Abstract

In this paper, we extend Landau’s notion of ‘exchange relations’ so as to make sense for arbitrary planar algebras, which need not necessarily be generated by its ‘2-boxes’. We show, as in Landau’s case, that these ‘higher exchange relation planar algebras’ are necessarily ‘finite dimensional’, and that examples of such planar algebras are given by all (even possibly reducible) depth two subfactors, as well as planar algebras associated to subfactors with principal graphs E₆ and E₈.

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