Singly generated planar algebras of small dimension, Part III
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- by Dietmar Bisch, Vaughan F. R. Jones and Zhengwei Liu PDF
- Trans. Amer. Math. Soc. 369 (2017), 2461-2476 Request permission
Abstract:
The first two authors classified subfactor planar algebra generated by a non-trivial 2-box subject to the condition that the dimension of 3-boxes is at most 12 in Part I; 13 in Part II of this series. They are the group planar algebra for $\mathbb {Z}_3$, the Fuss-Catalan planar algebra, and the group/subgroup planar algebra for $\mathbb {Z}_2\subset \mathbb {Z}_5\rtimes \mathbb {Z}_2$. In the present paper, we extend the classification to 14 dimensional 3-boxes. They are all Birman-Murakami-Wenzl algebras. Precisely it contains a depth 3 one from quantum $O(3)$, and a one-parameter family from quantum $Sp(4)$.References
- M. Asaeda and U. Haagerup, Exotic subfactors of finite depth with Jones indices $(5+\sqrt {13})/2$ and $(5+\sqrt {17})/2$, Comm. Math. Phys. 202 (1999), no.Β 1, 1β63. MR 1686551, DOI 10.1007/s002200050574
- Anna Beliakova and Christian Blanchet, Skein construction of idempotents in Birman-Murakami-Wenzl algebras, Math. Ann. 321 (2001), no.Β 2, 347β373. MR 1866492, DOI 10.1007/s002080100233
- Dietmar Bisch, A note on intermediate subfactors, Pacific J. Math. 163 (1994), no.Β 2, 201β216. MR 1262294
- Dietmar Bisch, Principal graphs of subfactors with small Jones index, Math. Ann. 311 (1998), no.Β 2, 223β231. MR 1625762, DOI 10.1007/s002080050185
- Dietmar Bisch and Vaughan Jones, Algebras associated to intermediate subfactors, Invent. Math. 128 (1997), no.Β 1, 89β157. MR 1437496, DOI 10.1007/s002220050137
- Dietmar Bisch and Vaughan Jones, Singly generated planar algebras of small dimension, Duke Math. J. 101 (2000), no.Β 1, 41β75. MR 1733737, DOI 10.1215/S0012-7094-00-10112-3
- Dietmar Bisch and Vaughan Jones, Singly generated planar algebras of small dimension. II, Adv. Math. 175 (2003), no.Β 2, 297β318. MR 1972635, DOI 10.1016/S0001-8708(02)00060-9
- Stephen Bigelow, Emily Peters, Scott Morrison, and Noah Snyder, Constructing the extended Haagerup planar algebra, Acta Math. 209 (2012), no.Β 1, 29β82. MR 2979509, DOI 10.1007/s11511-012-0081-7
- Joan S. Birman and Hans Wenzl, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1989), no.Β 1, 249β273. MR 992598, DOI 10.1090/S0002-9947-1989-0992598-X
- Frederick M. Goodman, Pierre de la Harpe, and Vaughan F. R. Jones, Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications, vol. 14, Springer-Verlag, New York, 1989. MR 999799, DOI 10.1007/978-1-4613-9641-3
- Uffe Haagerup, Principal graphs of subfactors in the index range $4<[M:N]<3+\sqrt 2$, Subfactors (Kyuzeso, 1993) World Sci. Publ., River Edge, NJ, 1994, pp.Β 1β38. MR 1317352
- Masaki Izumi, Vaughan F. R. Jones, Scott Morrison, and Noah Snyder, Subfactors of index less than 5, Part 3: Quadruple points, Comm. Math. Phys. 316 (2012), no.Β 2, 531β554. MR 2993924, DOI 10.1007/s00220-012-1472-5
- Masaki Izumi, Application of fusion rules to classification of subfactors, Publ. Res. Inst. Math. Sci. 27 (1991), no.Β 6, 953β994. MR 1145672, DOI 10.2977/prims/1195169007
- Vaughan F. R. Jones, Scott Morrison, and Noah Snyder, The classification of subfactors of index at most 5, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no.Β 2, 277β327. MR 3166042, DOI 10.1090/S0273-0979-2013-01442-3
- V. F. R. Jones, Planar algebras, I, arXiv:math.QA/9909027.
