×
Bisch, Dietmar; Jones, Vaughan F. R.; Liu, Zhengwei

Singly generated planar algebras of small dimension. III. (English) Zbl 1370.46036

Trans. Am. Math. Soc. 369, No. 4, 2461-2476 (2017).
Summary: 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 (see [D. Bisch and V. Jones, Duke Math. J. 101, No. 1, 41–75 (2000; Zbl 1075.46053)]); 13 in Part II (see [D. Bisch and V. Jones, Adv. Math. 175, No. 2, 297–318 (2003; Zbl 1041.46048)]) 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 \( Sp(4)\).

Cited in 1 Review
Cited in 9 Documents

MSC:

46L37 Subfactors and their classification
46L10 General theory of von Neumann algebras

Keywords:

subfactor planar algebra; non-trivial 2-box; dimension; Birman-Murakami-Wenzl algebras

Citations:

Zbl 1075.46053; Zbl 1041.46048
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Asaeda, M.; Haagerup, U., Exotic subfactors of finite depth with Jones indices \((5+\sqrt{13})/2\) and \((5+\sqrt{17})/2\), Comm. Math. Phys., 202, 1, 1-63 (1999) · Zbl 1014.46042 · doi:10.1007/s002200050574
[2] Beliakova, Anna; Blanchet, Christian, Skein construction of idempotents in Birman-Murakami-Wenzl algebras, Math. Ann., 321, 2, 347-373 (2001) · Zbl 1015.16041 · doi:10.1007/s002080100233
[3] Bisch, Dietmar, A note on intermediate subfactors, Pacific J. Math., 163, 2, 201-216 (1994) · Zbl 0814.46053
[4] Bisch, Dietmar, Principal graphs of subfactors with small Jones index, Math. Ann., 311, 2, 223-231 (1998) · Zbl 0927.46040 · doi:10.1007/s002080050185
[5] Bisch, Dietmar; Jones, Vaughan, Algebras associated to intermediate subfactors, Invent. Math., 128, 1, 89-157 (1997) · Zbl 0891.46035 · doi:10.1007/s002220050137
[6] Bisch, Dietmar; Jones, Vaughan, Singly generated planar algebras of small dimension, Duke Math. J., 101, 1, 41-75 (2000) · Zbl 1075.46053 · doi:10.1215/S0012-7094-00-10112-3
[7] Bisch, Dietmar; Jones, Vaughan, Singly generated planar algebras of small dimension. II, Adv. Math., 175, 2, 297-318 (2003) · Zbl 1041.46048 · doi:10.1016/S0001-8708(02)00060-9
[8] Bigelow, Stephen; Peters, Emily; Morrison, Scott; Snyder, Noah, Constructing the extended Haagerup planar algebra, Acta Math., 209, 1, 29-82 (2012) · Zbl 1270.46058 · doi:10.1007/s11511-012-0081-7
[9] Birman, Joan S.; Wenzl, Hans, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc., 313, 1, 249-273 (1989) · Zbl 0684.57004 · doi:10.2307/2001074
[10] Goodman, Frederick M.; de la Harpe, Pierre; Jones, Vaughan F. R., Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications 14, x+288 pp. (1989), Springer-Verlag, New York · Zbl 0698.46050 · doi:10.1007/978-1-4613-9641-3
[11] Haagerup, Uffe, Principal graphs of subfactors in the index range \(4<[M:N]<3+\sqrt 2\). Subfactors, Kyuzeso, 1993, 1-38 (1994), World Sci. Publ., River Edge, NJ · Zbl 0933.46058
[12] Izumi, Masaki; Jones, Vaughan F. R.; Morrison, Scott; Snyder, Noah, Subfactors of index less than 5, Part 3: Quadruple points, Comm. Math. Phys., 316, 2, 531-554 (2012) · Zbl 1272.46051 · doi:10.1007/s00220-012-1472-5
[13] Izumi, Masaki, Application of fusion rules to classification of subfactors, Publ. Res. Inst. Math. Sci., 27, 6, 953-994 (1991) · Zbl 0765.46048 · doi:10.2977/prims/1195169007
[14] Jones, Vaughan F. R.; Morrison, Scott; Snyder, Noah, The classification of subfactors of index at most 5, Bull. Amer. Math. Soc. (N.S.), 51, 2, 277-327 (2014) · Zbl 1301.46039 · doi:10.1090/S0273-0979-2013-01442-3
[15] V. F. R. Jones, Planar algebras, I, arXiv:math.QA/9909027. · Zbl 1328.46049
