Frobenius subalgebra lattices in tensor categories

This paper studies Frobenius subalgebra posets in abelian monoidal categories and shows that, under general conditions--satisfied in all semisimple tensor categories over the complex field--they collapse to lattices through a rigidity invariance perspective. Based on this, we extend Watatani's finiteness theorem for intermediate subfactors by proving that, under a weak positivity assumption--met by all semisimple tensor categories over the complex field--and a compatibility condition--fulfilled by all pivotal ones--the lattices arising from connected Frobenius algebras are finite. We also derive a non-semisimple version via semisimplification. Our approach relies on the concept of a formal angle, and the extension of key results--such as the planar algebraic exchange relation and Landau's theorems--to linear monoidal categories. Major applications of our findings include a stronger version of the Ino-Watatani result: we show that the finiteness of intermediate C*-algebras holds in a finite-index unital irreducible inclusion of C*-algebras without requiring the simple assumption. Moreover, for a finite-dimensional semisimple Hopf algebra H, we prove that H* is a Frobenius algebra object in Rep(H) and has a finite number of rigid invariant Frobenius subalgebras. Finally, we explore a range of applications, including abstract spin chains, vertex operator algebras and speculations on quantum arithmetic involving the generalization of Ore's theorem, Euler's totient and sigma functions, and RH.