From a Kac algebra subfactor to Drinfeld double

Abstract

Given a finite-index and finite-depth subfactor, we define the notion of \textit{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 $R^H\subset R$ produces Drinfeld double of $H$ where $H$ is a finite-dimensional Kac algebra acting outerly on the hyperfinite $I{I}_1$ factor $R$ and $R^H$ denotes the fixed-point subalgebra. More precisely, quantum double inclusion of $R^H\subset R$ is isomorphic to $R\subset R⋊D(H{)}^{cop}$ for some outer action of $D(H{)}^{cop}$ on $R$ where $D(H)$ denotes the Drinfeld double of $H$.

Minor modification in the abstract, added a short paragraph in the introduction, modified Figures 8, 9, 10, and 11, changed caption of a figure, added a new figure- Figure 13, added a new reference [19]