[16] Jones, V. F. R., Index for subfactors, Invent. Math., 72, 1, 1-25 (1983) · Zbl 0508.46040 · doi:10.1007/BF01389127
[17] Jones, Vaughan F. R., Quadratic tangles in planar algebras, Duke Math. J., 161, 12, 2257-2295 (2012) · Zbl 1257.46033 · doi:10.1215/00127094-1723608
[18] Kauffman, Louis H., An invariant of regular isotopy, Trans. Amer. Math. Soc., 318, 2, 417-471 (1990) · Zbl 0763.57004 · doi:10.2307/2001315
[19] Landau, Zeph A., Exchange relation planar algebras, Geom. Dedicata. Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000), 95, 183-214 (2002) · Zbl 1022.46039 · doi:10.1023/A:1021296230310
[20] Z. Liu, Exchange relation planar algebras of small rank, Trans. Amer. Math. Soc., to appear. · Zbl 1365.46053
[21] Morrison, Scott; Peters, Emily, The little desert? Some subfactors with index in the interval \((5,3+\sqrt{5})\), Internat. J. Math., 25, 8, 1450080, 51 pp. (2014) · Zbl 1314.46074 · doi:10.1142/S0129167X14500803
[22] Morrison, Scott; Penneys, David; Peters, Emily; Snyder, Noah, Subfactors of index less than 5, Part 2: Triple points, Internat. J. Math., 23, 3, 1250016, 33 pp. (2012) · Zbl 1246.46054 · doi:10.1142/S0129167X11007586
[23] Morrison, Scott; Peters, Emily; Snyder, Noah, Skein theory for the \(D_{2n}\) planar algebras, J. Pure Appl. Algebra, 214, 2, 117-139 (2010) · Zbl 1191.46051 · doi:10.1016/j.jpaa.2009.04.010
[24] Morrison, Scott; Snyder, Noah, Subfactors of index less than 5, Part 1: The principal graph odometer, Comm. Math. Phys., 312, 1, 1-35 (2012) · Zbl 1246.46055 · doi:10.1007/s00220-012-1426-y
[25] Murakami, Jun, The Kauffman polynomial of links and representation theory, Osaka J. Math., 24, 4, 745-758 (1987) · Zbl 0666.57006
[26] Ocneanu, Adrian, Quantized groups, string algebras and Galois theory for algebras. Operator algebras and applications, Vol.2, London Math. Soc. Lecture Note Ser. 136, 119-172 (1988), Cambridge Univ. Press, Cambridge · Zbl 0696.46048
[27] Peters, Emily, A planar algebra construction of the Haagerup subfactor, Internat. J. Math., 21, 8, 987-1045 (2010) · Zbl 1203.46039 · doi:10.1142/S0129167X10006380
[28] Popa, S., Classification of subfactors: the reduction to commuting squares, Invent. Math., 101, 1, 19-43 (1990) · Zbl 0757.46054 · doi:10.1007/BF01231494
[29] Popa, Sorin, Classification of amenable subfactors of type II, Acta Math., 172, 2, 163-255 (1994) · Zbl 0853.46059 · doi:10.1007/BF02392646
[30] Popa, Sorin, An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math., 120, 3, 427-445 (1995) · Zbl 0831.46069 · doi:10.1007/BF01241137
[31] Penneys, David; Tener, James E., Subfactors of index less than 5, Part 4: Vines, Internat. J. Math., 23, 3, 1250017, 18pp. pp. (2012) · Zbl 1246.46056 · doi:10.1142/S0129167X11007641
[32] Rowell, Eric C., On a family of non-unitarizable ribbon categories, Math. Z., 250, 4, 745-774 (2005) · Zbl 1137.17015 · doi:10.1007/s00209-005-0773-1
[33] Rowell, Eric C., Unitarizability of premodular categories, J. Pure Appl. Algebra, 212, 8, 1878-1887 (2008) · Zbl 1184.17007 · doi:10.1016/j.jpaa.2007.11.004
[34] Sawin, Stephen, Subfactors constructed from quantum groups, Amer. J. Math., 117, 6, 1349-1369 (1995) · Zbl 0874.46040 · doi:10.2307/2375022
[35] Sunder, V. S.; Vijayarajan, A. K., On the nonoccurrence of the Coxeter graphs \(\beta_{2n+1},\ D_{2n+1}\) and \(E_7\) as the principal graph of an inclusion of \({\rm II}_1\) factors, Pacific J. Math., 161, 1, 185-200 (1993) · Zbl 0798.43005
[36] Wenzl, Hans, Quantum groups and subfactors of type \(B, C\), and \(D\), Comm. Math. Phys., 133, 2, 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.