Quantum double inclusions associated to a family of Kac algebra subfactors

Abstract

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 $I{I}_1$ 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 ${\mathcal{N}}^m=((\cdots ⋊H^{-2}⋊H^{-1})\otimes (H^m⋊H^{m+1}\cdots ){)}^{\prime \prime },\mathcal{M}=(\cdots ⋊H^{-1}⋊H^0⋊H^1⋊\cdots {)}^{\prime \prime }$ and where for any integer $i$, $H^i$ denotes $H$ or $H^{\ast }$ 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 ${}^{\ast (m)}P(H^m)$ (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 $C^{\ast }$-algebra structure on $({\mathcal{N}}^m{)}^{\prime }\cap {\mathcal{M}}_2$, thus obtaining a family of weak Hopf $C^{\ast }$-algebras starting with a single Kac algebra $H$.

arXiv admin note: substantial text overlap with arXiv:1812.05071