We refer to [JS97] for the notion of subfactor. Yasuo Watatani proved the following result [W96, Theorem 2.2]:
Theorem: An irreducible finite index subfactor of a type $\mathrm{II}_1$ factor has finitely many intermediate subfactors.
We refer to [EGNO15] for the notion of tensor category. We are interested in extending Watatani's theorem to tensor categories. Dave Penneys reformulated it in this framework (private communication) using the following dictionnary:
- Finite index subfactor $\leftrightarrow$ Unitary Frobenius algebra in a unitary tensor category,
- Irreducible $\leftrightarrow$ Connected,
- Intermediate subfactor $\leftrightarrow$ Unitary Frobenius subalgebra.
So he got that:
Theorem: A connected unitary Frobenius algebra in a unitary tensor category has finitely many unitary Frobenius subalgebras.
Remark: The notion of Frobenius algebra (object) $A$ in a tensor category $\mathcal{C}$ is defined in [EGNO15, Definition 7.20.3], next see [EGNO15, Remark 9.4.7] for unitary, and finally, connected means that $\dim_{\mathbb{k}}(\hom_{\mathcal{C}}(1,A)) = 1$.
Question: How to prove above theorem directly in the tensor category framework? To what extent can the unitary assumption be relaxed?
Remark: It is also proved in [W96] that the set of intermediate subfactors forms a lattice. So we could also check whether the set of Frobenius subalgebras always forms a lattice.
A stronger version of Watatani's theorem was proved in [BDLR19], providing a bound for the cardinal of the lattice. Its proof involves a new notion of angle between two intermediate subfactors and Pimsner-Popa basis. Note that the arXiv version of [BDLR19] also contains a purely planar algebraic (short) proof only involving the notion of angle between two biprojections. Thus, a generalization of the notion of angle to between two Frobenius subalgebras may help answer the above question.
Finally, note that Watatani's theorem (together with the notion of angle and the bound) was also generalized in [BG21] to intermediate ${\mathrm C}^*$-subalgebras of any irreducible inclusion of simple unital ${\mathrm C}^*$-algebras with finite index conditional expectation.
References
[BDLR19] Bakshi, Keshab Chandra; Das, Sayan; Liu, Zhengwei; Ren, Yunxiang. An angle between intermediate subfactors and its rigidity. Trans. Amer. Math. Soc. 371 (2019), no. 8, 5973--5991; arXiv:1710.00285.
[BG21] Bakshi, Keshab Chandra; Gupta, Ved Prakash. Lattice of intermediate subalgebras. J. Lond. Math. Soc. (2) 104 (2021), no. 5, 2082--2127.
[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.
[JS97] Jones, V.; Sunder, V. S. Introduction to subfactors. London Mathematical Society Lecture Note Series, 234. Cambridge University Press, Cambridge, 1997. xii+162 pp.
[W96] Watatani, Yasuo. Lattices of intermediate subfactors. J. Funct. Anal. 140 (1996), no. 2, 312--334.
