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Quantum double inclusions associated to a family of Kac algebra subfactors

In \cite{Sde2018} we defined the notion of \textit{quantum double inclusion} associated to a finite-index and finite-depth subfactor and studied the quantum double inclusion associated to the Kac algebra subfactor $R^H \subset R$ 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. In this article we analyse quantum double inclusions associated to the family of Kac algebra subfactors given by $\{ R^H \subset R \rtimes \underbrace{H \rtimes H^* \rtimes \cdots}_{\text{$m$ times}} : m \geq 1 \}$. For each $m > 2$, we construct a model $\mathcal{N}^m \subset \mathcal{M}$ for the quantum double inclusion of $\{ R^H \subset R \rtimes \underbrace{H \rtimes H^* \rtimes \cdots}_{\text{$m-2$ times}} \}$ with $\mathcal{N}^m = ((\cdots \rtimes H^{-2} \rtimes H^{-1}) \otimes (H^m \rtimes H^{m+1} \cdots))^{\prime \prime}, \mathcal{M} = (\cdots \rtimes H^{-1} \rtimes H^0 \rtimes H^1 \rtimes \cdots)^{\prime \prime}$ and where for any integer $i$, $H^i$ denotes $H$ or $H^*$ according as $i$ is odd or even. In this article, we give an explicit description of $P^{\mathcal{N}^m \subset \mathcal{M}}$ ($m > 2$), the subfactor planar algebra associated to $\mathcal{N}^m \subset \mathcal{M}$, which turns out to be a planar subalgebra of $^{*(m)}\!P(H^m)$ (the adjoint of the $m$-cabling of the planar algebra of $H^m$). We then show that for $m > 2$, depth of $\mathcal{N}^m \subset \mathcal{M}$ is always two. Observing that $\mathcal{N}^m \subset \mathcal{M}$ is reducible for all $m > 2$, we explicitly describe the weak Hopf $C^*$-algebra structure on $(\mathcal{N}^m)^{\prime} \cap \mathcal{M}_2$, thus obtaining a family of weak Hopf $C^*$-algebras starting with a single Kac algebra $H$.