Left coideal subalgebras of H* as Frobenius subalgebra objects in Rep(H)

Ascertain whether, for every finite-dimensional Hopf algebra H, each left coideal subalgebra K of the dual Hopf algebra H* can be endowed with the structure of a connected Frobenius subalgebra object of H* in the tensor category Rep(H).

Background

The paper constructs H* as a connected Frobenius algebra object in Rep(H) using a Fourier-transform-based comultiplication and counit. In the unitary case, left coideal subalgebras of H* are known to correspond to Frobenius subalgebras.

Extending this correspondence beyond the unitary setting would potentially recover and generalize finiteness results for left coideal subalgebras and connect Hopf algebra structure to categorical Frobenius subalgebras.