We consider inclusions of type , where is a compact quantum group of Kac type acting on a factor , and on a Markov inclusion of finite dimensional -algebras . In the case , which basically covers all known examples, we show that the planar algebra of such a subfactor is of the form , with acting in some natural sense on the bipartite graph algebra .
On considère des inclusions du type , où est un groupe quantique compact de type Kac agissant sur un facteur de type , et sur une inclusion de Markov de -algèbres de dimension finie . Dans le cas , qui couvre essentiellement tous les exemples connus, on montre que l’algèbre planaire d’un tel sous-facteur est de la forme , avec agissant dans un certain sens naturel sur l’algèbre de graphe bipartite .
@article{AMBP_2018__25_2_247_0, author = {Teodor Banica}, title = {The planar algebra of a fixed point subfactor}, journal = {Annales math\'ematiques Blaise Pascal}, pages = {247--264}, publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal}, volume = {25}, number = {2}, year = {2018}, doi = {10.5802/ambp.376}, language = {en}, url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.376/} }
TY - JOUR AU - Teodor Banica TI - The planar algebra of a fixed point subfactor JO - Annales mathématiques Blaise Pascal PY - 2018 SP - 247 EP - 264 VL - 25 IS - 2 PB - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal UR - https://ambp.centre-mersenne.org/articles/10.5802/ambp.376/ DO - 10.5802/ambp.376 LA - en ID - AMBP_2018__25_2_247_0 ER -
%0 Journal Article %A Teodor Banica %T The planar algebra of a fixed point subfactor %J Annales mathématiques Blaise Pascal %D 2018 %P 247-264 %V 25 %N 2 %I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal %U https://ambp.centre-mersenne.org/articles/10.5802/ambp.376/ %R 10.5802/ambp.376 %G en %F AMBP_2018__25_2_247_0
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] Exotic subfactors of finite depth with Jones indices and , Commun. Math. Phys., Volume 202 (1999), pp. 1-63 | Zbl
[2] Compact Kac algebras and commuting squares, J. Funct. Anal., Volume 176 (2000) no. 1, pp. 80-99 | Zbl
[3] Subfactors associated to compact Kac algebras, Integral Equations Oper. Theory, Volume 39 (2001) no. 1, pp. 1-14 | Zbl
[4] Quantum groups and Fuss-Catalan algebras, Commun. Math. Phys., Volume 226 (2002) no. 1, pp. 221-232 | Zbl
[5] The planar algebra of a coaction, J. Oper. Theory, Volume 53 (2005) no. 1, pp. 119-158 | Zbl
[6] Quantum automorphism groups of homogeneous graphs, J. Funct. Anal., Volume 224 (2005) no. 2, pp. 243-280 | Zbl
[7] Quantum groups acting on points, J. Reine Angew. Math., Volume 626 (2009), pp. 74-114 | Zbl
[8] Quantum groups, Proceedings of the International Congress of Mathematicians (Berkely, 1986), American Mathematical Society, 1987, pp. 798-820 | Zbl
[9] Quantum symmetries on operator algebras, Oxford Mathematical Monographs, Clarendon Press, 1998, xv+829 pages | Zbl
[10] Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications, 14, Springer, 1989, vi+288 pages | Zbl
[11] Principal graphs of subfactors in the index range , Subfactors. Proceedings of the Taniguchi symposium on operator algebras (Kyuzeso, 1993), World Scientific, 1994, pp. 1-38 | Zbl
[12] A -difference analog of and the Yang-Baxter equation, Lett. Math. Phys., Volume 10 (1985), pp. 63-69 | Zbl
[13] Index for subfactors, Invent. Math., Volume 72 (1983), pp. 1-25 | Zbl
[14] Planar algebras I (1999) (https://arxiv.org/abs/math/9909027) | Zbl
[15] 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] 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] On an inner product in modular categories, J. Am. Math. Soc., Volume 9 (1996) no. 4, pp. 1135-1170 | Zbl
[18] Entropy and index for subfactors, Ann. Sci. Éc. Norm. Supér., Volume 19 (1986) no. 1, pp. 57-106 | Zbl
[19] An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math., Volume 120 (1995) no. 3, pp. 427-445 | Zbl
[20] 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] 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] C-tensor categories from quantum groups, J. Am. Math. Soc., Volume 11 (1998) no. 2, pp. 261-282 | Zbl
[23] Quantum field theory and the Jones polynomial, Commun. Math. Phys., Volume 121 (1989) no. 3, pp. 351-399 | Zbl
[24] Compact matrix pseudogroups, Commun. Math. Phys., Volume 111 (1987), pp. 613-665 | Zbl
[25] Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math., Volume 93 (1988) no. 1, pp. 35-76 | Zbl
[26] Standard -lattices from quantum groups, Invent. Math., Volume 134 (1998) no. 3, pp. 455-487 | Zbl
Cited by Sources: