Realizations of algebra objects and discrete subfactors

Abstract

We give a characterization of extremal irreducible discrete subfactors (N⊆M,E) where N is type II1 in terms of connected W*-algebra objects in rigid C*-tensor categories. We prove an equivalence of categories where the morphisms for discrete inclusions are normal N−N bilinear ucp maps which preserve the state τ∘E, and the morphisms for W*-algebra objects are categorical ucp morphisms.As an application, we get a well-behaved notion of the standard invariant of an extremal irreducible discrete subfactor, together with a subfactor reconstruction theorem. Thus our equivalence provides many new examples of discrete inclusions (N⊆M,E), in particular, examples where M is type III coming from non Kac-type discrete quantum groups and associated module W*-categories. Finally, we obtain a Galois correspondence between intermediate subfactors of an extremal irreducible discrete inclusion and intermediate W*-algebra objects.