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Singly generated planar algebras of small dimension. II. (English) Zbl 1041.46048
The following theorem is proved: Let $${\mathcal P}= \{P_k\}_{k\geq 0}$$ be a spherical $$C^*$$-planar algebra generated by a non-trivial element in $$P_2$$ (i.e. an element not contained in the Temperley-Lieb subalgebra of $$P_2$$) subject to the conditions $$\dim P_2= 3$$ and $$\dim P_3= 13$$. Then $${\mathcal P}$$ is the standard invariant of the crossed product subfactor $$R\ltimes \mathbb{Z}_2\subset R\ltimes D_5$$. Thus there is precisely one spherical $$C^*$$-planar algebra $${\mathcal P}$$ subject to the above conditions.
This somewhat unexpected result complements the classification of planar algebras arising from subfactors with $$\dim P_3\leq 12$$ which the authors gave in Part I of this paper [Duke Math. J. 101, 41–75 (2000; Zbl 1075.46053)].

##### MSC:
 46L37 Subfactors and their classification 46L10 General theory of von Neumann algebras
##### Keywords:
subfactor; planar algebra
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##### References:
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