Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 097, 21 pages      arXiv:1606.00569      https://doi.org/10.3842/SIGMA.2016.097

Fixed Point Algebras for Easy Quantum Groups

Olivier Gabriel a and Moritz Weber b
a) University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark
b) Fachbereich Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saabrücken, Germany

Received June 13, 2016, in final form September 26, 2016; Published online October 01, 2016

Abstract
Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their $K$-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the $K$-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group $S_n^+$, the free orthogonal quantum group $O_n^+$ and the quantum reflection groups $H_n^{s+}$. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups, which are related to Hopf-Galois extensions.

Key words: $K$-theory; Kirchberg algebras; easy quantum groups; noncrossing partitions; fusion rules; free actions; free orthogonal quantum groups; quantum permutation groups; quantum reflection groups.

pdf (548 kb)   tex (37 kb)

References

  1. Banica T., Théorie des représentations du groupe quantique compact libre ${\rm O}(n)$, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 241-244, math.QA/9806063.
  2. Banica T., Symmetries of a generic coaction, Math. Ann. 314 (1999), 763-780, math.QA/9811060.
  3. Banica T., Belinschi S.T., Capitaine M., Collins B., Free Bessel laws, Canad. J. Math. 63 (2011), 3-37, arXiv:0710.5931.
  4. Banica T., Bichon J., Collins B., The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22 (2007), 345-384, math.RT/0701859.
  5. Banica T., Speicher R., Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), 1461-1501, arXiv:0808.2628.
  6. Banica T., Vergnioux R., Fusion rules for quantum reflection groups, J. Noncommut. Geom. 3 (2009), 327-359, arXiv:0805.4801.
  7. Baum P.F., De Commer K., Hajac P.M., Free actions of compact quantum group on unital ${C}^*$-algebras, arXiv:1304.2812.
  8. Baumgärtel H., Lledó F., Duality of compact groups and Hilbert $C^*$-systems for $C^*$-algebras with a nontrivial center, Internat. J. Math. 15 (2004), 759-812, math.OA/0311170.
  9. Bichon J., Free wreath product by the quantum permutation group, Algebr. Represent. Theory 7 (2004), 343-362, math.QA/0107029.
  10. Carey A.L., Paolucci A., Zhang R.B., Quantum group actions on the Cuntz algebra, Ann. Henri Poincaré 1 (2000), 1097-1122, q-alg/9705020.
  11. Cuntz J., Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), 173-185.
  12. Cuntz J., Regular actions of Hopf algebras on the $C^*$-algebra generated by a Hilbert space, in Operator Algebras, Mathematical Physics, and Low-Dimensional Topology (Istanbul, 1991), Res. Notes Math., Vol. 5, A K Peters, Wellesley, MA, 1993, 87-100.
  13. Dąbrowski L., Hadfield T., Hajac P.M., Equivariant join and fusion of noncommutative algebras, SIGMA 11 (2015), 082, 7 pages, arXiv:1407.6020.
  14. De Commer K., Yamashita M., A construction of finite index $C^*$-algebra inclusions from free actions of compact quantum groups, Publ. Res. Inst. Math. Sci. 49 (2013), 709-735, arXiv:1201.4022.
  15. Doplicher S., Roberts J.E., Duals of compact Lie groups realized in the Cuntz algebras and their actions on $C^*$-algebras, J. Funct. Anal. 74 (1987), 96-120.
  16. Doplicher S., Roberts J.E., Endomorphisms of $C^*$-algebras, cross products and duality for compact groups, Ann. of Math. 130 (1989), 75-119.
  17. Ellwood D.A., A new characterisation of principal actions, J. Funct. Anal. 173 (2000), 49-60.
  18. Freslon A., Weber M., On the representation theory of partition (easy) quantum groups, J. Reine Angew. Math., to appear, arXiv:1308.6390.
