Planar depth and planar subalgebras

Abstract

We consider the notion of planar depth of a planar algebra, viz., the smallest n for which the planar algebra is generated by its 'n-boxes'. We establish a simple result which yields a sufficient condition, in terms of the principal graph of the planar algebra, for the planar depth to be bounded by k. This suffices to determine the planar depth of the E₆, E ₈ and the 5+√13/2 subfactors. We then consider a planar subalgebra of the 'group planar algebra' which is naturally associated with a group θ of automorphisms of the given group G. We show that this planar algebra corresponds to the 'subgroup-subfactor' associated with the inclusion θ⊂(G⋊θ) (given by the semi-direct product extension). We conclude with a discussion of the planar depth of this planar algebra P^θ in some examples.