×

Quantum isometry groups of symmetric groups. (English) Zbl 1279.58003

Summary: We identify the quantum isometry groups of spectral triples built on the symmetric groups with length functions arising from the nearest-neighbor transpositions as generators. It turns out that they are isomorphic to certain “doubling” of the group algebras of the respective symmetric groups. We discuss the doubling procedure in the context of regular multiplier Hopf algebras. In the last section we study the dependence of the isometry group of \(S_n\) on the choice of generators in the case \(n=3\). We show that two different choices of generators lead to nonisomorphic quantum isometry groups which exhaust the list of noncommutative noncocommutative semisimple Hopf algebras of dimension 12. This provides noncommutative geometric interpretation of these Hopf algebras.

MSC:

58B32 Geometry of quantum groups
16T05 Hopf algebras and their applications
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L65 Quantizations, deformations for selfadjoint operator algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] DOI: 10.1016/j.jfa.2010.11.016 · Zbl 1279.46050 · doi:10.1016/j.jfa.2010.11.016
[2] DOI: 10.1007/s00220-008-0611-5 · Zbl 1159.81028 · doi:10.1007/s00220-008-0611-5
[3] DOI: 10.1016/j.jfa.2009.07.006 · Zbl 1180.58005 · doi:10.1016/j.jfa.2009.07.006
[4] DOI: 10.1016/j.jfa.2010.02.009 · Zbl 1210.58005 · doi:10.1016/j.jfa.2010.02.009
[5] DOI: 10.1016/j.geomphys.2010.05.007 · Zbl 1194.58004 · doi:10.1016/j.geomphys.2010.05.007
[6] Dixmier J., C*-Algebras and their Representations (1977)
[7] DOI: 10.1080/00927879908826688 · Zbl 0951.16013 · doi:10.1080/00927879908826688
[8] Fukuda N., Tsukuba J. Math. 21 pp 43–
[9] DOI: 10.1007/s00220-008-0461-1 · Zbl 1228.81188 · doi:10.1007/s00220-008-0461-1
[10] DOI: 10.1142/S0129167X97000500 · Zbl 1009.46038 · doi:10.1142/S0129167X97000500
[11] Pedersen G. K., C*-Algebras and their Automorphism Groups (1979)
[12] DOI: 10.1007/BF02099436 · Zbl 0853.46074 · doi:10.1007/BF02099436
[13] DOI: 10.1142/S0129055X98000173 · Zbl 0918.17005 · doi:10.1142/S0129055X98000173
[14] DOI: 10.1007/BF02473358 · Zbl 0703.22018 · doi:10.1007/BF02473358
[15] Sołtan P. M., Ill. J. Math. 49 pp 1245–
[16] DOI: 10.1016/j.geomphys.2008.11.007 · Zbl 1160.58007 · doi:10.1016/j.geomphys.2008.11.007
[17] DOI: 10.1142/S0219025709003768 · Zbl 1178.58002 · doi:10.1142/S0219025709003768
[18] Sołtan P. M., J. Noncommut. Geom. 4 pp 1–
[19] DOI: 10.1007/978-1-4612-6188-9 · doi:10.1007/978-1-4612-6188-9
[20] DOI: 10.1017/CBO9780511662393 · doi:10.1017/CBO9780511662393
[21] DOI: 10.1090/S0002-9947-1994-1220906-5 · doi:10.1090/S0002-9947-1994-1220906-5
[22] DOI: 10.1006/aima.1998.1775 · Zbl 0933.16043 · doi:10.1006/aima.1998.1775
[23] DOI: 10.1007/BF02100032 · Zbl 0743.46080 · doi:10.1007/BF02100032
[24] S. L. Woronowicz, Symétries Quantiques, les Houches, Session LXIV 1995, eds. A. Connes, K. Gawedzki and J. Zinn-Justin (Elsevier, 1998) pp. 845–884.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.