The planar algebra of a fixed point subfactor
Annales mathématiques Blaise Pascal, Volume 25 (2018) no. 2, pp. 247-264.

We consider inclusions of type (PA) G (PB) G , where G is a compact quantum group of Kac type acting on a II 1 factor P, and on a Markov inclusion of finite dimensional C * -algebras AB. In the case [A,B]=0, which basically covers all known examples, we show that the planar algebra of such a subfactor is of the form P(AB) G , with G acting in some natural sense on the bipartite graph algebra P(AB).

On considère des inclusions du type (PA) G (PB) G , où G est un groupe quantique compact de type Kac agissant sur un facteur de type II 1 P, et sur une inclusion de Markov de C * -algèbres de dimension finie AB. Dans le cas [A,B]=0, qui couvre essentiellement tous les exemples connus, on montre que l’algèbre planaire d’un tel sous-facteur est de la forme P(AB) G , avec G agissant dans un certain sens naturel sur l’algèbre de graphe bipartite P(AB).

Published online:
DOI: 10.5802/ambp.376
Classification: 46L65, 46L37
Keywords: Compact quantum group, Fixed point subfactor
Keywords: Groupe quantique compact, Sous-facteur de points fixes
Teodor Banica 1

1 Department of Mathematics Cergy-Pontoise University 95000 Cergy-Pontoise, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Teodor Banica. The planar algebra of a fixed point subfactor. Annales mathématiques Blaise Pascal, Volume 25 (2018) no. 2, pp. 247-264. doi : 10.5802/ambp.376. https://ambp.centre-mersenne.org/articles/10.5802/ambp.376/

[1] Marta Asaeda; Uffe Haagerup Exotic subfactors of finite depth with Jones indices (5+13)/2 and (5+17)/2, Commun. Math. Phys., Volume 202 (1999), pp. 1-63 | Zbl

[2] Teodor Banica Compact Kac algebras and commuting squares, J. Funct. Anal., Volume 176 (2000) no. 1, pp. 80-99 | Zbl

[3] Teodor Banica Subfactors associated to compact Kac algebras, Integral Equations Oper. Theory, Volume 39 (2001) no. 1, pp. 1-14 | Zbl

[4] Teodor Banica Quantum groups and Fuss-Catalan algebras, Commun. Math. Phys., Volume 226 (2002) no. 1, pp. 221-232 | Zbl

[5] Teodor Banica The planar algebra of a coaction, J. Oper. Theory, Volume 53 (2005) no. 1, pp. 119-158 | Zbl

[6] Teodor Banica Quantum automorphism groups of homogeneous graphs, J. Funct. Anal., Volume 224 (2005) no. 2, pp. 243-280 | Zbl

[7] Teodor Banica; Julien Bichon Quantum groups acting on 4 points, J. Reine Angew. Math., Volume 626 (2009), pp. 74-114 | Zbl

[8] Vladimir G Drinfeld Quantum groups, Proceedings of the International Congress of Mathematicians (Berkely, 1986), American Mathematical Society, 1987, pp. 798-820 | Zbl

[9] David E. Evans; Yasuyuki Kawahigashi Quantum symmetries on operator algebras, Oxford Mathematical Monographs, Clarendon Press, 1998, xv+829 pages | Zbl

[10] Frederick Goodman; Pierre De la Harpe; Vaughan F. R. Jones Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications, 14, Springer, 1989, vi+288 pages | Zbl

[11] Uffe Haagerup Principal graphs of subfactors in the index range 4<[M:N]<3+2, Subfactors. Proceedings of the Taniguchi symposium on operator algebras (Kyuzeso, 1993), World Scientific, 1994, pp. 1-38 | Zbl

[12] Michio Jimbo A q-difference analog of U(𝔤) and the Yang-Baxter equation, Lett. Math. Phys., Volume 10 (1985), pp. 63-69 | Zbl

[13] Vaughan F. R. Jones Index for subfactors, Invent. Math., Volume 72 (1983), pp. 1-25 | Zbl

[14] Vaughan F. R. Jones Planar algebras I (1999) (https://arxiv.org/abs/math/9909027) | Zbl

[15] Vaughan F. R. Jones The planar algebra of a bipartite graph, Knots in Hellas ’98 (Series on Knots and Everything), Volume 24, World Scientific, 2000, pp. 94-117 | Zbl

[16] Vaughan F. R. Jones The annular structure of subfactors, Essays on geometry and related topics (Monographies de l’Enseignement Mathématique), Volume 38, L’Enseignement Mathématique, 2001, pp. 401-463 | Zbl

[17] Jr. Kirillov On an inner product in modular categories, J. Am. Math. Soc., Volume 9 (1996) no. 4, pp. 1135-1170 | Zbl

[18] M. Pimsner; Sorin Popa Entropy and index for subfactors, Ann. Sci. Éc. Norm. Supér., Volume 19 (1986) no. 1, pp. 57-106 | Zbl

[19] Sorin Popa An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math., Volume 120 (1995) no. 3, pp. 427-445 | Zbl

[20] H. V. N. Temperley; Elliot H. Lieb Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem, Proc. R. Soc. Lond., Ser. A, Volume 322 (1971), pp. 251-280 | Zbl

[21] Antony Wassermann Coactions and Yang-Baxter equations for ergodic actions and subfactors, Operator algebras and applications. Volume II: Mathematical physics and subfactors (Warwick, 1987) (London Mathematical Society Lecture Note Series), Volume 136, Cambridge University Press, 1988, pp. 203-236 | Zbl

[22] Hans Wenzl C * -tensor categories from quantum groups, J. Am. Math. Soc., Volume 11 (1998) no. 2, pp. 261-282 | Zbl

[23] Edward Witten Quantum field theory and the Jones polynomial, Commun. Math. Phys., Volume 121 (1989) no. 3, pp. 351-399 | Zbl

[24] Stanisław L. Woronowicz Compact matrix pseudogroups, Commun. Math. Phys., Volume 111 (1987), pp. 613-665 | Zbl

[25] Stanisław L. Woronowicz Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math., Volume 93 (1988) no. 1, pp. 35-76 | Zbl

[26] Feng Xu Standard λ-lattices from quantum groups, Invent. Math., Volume 134 (1998) no. 3, pp. 455-487 | Zbl

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