2010/09/01 by Das, Paramita, Ghosh, Shamindra Kumar, Gupta, Ved Prakash
#46L37 #FOS: Mathematics #Operator Algebras (math.OA) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.1009.0186
We analyze the effect of pivotal structures (on a 2-category) on the planar algebra associated to a 1-cell as in \citeGho08 and come up with the notion of \em perturbations of planar algebras by weights (a concept that appeared earlier in Michael Burns' thesis \citeBur03); we establish a one-to-one correspondence between weights and pivotal structures. Using the construction of \citeGho08, to each bifinite bimodule over II1-factors, we associate a \em 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 (\citeJon) (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.