  19. Gabriel O., Fixed points of compact quantum groups actions on Cuntz algebras, Ann. Henri Poincaré 15 (2014), 1013-1036, arXiv:1210.5630.
  20. Gelaki S., Nikshych D., Nilpotent fusion categories, Adv. Math. 217 (2008), 1053-1071, math.QA/0610726.
  21. Hajac P.M., Krähmer U., Matthes R., Zieliński B., Piecewise principal comodule algebras, J. Noncommut. Geom. 5 (2011), 591-614, arXiv:0707.1344.
  22. Kashina Y., Sommerhäuser Y., Zhu Y., On higher Frobenius-Schur indicators, Mem. Amer. Math. Soc. 181 (2006), viii+65 pages, math.RA/0311199.
  23. Kirchberg E., The classification of purely infinite $C^*$-algebras using Kasparov theory, Preprint, 1994.
  24. Kirchberg E., Phillips N.C., Embedding of exact $C^*$-algebras in the Cuntz algebra ${\mathcal O}_2$, J. Reine Angew. Math. 525 (2000), 17-53, funct-an/9712002.
  25. Konishi Y., Nagisa M., Watatani Y., Some remarks on actions of compact matrix quantum groups on $C^*$-algebras, Pacific J. Math. 153 (1992), 119-127.
  26. Kreimer H.F., Takeuchi M., Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J. 30 (1981), 675-692.
  27. Marciniak M., Actions of compact quantum groups on $C^*$-algebras, Proc. Amer. Math. Soc. 126 (1998), 607-616.
  28. Montgomery S., Hopf Galois theory: a survey, in New Topological Contexts for Galois Theory and Algebraic Geometry (BIRS 2008), Geom. Topol. Monogr., Vol. 16, Geom. Topol. Publ., Coventry, 2009, 367-400.
  29. Müger M., On the center of a compact group, Int. Math. Res. Not. 2004 (2004), 2751-2756.
  30. Neshveyev S., Tuset L., Compact quantum groups and their representation categories, Cours Spécialisés, Vol. 20, Société Mathématique de France, Paris, 2013, available at http://folk.uio.no/sergeyn/papers/CQGRC.pdf.
  31. Paolucci A., Coactions of Hopf algebras on Cuntz algebras and their fixed point algebras, Proc. Amer. Math. Soc. 125 (1997), 1033-1042.
  32. Pinzari C., Simple $C^*$-algebras associated with compact groups and $K$-theory, J. Funct. Anal. 123 (1994), 46-58.
  33. Pinzari C., Roberts J.E., A duality theorem for ergodic actions of compact quantum groups on $C^*$-algebras, Comm. Math. Phys. 277 (2008), 385-421, math.OA/0607188.
  34. Pinzari C., Roberts J.E., A rigidity result for extensions of braided tensor $C^*$-categories derived from compact matrix quantum groups, Comm. Math. Phys. 306 (2011), 647-662, arXiv:1007.4480.
  35. Rørdam M., Classification of nuclear, simple $C^*$-algebras, in Classification of Nuclear $C^*$-algebras. Entropy in Operator Algebras, Encyclopaedia Math. Sci., Vol. 126, Springer, Berlin, 2002, 1-145.
  36. Tarrago P., Weber M., The classification of tensor categories of two-colored noncrossing partitions, arXiv:1509.00988.
  37. Tarrago P., Weber M., Unitary easy quantum groups: the free case and the group case, Int. Math. Res. Not., to appear, arXiv:1512.00195.
  38. Tikuisis A., White S., Winter W., Quasidiagonality of nuclear $C^*$-algebras, Ann. of Math., to appear, arXiv:1509.08318.
  39. Wang S., Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), 671-692.
  40. Wang S., Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195-211, math.OA/9807091.
  41. Woronowicz S.L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-665.
  42. Woronowicz S.L., Twisted ${\rm SU}(2)$ group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), 117-181.
  43. Woronowicz S.L., Tannaka-Krein duality for compact matrix pseudogroups. Twisted ${\rm SU}(N)$ groups, Invent. Math. 93 (1988), 35-76.
  44. Woronowicz S.L., Compact quantum groups, in Symétries Quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, 845-884.

Previous article  Next article   Contents of Volume 12 (2016)