On two-coloured noncrossing partition quantum groups

Freslon, Amaury

doi:10.1090/tran/7846

On two-coloured noncrossing partition quantum groups
HTML articles powered by AMS MathViewer

by Amaury Freslon PDF

Trans. Amer. Math. Soc. 372 (2019), 4471-4508 Request permission

Abstract:

We classify compact quantum groups associated to noncrossing partitions coloured with two elements $x$ and $y$ which are their own inverses. Together with the work of P. Tarrago and M. Weber, this completes the classification of all noncrossing partition quantum groups on two colours. We also give some general results on the class of all noncrossing partition quantum groups and suggest some wider classification statements.

References

Teodor Banica, Le groupe quantique compact libre $\textrm {U}(n)$, Comm. Math. Phys. 190 (1997), no. 1, 143–172 (French, with English summary). MR 1484551, DOI 10.1007/s002200050237
Teodor Banica, Stephen Curran, and Roland Speicher, De Finetti theorems for easy quantum groups, Ann. Probab. 40 (2012), no. 1, 401–435. MR 2917777, DOI 10.1214/10-AOP619
Teodor Banica and Roland Speicher, Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), no. 4, 1461–1501. MR 2554941, DOI 10.1016/j.aim.2009.06.009
Teodor Banica and Roland Vergnioux, Fusion rules for quantum reflection groups, J. Noncommut. Geom. 3 (2009), no. 3, 327–359. MR 2511633, DOI 10.4171/JNCG/39
Teodor Banica and Roland Vergnioux, Invariants of the half-liberated orthogonal group, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 6, 2137–2164 (English, with English and French summaries). MR 2791653, DOI 10.5802/aif.2579
Julien Bichon, Free wreath product by the quantum permutation group, Algebr. Represent. Theory 7 (2004), no. 4, 343–362. MR 2096666, DOI 10.1023/B:ALGE.0000042148.97035.ca
G. Cébron and M. Weber, Quantum groups based on spatial partitions, arXiv:1609.02321, 2016.
Amaury Freslon, Fusion (semi)rings arising from quantum groups, J. Algebra 417 (2014), 161–197. MR 3244644, DOI 10.1016/j.jalgebra.2014.06.029
Amaury Freslon, On the partition approach to Schur-Weyl duality and free quantum groups, Transform. Groups 22 (2017), no. 3, 707–751. MR 3682834, DOI 10.1007/s00031-016-9410-9
Amaury Freslon and Moritz Weber, On the representation theory of partition (easy) quantum groups, J. Reine Angew. Math. 720 (2016), 155–197. MR 3565972, DOI 10.1515/crelle-2014-0049
Daniel Gromada, Classification of globally colorized categories of partitions, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 21 (2018), no. 4, 1850029, 25. MR 3897130, DOI 10.1142/S0219025718500297
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, DOI 10.1016/j.jfa.2014.07.002
Sergey Neshveyev and Lars Tuset, Compact quantum groups and their representation categories, Cours Spécialisés [Specialized Courses], vol. 20, Société Mathématique de France, Paris, 2013. MR 3204665
Sven Raum and Moritz Weber, The full classification of orthogonal easy quantum groups, Comm. Math. Phys. 341 (2016), no. 3, 751–779. MR 3452270, DOI 10.1007/s00220-015-2537-z
R. Speicher and M. Weber, Quantum groups with partial commutation relations, Indiana Univ. Math. J. (2019).
Pierre Tarrago and Moritz Weber, Unitary easy quantum groups: the free case and the group case, Int. Math. Res. Not. IMRN 18 (2017), 5710–5750. MR 3704744, DOI 10.1093/imrn/rnw185
Pierre Tarrago and Moritz Weber, The classification of tensor categories of two-colored noncrossing partitions, J. Combin. Theory Ser. A 154 (2018), 464–506. MR 3718074, DOI 10.1016/j.jcta.2017.09.003
Shuzhou Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), no. 3, 671–692. MR 1316765, DOI 10.1007/BF02101540
Moritz Weber, On the classification of easy quantum groups, Adv. Math. 245 (2013), 500–533. MR 3084436, DOI 10.1016/j.aim.2013.06.019
S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613–665. MR 901157, DOI 10.1007/BF01219077
S. L. Woronowicz, Tannaka-Kreĭn duality for compact matrix pseudogroups. Twisted $\textrm {SU}(N)$ groups, Invent. Math. 93 (1988), no. 1, 35–76. MR 943923, DOI 10.1007/BF01393687
S. L. Woronowicz, Compact quantum groups, Symétries quantiques (Les Houches, 1995) North-Holland, Amsterdam, 1998, pp. 845–884. MR 1616348