Perturbations of Planar Algebras

Abstract

We analyze the effect of pivotal structures (on a $2$-category) on the planar algebra associated to a $1$-cell as in [8] and come up with the notion of perturbations of planar algebras by weights (a concept that appeared earlier in Michael Burns' thesis [6]); we establish a one-to-one correspondence between weights and pivotal structures. Using the construction of [8], to each bifinite bimodule over $\mathit{II}_1$-factors, we associate a bimodule planar algebra in such a way that extremality of the bimodule corresponds to sphericality of the planar algebra. As a consequence of this, we reproduce an extension of Jones' theorem ([13]) (of associating 'subfactor planar algebras' to extremal subfactors). Conversely, given a bimodule planar algebra, we construct a bifinite bimodule whose associated bimodule planar algebra is the one which we start with, using perturbations and Jones-Walker-Shlyakhtenko-Kodiyalam-Sunder method of reconstructing an extremal subfactor from a subfactor planar algebra. The perturbation technique helps us to construct an example of a family of non-spherical planar algebras starting from a particular spherical one; we also show that this family is associated to a known family of subfactors constructed by Jones.