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Growing out of the initial connections between subfactors and knot theory that gave rise to the Jones polynomial, Jones' axiomatization of the standard invariant of an extremal finite index $II_1$ subfactor as a spherical $C^*$-planar algebra, presented in arXiv:math.QA/9909027, is the most elegant and powerful description available. We make the natural extension of this axiomatization to the case of finite index subfactors of arbitrary type. We also provide the first steps toward a limited planar structure in the infinite index case. The central role of rotations, which provide the main non-trivial part of the planar structure, is a recurring theme throughout this work. In the finite index case the axioms of a $C^*$-planar algebra need to be weakened to disallow rotation of internal discs, giving rise to the notion of a rigid $C^*$-planar algebra. We show that the standard invariant of any finite index subfactor has a rigid $C^*$-planar algebra structure. We then show that rotations can be re-introduced with associated correction terms entirely controlled by the Radon-Nikodym derivative of the two canonical states on the first relative commutant, $N' \cap M$. By deforming a rigid

Author

Burns, Michael

Published

2011

Language

EN