Frobenius subalgebra lattices in tensor categories

Abstract

This paper generalizes Watatani's finiteness theorem for intermediate subfactors to a wide class of monoidal categories. We characterize the sublattices of Frobenius subalgebra posets in abelian monoidal categories by introducing a notion of ambient selfduality. By extending several key results -- such as the planar algebraic exchange relation and Landau's theorems -- to linear monoidal categories, we establish a structural rigidity property for a formal angle associated to every coherent pair of Frobenius subalgebras (that is, whose intersection and sum are ambiently selfdual). Furthermore, within a weak positivity framework, we deduce that such coherent sublattices are finite for any connected Frobenius algebra. This significantly generalizes Watatani's theorem, since the unitary Frobenius subalgebra lattices are inherently coherent through a property termed rigidity invariance. Applications of this work include a unified framework that encompasses several previously unrelated finiteness results. Specifically, we recover the finiteness of the left coideal subalgebra lattice of a finite-dimensional semisimple Hopf algebra (Etingof-Walton theorem) modulo a well-supported coherence hypothesis, as well as the finiteness of the intermediate C*-algebra lattice for a finite-index unital irreducible inclusion of C*-algebras (relaxing simplicity in Ino-Watatani theorem) under an E-compatibility condition shown to be unavoidable. Furthermore, we present a variety of new applications involving abstract spin chains and vertex operator algebras, alongside speculations on quantum arithmetic that include extensions of Ore's theorem, Euler's totient and sigma functions, and RH.

52 pages. Significantly revised. Removed semisimplicity assumption via new concepts of ambient selfduality and coherent sublattices. Major updates in the C*-algebras and Hopf algebras sections. Comments are welcome!