Singly generated planar algebras of small dimension. II. (Q1874470)

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)].