Frobenius subalgebra lattices in tensor categories

TL;DR

This paper investigates Frobenius subalgebra lattices in tensor categories, establishing their finiteness under certain conditions, and extends classical results to broader categorical contexts with applications in operator algebras and quantum algebra.

Contribution

It introduces a framework for understanding Frobenius subalgebra lattices in tensor categories, proving their finiteness and extending key theorems to non-semisimple cases with diverse applications.

Findings

Frobenius subalgebra lattices collapse to finite lattices in semisimple tensor categories.

Extension of Watatani's finiteness theorem to broader categorical settings.

Application of results to C*-algebra inclusions, Hopf algebras, and quantum algebra contexts.

Abstract

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.