On the symmetric enveloping algebra of planar algebra subfactors
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- by Stephen Curran, Vaughan F. R. Jones and Dimitri Shlyakhtenko PDF
- Trans. Amer. Math. Soc. 366 (2014), 113-133 Request permission
Abstract:
We give a diagrammatic description of Popa’s symmetric enveloping algebras associated to planar algebra subfactors. As an application we construct a natural family of derivations on these factors and compute a certain free entropy dimension type quantity.References
- Marcelo Aguiar, Infinitesimal Hopf algebras, New trends in Hopf algebra theory (La Falda, 1999) Contemp. Math., vol. 267, Amer. Math. Soc., Providence, RI, 2000, pp. 1–29. MR 1800704, DOI 10.1090/conm/267/04262
- Alain Connes and Dimitri Shlyakhtenko, $L^2$-homology for von Neumann algebras, J. Reine Angew. Math. 586 (2005), 125–168. MR 2180603, DOI 10.1515/crll.2005.2005.586.125
- P. Di Francesco, O. Golinelli, and E. Guitter, Meanders and the Temperley-Lieb algebra, Comm. Math. Phys. 186 (1997), no. 1, 1–59. MR 1462755, DOI 10.1007/BF02885671
- Ken Dykema, Free products of hyperfinite von Neumann algebras and free dimension, Duke Math. J. 69 (1993), no. 1, 97–119. MR 1201693, DOI 10.1215/S0012-7094-93-06905-0
- Ken Dykema, Interpolated free group factors, Pacific J. Math. 163 (1994), no. 1, 123–135. MR 1256179, DOI 10.2140/pjm.1994.163.123
- 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
- A. Guionnet, V. F. R. Jones, and D. Shlyakhtenko, Random matrices, free probability, planar algebras and subfactors, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, pp. 201–239. MR 2732052
- A. Guionnet, V. Jones, and D. Shlyakhtenko, A semi-finite algebra associated to a subfactor planar algebra, J. Funct. Anal. 261 (2011), no. 5, 1345–1360. MR 2807103, DOI 10.1016/j.jfa.2011.05.004
- A. Guionnet, V. F. R. Jones, D. Shlyakhtenko, and P. Zinn-Justin, Loop models, random matrices and planar algebras, Comm. Math. Phys. 316 (2012), no. 1, 45–97. MR 2989453, DOI 10.1007/s00220-012-1573-1
- Vaughan Jones, Dimitri Shlyakhtenko, and Kevin Walker, An orthogonal approach to the subfactor of a planar algebra, Pacific J. Math. 246 (2010), no. 1, 187–197. MR 2645882, DOI 10.2140/pjm.2010.246.187
- Vijay Kodiyalam and V. S. Sunder, From subfactor planar algebras to subfactors, Internat. J. Math. 20 (2009), no. 10, 1207–1231. MR 2574313, DOI 10.1142/S0129167X0900573X
- Vijay Kodiyalam and V. S. Sunder, On the Guionnet-Jones-Shlyakhtenko construction for graphs, J. Funct. Anal. 260 (2011), no. 9, 2635–2673. MR 2772347, DOI 10.1016/j.jfa.2011.01.018
- 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. 119–172. MR 996454
- Sorin Popa, Symmetric enveloping algebras, amenability and AFD properties for subfactors, Math. Res. Lett. 1 (1994), no. 4, 409–425. MR 1302385, DOI 10.4310/MRL.1994.v1.n4.a2
- Sorin Popa, An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math. 120 (1995), no. 3, 427–445. MR 1334479, DOI 10.1007/BF01241137
- Sorin Popa, Some properties of the symmetric enveloping algebra of a subfactor, with applications to amenability and property T, Doc. Math. 4 (1999), 665–744. MR 1729488
- Dimitri Shlyakhtenko, Free probability, planar algebras, subfactors and random matrices, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1603–1623. MR 2827857
- Dimitri Shlyakhtenko, Lower estimates on microstates free entropy dimension, Anal. PDE 2 (2009), no. 2, 119–146. MR 2547131, DOI 10.2140/apde.2009.2.119
- Dan Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104 (1991), no. 1, 201–220. MR 1094052, DOI 10.1007/BF01245072
- Dan Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. V. Noncommutative Hilbert transforms, Invent. Math. 132 (1998), no. 1, 189–227. MR 1618636, DOI 10.1007/s002220050222
- Dan Voiculescu, Free analysis questions. I. Duality transform for the coalgebra of $\partial _{X\colon B}$, Int. Math. Res. Not. 16 (2004), 793–822. MR 2036956, DOI 10.1155/S1073792804132443
- Dan-Virgil Voiculescu, Free analysis questions II: the Grassmannian completion and the series expansions at the origin, J. Reine Angew. Math. 645 (2010), 155–236. MR 2673426, DOI 10.1515/CRELLE.2010.063
Additional Information
- Stephen Curran
- Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095
- Address at time of publication: The D. E. Shaw Group, New York, New York 10036
- Email: curransr@math.ucla.edu, Stephen.Curran@deshaw.com
- Vaughan F. R. Jones
- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
- MR Author ID: 95565
- Email: vfr@math.berkeley.edu
- Dimitri Shlyakhtenko
- Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095
- MR Author ID: 606307
- ORCID: 0000-0002-0221-7508
- Email: shlyakht@math.ucla.edu
- Received by editor(s): October 3, 2011
- Published electronically: September 19, 2013
- Additional Notes: The first author’s research was supported by an NSF postdoctoral fellowship and NSF grant DMS-0900776.
The second author’s research was supported by NSF grant DMS-0856316.
The third author’s research was supported by NSF grant DMS-0900776
All the authors were also supported by DARPA Award 0011-11-0001. - © Copyright 2013 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 366 (2014), 113-133
- MSC (2010): Primary 46L37, 46L54
- DOI: https://doi.org/10.1090/S0002-9947-2013-05910-7
- MathSciNet review: 3118393