Quantum double inclusions associated to a family of Kac algebra subfactors

We defined the notion of the quantum double inclusion [S. De, J. Math. Phys. 60, 071701 (2019)1 associated with a finite -index and finitedepth subfactor, which is closely related to that of Ocneanu's asymptotic inclusion, and studied the quantum double inclusion associated with the Kac algebra subfactor R11 ci R, where H is a finite dimensionalKac algebra acting outerly on the hyperfinite Hi factor 11 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 {R" aRxH>4.1 H* >4 : m> 1}. For each m > 2, we construct a model Arm c [A,41. for the quantum double inclusion of RH- a 11 >4 44, [me, >4 H-2 >4 H-4)H H*.>"1 " " ", where Arm = (H"' H''>4 " " "))",./141 = H -1i le >c Hi x " " ")", and for any integer i, the notation rn-2 times H' 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 Arm c M (m > 2) which turns out to be a planar subalgebra of *(m)P(Hm) (the adjoint of the m -cabling of the planar algebra of H"'). We then show that for each m > 2, the depth of Arm c lulls always two. Observing that Arm c.,4141 is reducible for all ma 2, we study in great detail the weak Kac algebra structure of the relative commutant (Am)' nI\42.