From a Kac algebra subfactor to Drinfeld double

Abstract

Given a finite-index and finite-depth subfactor, we define the notion of quantum double inclusion—a certain unital inclusion of von Neumann algebras constructed from the given subfactor—which is closely related to that of Ocneanu’s asymptotic inclusion. We show that the quantum double inclusion when applied to the Kac algebra subfactor RH ⊂ R produces Drinfeld double of H, where H is a finite-dimensional Kac algebra acting outerly on the hyperfinite II1 factor R and RH denotes the fixed-point subalgebra. More precisely, quantum double inclusion of RH ⊂ R is isomorphic to R ⊂ R ⋊ D(H)cop for some outer action of D(H)cop on R, where D(H) denotes the Drinfeld double of H.