Singly generated planar algebras of small dimension. II.
type: Item
instance of: scholarly article
title: Singly generated planar algebras of small dimension. II.
zbMATH Open document ID: 1041.46048
DOI: 10.1016/S0001-8708(02)00060-9
author: Dietmar Bisch
author: V. F. R. Jones
published in: Advances in Mathematics
publication date: 2003-05-25T00:00:00Z
review text: 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)].
reviewed by: Wu Liangsen
Mathematics Subject Classification ID: 46L37
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 1915639
zbMATH Keywords: subfactor
zbMATH Keywords: planar algebra
MaRDI profile type: Publication
cites work: Exotic subfactors of finite depth with Jones indices \((5+\sqrt{13})/2\) and \((5+\sqrt{17})/2\)
cites work: Braids, Link Polynomials and a New Algebra
cites work: Q4356031
cites work: Principal graphs of subfactors with small Jones index
cites work: Algebras associated to intermediate subfactors
cites work: Singly generated planar algebras of small dimension.
cites work: Metaplectic link invariants
cites work: Q4693152
cites work: Q4245127
cites work: Application of fusion rules to classification of subfactors
cites work: Index for subfactors
cites work: On a certain value of the Kauffman polynomial
cites work: The Kauffman polynomial of links and representation theory
cites work: Entropy and index for subfactors
cites work: Markov traces on universal Jones algebras and subfactors of finite index
cites work: An axiomatization of the lattice of higher relative commutants of a subfactor
cites work: On the non-occurrence of the Coxeter graphs \(\beta_{2n+1}\), \(D_{2n+1}\) and \(E_ 7\) as the principal graph of an inclusion of II\(_ 1\) factors
cites work: Quantum groups and subfactors of type B, C, and D
description: scientific article; zbMATH DE number 1915639