Abstract
In recent joint work with Jones and Shlyakhtenko, we have given a diagrammatic description of Popa's symmetric enveloping inclusion for planar algebra subfactors. In this paper, we give a diagrammatic construction of the associated Jones tower, in the case that the planar algebra is finite-depth. We then use this construction to describe the planar algebra of the symmetric enveloping inclusion, which is known to be isomorphic to the planar algebra of Ocneanu's asymptotic inclusion by a result of Popa. As an application we give a planar algebraic computation of the (reduced) fusion algebra of the asymptotic inclusion, recovering some well-known results of Ocneanu and Evans–Kawahigashi.
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