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Planar Algebras in Braided Tensor Categories
About this Title
André Gil Henriques, David Penneys and James Tener
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 282, Number 1392
ISBNs: 978-1-4704-5540-8 (print); 978-1-4704-7348-8 (online)
DOI: https://doi.org/10.1090/memo/1392
Published electronically: January 3, 2023
Table of Contents
Chapters
- 1. Introduction
- 2. Anchored planar algebras
- 3. The main theorem and examples
- 4. Constructing anchored planar algebras
- 5. Anchored planar algebras from module tensor categories
- 6. Module tensor categories from anchored planar algebras
- 7. Equivalence of categories
- A. An associativity type relation
- B. Anchored planar tangles with coupons
- C. The tube string calculus for the categorified trace
Abstract
We generalize Jones’ planar algebras by internalising the notion to a pivotal braided tensor category $\mathcal {C}$. To formulate the notion, the planar tangles are now equipped with additional ‘anchor lines’ which connect the inner circles to the outer circle. We call the resulting notion an anchored planar algebra. If we restrict to the case when $\mathcal {C}$ is the category of vector spaces, then we recover the usual notion of a planar algebra.
Building on our previous work on categorified traces, we prove that there is an equivalence of categories between anchored planar algebras in $\mathcal {C}$ and pivotal module tensor categories over $\mathcal {C}$ equipped with a chosen self-dual generator. Even in the case of usual planar algebras, the precise formulation of this theorem, as an equivalence of categories, has not appeared in the literature. Using our theorem, we describe many examples of anchored planar algebras.
- N. Afzaly, S. Morrison, and D. Penneys, The classification of subfactors with index at most $5\frac {1}{4}$, 2015, arXiv:1509.00038, to appear Mem. Amer. Math. Soc.
- Jens Böckenhauer, David E. Evans, and Yasuyuki Kawahigashi, Longo-Rehren subfactors arising from $\alpha$-induction, Publ. Res. Inst. Math. Sci. 37 (2001), no. 1, 1–35. MR 1815993
- Arnaud Brothier, Michael Hartglass, and David Penneys, Rigid $C^*$-tensor categories of bimodules over interpolated free group factors, J. Math. Phys. 53 (2012), no. 12, 123525, 43. MR 3405915, DOI 10.1063/1.4769178
- Stephen Bigelow, Emily Peters, Scott Morrison, and Noah Snyder, Constructing the extended Haagerup planar algebra, Acta Math. 209 (2012), no. 1, 29–82. MR 2979509, DOI 10.1007/s11511-012-0081-7
- J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. MR 420609
- David Clark, Scott Morrison, and Kevin Walker, Fixing the functoriality of Khovanov homology, Geom. Topol. 13 (2009), no. 3, 1499–1582. MR 2496052, DOI 10.2140/gt.2009.13.1499
- Alexei Davydov, Michael Müger, Dmitri Nikshych, and Victor Ostrik, The Witt group of non-degenerate braided fusion categories, J. Reine Angew. Math. 677 (2013), 135–177. MR 3039775, DOI 10.1515/crelle.2012.014
- David E. Evans and Terry Gannon, Near-group fusion categories and their doubles, Adv. Math. 255 (2014), 586–640. MR 3167494, DOI 10.1016/j.aim.2013.12.014
- David E. Evans and Mathew Pugh, $A_2$-planar algebras I, Quantum Topol. 1 (2010), no. 4, 321–377. MR 2733244, DOI 10.4171/QT/8
- Z. Fiedorowicz, Symmetric bar construction, Available at https://people.math.osu.edu/fiedorowicz.1/.
- Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125
- Jürgen Fuchs, Ingo Runkel, and Christoph Schweigert, TFT construction of RCFT correlators. III. Simple currents, Nuclear Phys. B 694 (2004), no. 3, 277–353. MR 2076134, DOI 10.1016/j.nuclphysb.2004.05.014
- Satoshi Goto, On Ocneanu’s theory of double triangle algebras for subfactors and classification of irreducible connections on the Dynkin diagrams, Expo. Math. 28 (2010), no. 3, 218–253. MR 2670999, DOI 10.1016/j.exmath.2009.11.001
- André G. Henriques, What Chern-Simons theory assigns to a point, Proc. Natl. Acad. Sci. USA 114 (2017), no. 51, 13418–13423. MR 3747830, DOI 10.1073/pnas.1711591114
- André Henriques and David Penneys, Bicommutant categories from fusion categories, Selecta Math. (N.S.) 23 (2017), no. 3, 1669–1708. MR 3663592, DOI 10.1007/s00029-016-0251-0
- André Henriques, David Penneys, and James Tener, Categorified trace for module tensor categories over braided tensor categories, Doc. Math. 21 (2016), 1089–1149. MR 3578212
- Masaki Izumi, On type $\textrm {II}$ and type $\textrm {III}$ principal graphs of subfactors, Math. Scand. 73 (1993), no. 2, 307–319. MR 1269266, DOI 10.7146/math.scand.a-12473
- Masaki Izumi, The structure of sectors associated with Longo-Rehren inclusions. II. Examples, Rev. Math. Phys. 13 (2001), no. 5, 603–674. MR 1832764, DOI 10.1142/S0129055X01000818
- Masaki Izumi, A Cuntz algebra approach to the classification of near-group categories, Proceedings of the 2014 Maui and 2015 Qinhuangdao conferences in honour of Vaughan F. R. Jones’ 60th birthday, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 46, Austral. Nat. Univ., Canberra, 2017, pp. 222–343. MR 3635673
- Arthur Jaffe and Zhengwei Liu, Planar para algebras, reflection positivity, Comm. Math. Phys. 352 (2017), no. 1, 95–133. MR 3623255, DOI 10.1007/s00220-016-2779-4
- A. Jaffe, Z. Liu, and A. Wozniakowski, Qudit isotopy, 2016, arXiv:1602.02671.
- 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. 2, 277–327. MR 3166042, DOI 10.1090/S0273-0979-2013-01442-3
- V. F. R. Jones, Planar algebras I, 1999, arXiv:math.QA/9909027.
- Vaughan F. R. Jones and David Penneys, The embedding theorem for finite depth subfactor planar algebras, Quantum Topol. 2 (2011), no. 3, 301–337. MR 2812459, DOI 10.4171/QT/23
- Jacob Lurie, On the classification of topological field theories, Current developments in mathematics, 2008, Int. Press, Somerville, MA, 2009, pp. 129–280. MR 2555928
- J. P. May, The geometry of iterated loop spaces, Lecture Notes in Mathematics, Vol. 271, Springer-Verlag, Berlin-New York, 1972. MR 420610
- Scott Morrison, Emily Peters, and Noah Snyder, Skein theory for the $D_{2n}$ planar algebras, J. Pure Appl. Algebra 214 (2010), no. 2, 117–139. MR 2559686, DOI 10.1016/j.jpaa.2009.04.010
- Michael Müger, From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors, J. Pure Appl. Algebra 180 (2003), no. 1-2, 159–219. MR 1966525, DOI 10.1016/S0022-4049(02)00248-7
- David Penneys, A cyclic approach to the annular Temperley-Lieb category, J. Knot Theory Ramifications 21 (2012), no. 6, 1250049, 40. MR 2903179, DOI 10.1142/S0218216511010012
- Emily Peters, A planar algebra construction of the Haagerup subfactor, Internat. J. Math. 21 (2010), no. 8, 987–1045. MR 2679382, DOI 10.1142/S0129167X10006380
- S. Popa, Classification of subfactors: the reduction to commuting squares, Invent. Math. 101 (1990), no. 1, 19–43. MR 1055708, DOI 10.1007/BF01231494
- Sorin Popa, Classification of subfactors and their endomorphisms, CBMS Regional Conference Series in Mathematics, vol. 86, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1995. MR 1339767, DOI 10.1090/cbms/086
- Kenichi Shimizu, Frobenius-Schur indicators in Tambara-Yamagami categories, J. Algebra 332 (2011), 543–564. MR 2774703, DOI 10.1016/j.jalgebra.2011.02.002
- Daisuke Tambara and Shigeru Yamagami, Tensor categories with fusion rules of self-duality for finite abelian groups, J. Algebra 209 (1998), no. 2, 692–707. MR 1659954, DOI 10.1006/jabr.1998.7558
- S. Yamagami, Representations of multicategories of planar diagrams and tensor categories, 2012, arXiv:1207.1923.