Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The free wreath product of a discrete group by a quantum automorphism group


Author: Lorenzo Pittau
Journal: Proc. Amer. Math. Soc. 144 (2016), 1985-2001
MSC (2010): Primary 46L65, 81R50
DOI: https://doi.org/10.1090/proc/12975
Published electronically: January 27, 2016
MathSciNet review: 3460161
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathbb{G}$ be the quantum automorphism group of a finite dimensional C*-algebra $ (B,\psi )$ and $ \Gamma $ a discrete group. We want to compute the fusion rules of $ \widehat {\Gamma }\wr _* \mathbb{G}$. First of all, we will revise the representation theory of $ \mathbb{G}$ and, in particular, we will describe the spaces of intertwiners by using noncrossing partitions. It will allow us to find the fusion rules of the free wreath product in the general case of a state $ \psi $. We will also prove the simplicity of the reduced C*-algebra, when $ \psi $ is a trace, as well as the Haagerup property of $ L^\infty (\widehat {\Gamma }\wr _* \mathbb{G})$, when $ \Gamma $ is moreover finite.


References [Enhancements On Off] (What's this?)

  • [Avi82] Daniel Avitzour, Free products of $ C^{\ast } $-algebras, Trans. Amer. Math. Soc. 271 (1982), no. 2, 423-435. MR 654842 (83h:46070), https://doi.org/10.2307/1998890
  • [Ban99] Teodor Banica, Symmetries of a generic coaction, Math. Ann. 314 (1999), no. 4, 763-780. MR 1709109 (2001g:46146), https://doi.org/10.1007/s002080050315
  • [Ban02] Teodor Banica, Quantum groups and Fuss-Catalan algebras, Comm. Math. Phys. 226 (2002), no. 1, 221-232. MR 1889999 (2002k:46178), https://doi.org/10.1007/s002200200613
  • [BC07] Teodor Banica and Benoît Collins, Integration over compact quantum groups, Publ. Res. Inst. Math. Sci. 43 (2007), no. 2, 277-302. MR 2341011 (2008m:46137)
  • [Bic04] Julien Bichon, Free wreath product by the quantum permutation group, Algebr. Represent. Theory 7 (2004), no. 4, 343-362. MR 2096666 (2005j:46043), https://doi.org/10.1023/B:ALGE.0000042148.97035.ca
  • [Boc93] Florin Boca, On the method of constructing irreducible finite index subfactors of Popa, Pacific J. Math. 161 (1993), no. 2, 201-231. MR 1242197 (94h:46096)
  • [Bra13] Michael Brannan, Reduced operator algebras of trace-perserving quantum automorphism groups, Doc. Math. 18 (2013), 1349-1402. MR 3138849
  • [BS09] Teodor Banica and Roland Speicher, Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), no. 4, 1461-1501. MR 2554941 (2010j:46125), https://doi.org/10.1016/j.aim.2009.06.009
  • [BV09] Teodor Banica and Roland Vergnioux, Fusion rules for quantum reflection groups, J. Noncommut. Geom. 3 (2009), no. 3, 327-359. MR 2511633 (2010i:46109), https://doi.org/10.4171/JNCG/39
  • [DCFY14] Kenny De Commer, Amaury Freslon, and Makoto Yamashita, CCAP for universal discrete quantum groups, Comm. Math. Phys. 331 (2014), no. 2, 677-701. With an appendix by Stefaan Vaes. MR 3238527, https://doi.org/10.1007/s00220-014-2052-7
  • [DHR97] Ken Dykema, Uffe Haagerup, and Mikael Rørdam, The stable rank of some free product $ C^*$-algebras, Duke Math. J. 90 (1997), no. 1, 95-121. MR 1478545 (99g:46077a), https://doi.org/10.1215/S0012-7094-97-09004-9
  • [Lem14] François Lemeux, The fusion rules of some free wreath product quantum groups and applications, J. Funct. Anal. 267 (2014), no. 7, 2507-2550. MR 3250372, https://doi.org/10.1016/j.jfa.2014.07.002
  • [Wan95] Shuzhou Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), no. 3, 671-692. MR 1316765 (95k:46104)
  • [Wan98] Shuzhou Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), no. 1, 195-211. MR 1637425 (99h:58014), https://doi.org/10.1007/s002200050385
  • [Wor87] S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613-665. MR 901157 (88m:46079)
  • [Wor88] S. L. Woronowicz, Tannaka-Kreĭn duality for compact matrix pseudogroups. Twisted $ {\rm SU}(N)$ groups, Invent. Math. 93 (1988), no. 1, 35-76. MR 943923 (90e:22033), https://doi.org/10.1007/BF01393687
  • [Wor98] S. L. Woronowicz, Compact quantum groups, Symétries quantiques (Les Houches, 1995) North-Holland, Amsterdam, 1998, pp. 845-884. MR 1616348 (99m:46164)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46L65, 81R50

Retrieve articles in all journals with MSC (2010): 46L65, 81R50


Additional Information

Lorenzo Pittau
Affiliation: Université de Cergy-Pontoise, 95000, Cergy-Pontoise, France — and — Univ. Paris Diderot, Sorbonne Paris Cité, IMJ-PRG, UMR 7586 CNRS, Sorbonne Universités, UPMC Univ Paris 06, F-75013, Paris, France
Email: lorenzo.pittau@u-cergy.fr

DOI: https://doi.org/10.1090/proc/12975
Received by editor(s): October 6, 2014
Received by editor(s) in revised form: May 14, 2015
Published electronically: January 27, 2016
Communicated by: Ken Ono
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society