- V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no.Β 1, 1β25. MR 696688, DOI 10.1007/BF01389127
- Vaughan F. R. Jones, Quadratic tangles in planar algebras, Duke Math. J. 161 (2012), no.Β 12, 2257β2295. MR 2972458, DOI 10.1215/00127094-1723608
- Louis H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), no.Β 2, 417β471. MR 958895, DOI 10.1090/S0002-9947-1990-0958895-7
- Zeph A. Landau, Exchange relation planar algebras, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000), 2002, pp.Β 183β214. MR 1950890, DOI 10.1023/A:1021296230310
- Z. Liu, Exchange relation planar algebras of small rank, Trans. Amer. Math. Soc., to appear.
- Scott Morrison and Emily Peters, The little desert? Some subfactors with index in the interval $(5,3+\sqrt {5})$, Internat. J. Math. 25 (2014), no.Β 8, 1450080, 51. MR 3254427, DOI 10.1142/S0129167X14500803
- Scott Morrison, David Penneys, Emily Peters, and Noah Snyder, Subfactors of index less than 5, Part 2: Triple points, Internat. J. Math. 23 (2012), no.Β 3, 1250016, 33. MR 2902285, DOI 10.1142/S0129167X11007586
- Scott Morrison, Emily Peters, and Noah Snyder, Skein theory for the $D_{2n}$ planar algebras, J. Pure Appl. Algebra 214 (2010), no.Β 2, 117β139. MR 2559686, DOI 10.1016/j.jpaa.2009.04.010
- Scott Morrison and Noah Snyder, Subfactors of index less than 5, Part 1: The principal graph odometer, Comm. Math. Phys. 312 (2012), no.Β 1, 1β35. MR 2914056, DOI 10.1007/s00220-012-1426-y
- Jun Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), no.Β 4, 745β758. MR 927059
- Adrian Ocneanu, Quantized groups, string algebras and Galois theory for algebras, Operator algebras and applications, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 136, Cambridge Univ. Press, Cambridge, 1988, pp.Β 119β172. MR 996454
- Emily Peters, A planar algebra construction of the Haagerup subfactor, Internat. J. Math. 21 (2010), no.Β 8, 987β1045. MR 2679382, DOI 10.1142/S0129167X10006380
- S. Popa, Classification of subfactors: the reduction to commuting squares, Invent. Math. 101 (1990), no.Β 1, 19β43. MR 1055708, DOI 10.1007/BF01231494
- Sorin Popa, Classification of amenable subfactors of type II, Acta Math. 172 (1994), no.Β 2, 163β255. MR 1278111, DOI 10.1007/BF02392646
- Sorin Popa, An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math. 120 (1995), no.Β 3, 427β445. MR 1334479, DOI 10.1007/BF01241137
- David Penneys and James E. Tener, Subfactors of index less than 5, Part 4: Vines, Internat. J. Math. 23 (2012), no.Β 3, 1250017, 18. MR 2902286, DOI 10.1142/S0129167X11007641
- Eric C. Rowell, On a family of non-unitarizable ribbon categories, Math. Z. 250 (2005), no.Β 4, 745β774. MR 2180373, DOI 10.1007/s00209-005-0773-1
- Eric C. Rowell, Unitarizability of premodular categories, J. Pure Appl. Algebra 212 (2008), no.Β 8, 1878β1887. MR 2414692, DOI 10.1016/j.jpaa.2007.11.004
- Stephen Sawin, Subfactors constructed from quantum groups, Amer. J. Math. 117 (1995), no.Β 6, 1349β1369. MR 1363071, DOI 10.2307/2375022
- V. S. Sunder and A. K. Vijayarajan, On the nonoccurrence of the Coxeter graphs $\beta _{2n+1},\ D_{2n+1}$ and $E_7$ as the principal graph of an inclusion of $\textrm {II}_1$ factors, Pacific J. Math. 161 (1993), no.Β 1, 185β200. MR 1237144
- Hans Wenzl, Quantum groups and subfactors of type $B$, $C$, and $D$, Comm. Math. Phys. 133 (1990), no.Β 2, 383β432. MR 1090432
Additional Information
- Dietmar Bisch
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 259989
- Email: dietmar.bisch@vanderbilt.edu
- Vaughan F. R. Jones
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 95565
- Email: vaughan.f.jones@vanderbilt.edu
- Zhengwei Liu
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 1095405
- Email: zhengweiliu@fas.harvard.edu
- Received by editor(s): November 12, 2014
- Received by editor(s) in revised form: April 8, 2015
- Published electronically: July 20, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 2461-2476
- MSC (2010): Primary 46L37; Secondary 46L10
- DOI: https://doi.org/10.1090/tran/6719
- MathSciNet review: 3592517