# Quantum double inclusions associated to a family of Kac algebra subfactors

## Article Type

Research Article

## Publication Title

Journal of Mathematical Physics

## Abstract

We defined the notion of the quantum double inclusion [S. De, J. Math. Phys. 60, 071701 (2019)] associated with a finite-index and finite-depth subfactor, which is closely related to that of Ocneanu's asymptotic inclusion, and studied the quantum double inclusion associated with the Kac algebra subfactor RH ⊂ R, where H is a finite-dimensional Kac algebra acting outerly on the hyperfinite II 1 factor R and RH denotes the fixed-point subalgebra. In this article, we analyze quantum double inclusions associated with the family of Kac algebra subfactors given by {RH ⊂ R ⋊ H ⋊ H∗ ⋊ ⋯ m ≥ 1}. For each m > 2, we construct a model N m ⊂ M for the quantum double inclusion of R H ⊂ R ⋊ H ⋊ H∗ ⋊ ⋯ m-2 times, where N m = ((⋯ ⋊ H-2 ⋊ H-1) ⊗ (H m ⋊ H m + 1 ⋊ ⋯)) ′ ′, M = (⋯ ⋊ H-1 ⋊ H 0 ⋊ H 1 ⋊ ⋯) ′ ′, and for any integer i, the notation H i stands for H or H∗ according as i is odd or even. In this article, we give an explicit description of the subfactor planar algebra associated with N m ⊂ M (m > 2) which turns out to be a planar subalgebra of P∗ (m) (H m) (the adjoint of the m-cabling of the planar algebra of H m). We then show that for each m > 2, the depth of N m ⊂ M is always two. Observing that N m ⊂ M is reducible for all m > 2, we study in great detail the weak Kac algebra structure of the relative commutant (N m) ′ ∩ M 2.

## DOI

10.1063/1.5132346

## Publication Date

1-1-2020

## Recommended Citation

De, Sandipan, "Quantum double inclusions associated to a family of Kac algebra subfactors" (2020). *Journal Articles*. 533.

https://digitalcommons.isical.ac.in/journal-articles/533

## Comments

Open Access, Green