# From a Kac algebra subfactor to Drinfeld double

## TL;DR

This paper introduces the concept of quantum double inclusion for finite-depth subfactors, demonstrating its connection to the Drinfeld double of a finite-dimensional Kac algebra acting on the hyperfinite II_1 factor.

## Contribution

It defines quantum double inclusion for subfactors and shows its equivalence to a crossed product involving the Drinfeld double of a Kac algebra.

## Key findings

- Quantum double inclusion relates to Ocneanu's asymptotic inclusion.
- For a Kac algebra subfactor, the quantum double inclusion yields the Drinfeld double.
- The construction produces an isomorphism with a crossed product involving the 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 $II_1$ factor $R$ and $R^H$ denotes the fixed-point subalgebra. More precisely, quantum double inclusion of $R^H \subset R$ is isomorphic to $R \subset R \rtimes D(H)^{cop}$ for some outer action of $D(H)^{cop}$ on $R$ where $D(H)$ denotes the Drinfeld double of $H$.