normal left coideal subalgebras of $ H^* $, which are also in correspondence with the tensor subcategories of the finite tensor category $ \text{Rep}(H^*) $ as per [B12]. If $ H $ is also semisimple, it follows that $ \text{Rep}(H^*) $ forms a fusion category, which indeed has a finite number of fusion subcategories. Thus, the original question can be reframed as whether a finite tensor category with a fiber functor over the complex field has a finite number of tensor subcategories.
Reference
[B12] Burciu, Sebastian. Kernels of representations and coideal subalgebras of Hopf algebras. Glasg. Math. J. 54 (2012), no. 1, 107--119.
[EW14] Etingof, Pavel; Walton, Chelsea. Semisimple Hopf actions on commutative domains. Adv. Math. 251 (2014), 47--61.
[S17] Skryabin, Serge. Finiteness of the number of coideal subalgebras. Proc. Amer. Math. Soc. 145 (2017), no. 7, 2859--2869.
[T79] Takeuchi, Mitsuhiro. Relative Hopf modules—equivalences and freeness criteria. J. Algebra 60 (1979), no. 2, 452--471.
