Exchange relations and Frobenius subalgebras
Mainak Ghosh, Sebastien Palcoux
TL;DR
This work generalizes the Bisch–Jones correspondence from intermediate subfactors to Frobenius subalgebras in abelian monoidal categories by introducing and exploiting exchange relations. The authors develop a graphical, categorical framework for biprojections and exchange relations on Frobenius algebras, proving that a unital selfdual idempotent $b$ on $X$ arises from a Frobenius subalgebra if and only if $b$ satisfies the exchange relations, with unitary and C$^*$-compatible versions. They provide concrete Vec/$M_n(\mathbb{C})$ analyses, classifications for low dimensions, and an application showing averaging operators on C$^*$-correspondences correspond precisely to intermediate C$^*$-subalgebras in irreducible finite-index inclusions. An appendix documents computer-assisted verifications that support the theoretical results in low-dimensional cases. The results bridge subfactor theory with abelian monoidal categories, enabling categorical analogues of intermediate-subobject phenomena and operator-algebraic characterizations via averaging operators.
Abstract
Bisch and Jones established a bijection between the intermediate subfactors of an irreducible subfactor and certain idempotents satisfying exchange relations. In this paper, we generalize this result to abelian monoidal categories through Frobenius subalgebras. As an application, we show that certain morphisms on a C*-correspondence arise from intermediate C*-subalgebras if and only if they are averaging operators.