Finiteness of Frobenius subalgebra lattices in fusion categories

Prove that for every connected Frobenius algebra object X in any fusion category, the lattice of Frobenius subalgebras of X is finite.

Background

The paper establishes finiteness of the Frobenius subalgebra lattice under positivity assumptions, notably for pseudo-unitary fusion categories, and via semisimplification for spherical tensor categories. These results generalize Watatani’s theorem from subfactor theory to the tensor-categorical setting.

Conjecture 8.10 asks whether this finiteness persists in full generality for fusion categories, beyond the pseudo-unitary or spherical assumptions. A counterexample would require a fusion category not Grothendieck equivalent to a pseudo-unitary one, which the authors note are rare in the literature.

References

Conjecture 8.10. Let X be a connected Frobenius algebra in a fusion category. Then its Frobenius subalgebra lattice is finite.

— Frobenius subalgebra lattices in tensor categories  (2502.19876 - Ghosh et al., 27 Feb 2025) in Conjecture 8.10, Section 8

For which nonsemisimple tensor categories, and for which objects $X$ in them, does $X$ have only finitely many subobject isomorphism classes, so that the general argument also controls arbitrary separable algebra subobjects without compatible Frobenius structures?

— Local Ocneanu rigidity for separable algebra objects  (2609.10440 - Azzouz et al., 9 Sep 2026) in Section “Scope, sharpness, and open directions,” subsection “Further questions,” item 1

In positive tensor categories, or more specifically integral fusion categories, can $N_F{\mathrm{inn}(X)\leq2{r_X}$ be replaced asymptotically by a uniform bound $c{r_X}$ with $c<2$, or even a subexponential bound? For irreducible subfactors, how do the best asymptotic exchange and angle bounds compare when both are measured in the Jones index?

— Local Ocneanu rigidity for separable algebra objects  (2609.10440 - Azzouz et al., 9 Sep 2026) in Section “Scope, sharpness, and open directions,” subsection “Further questions,” item 2

Can arbitrary separable algebra subobjects of $X$, without a chosen compatible Frobenius structure, be encoded by an intrinsic quadratic system whose dimension is controlled by $\dim_{}\End_{}(X)$?

— Local Ocneanu rigidity for separable algebra objects  (2609.10440 - Azzouz et al., 9 Sep 2026) in Section “Scope, sharpness, and open directions,” subsection “Further questions,” item 3