zbMATH — the first resource for mathematics

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)].

46L37 Subfactors and their classification
46L10 General theory of von Neumann algebras
Full Text: DOI
[1] Asaeda, M.; Haagerup, U., Exotic subfactors of finite depth with Jones indices \((5+13)/2\) and \((5+17)/2\), Comm. math. phys., 202, 1-63, (1999) · Zbl 1014.46042
[2] Birman, J.; Wenzl, H., Braids, link polynomials and a new algebra, Trans. amer. math. soc., 313, 249-273, (1989) · Zbl 0684.57004
[3] D. Bisch Bimodules, higher relative commutants and the fusion algebra associated to a subfactor in: The Fields Institute for Research in Mathematical Sciences Communications Series, Vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 13-63. · Zbl 0894.46046
[4] Bisch, D., Principal graphs of subfactors with small Jones index, Math. annalen, 311, 223-231, (1998) · Zbl 0927.46040
[5] Bisch, D.; Jones, V.F.R., Algebras associated to intermediate subfactors, Invent. math., 128, 89-157, (1997) · Zbl 0891.46035
[6] Bisch, D.; Jones, V.F.R., Singly generated planar algebras of small dimension, Duke math. J., 101, 41-75, (2000) · Zbl 1075.46053
[7] Fateev, V.; Zamolodchikov, A., Self-dual solutions of the star-triangle relations in \(ZN\) models, Phys. lett., 92A, 37-39, (1982)
[8] Goldschmidt, D.; Jones, V., Metaplectic link invariants, Geom. dedicata, 31, 165-191, (1989) · Zbl 0678.57007
[9] F. Goodman, P. de la Harpe, V. Jones, Coxeter Graphs and Towers of Algebras, Springer, MSRI Publications, 1989. · Zbl 0698.46050
[10] U. Haagerup, Principal Graphs of Subfactors in the Index Range \(4<[M:N]<3 + 2\) Subfactors (Kyuzeso, 1993), World Scientific Publishing, River Edge, NJ, 1994, pp. 1-38. · Zbl 0933.46058
[11] Izumi, M., Applications of fusion rules to classification of subfactors, Publ. RIMS, Kyoto univ., 27, 953-994, (1991) · Zbl 0765.46048
[12] Jones, V.F.R., Index for subfactors, Invent. math., 72, 1-25, (1983) · Zbl 0508.46040
[13] Jones, V.F.R., On a certain value of the kauffman polynomial, Comm. math. phys., 125, 459-467, (1989) · Zbl 0695.57003
[14] V.F.R. Jones Planar algebras I, preprint, see http://www.math.berkeley.edu/ vfr.
[15] Z. Landau Exchange relation planar algebras, preprint, 2000.
[16] Murakami, J., The kauffman polynomial of links and representation theory, Osaka J. math., 24, 745-758, (1987) · Zbl 0666.57006
[17] Pimsner, M.; Popa, S., Entropy and index for subfactors, Ann. scient. ec. norm. sup., 19, 57-106, (1986) · Zbl 0646.46057
[18] Popa, S., Markov traces on universal Jones algebras and subfactors of finite index, Invent. math., 111, 375-405, (1993) · Zbl 0787.46047
[19] Popa, S., Classification of amenable subfactors of type II, Acta math., 172, 352-445, (1994)
[20] Popa, S., An axiomatizaton of the lattice of higher relative commutants, Invent. math., 120, 427-445, (1995) · Zbl 0831.46069
[21] Sunder, V.S.; Vijayarajan, A.K., On the nonoccurrence of the Coxeter graphs β2n+1, D2n+1 and E7 as the principal graph of an inclusion of II1 factors, Pacific J. math., 161, 185-200, (1993) · Zbl 0798.43005
[22] Wenzl, H., Quantum groups and subfactors of type B, C and D, Comm. math. phys., 133, 383-432, (1990) · Zbl 0744.17021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.