Higher l 2-Betti Numbers of Universal Quantum Groups
Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 225-235

Voir la notice de l'article provenant de la source Cambridge University Press

We calculate all ${{\ell }^{2}}$ -Betti numbers of the universal discrete Kac quantum groups $\widehat{\text{U}}_{n}^{+}$ as well as their half-liberated counterparts $\widehat{\text{U}}_{n}^{*}$ .
DOI : 10.4153/CMB-2017-036-3
Mots-clés : 16T05, 46L65, 20G42, l 2-Betti number, free unitary quantum group, half-liberated unitary quantum group, free product formula, extension
Bichon, Julien; Kyed, David; Raum, Sven. Higher l 2-Betti Numbers of Universal Quantum Groups. Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 225-235. doi: 10.4153/CMB-2017-036-3
@article{10_4153_CMB_2017_036_3,
     author = {Bichon, Julien and Kyed, David and Raum, Sven},
     title = {Higher l {2-Betti} {Numbers} of {Universal} {Quantum} {Groups}},
     journal = {Canadian mathematical bulletin},
     pages = {225--235},
     year = {2018},
     volume = {61},
     number = {2},
     doi = {10.4153/CMB-2017-036-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-036-3/}
}
TY  - JOUR
AU  - Bichon, Julien
AU  - Kyed, David
AU  - Raum, Sven
TI  - Higher l 2-Betti Numbers of Universal Quantum Groups
JO  - Canadian mathematical bulletin
PY  - 2018
SP  - 225
EP  - 235
VL  - 61
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-036-3/
DO  - 10.4153/CMB-2017-036-3
ID  - 10_4153_CMB_2017_036_3
ER  - 
%0 Journal Article
%A Bichon, Julien
%A Kyed, David
%A Raum, Sven
%T Higher l 2-Betti Numbers of Universal Quantum Groups
%J Canadian mathematical bulletin
%D 2018
%P 225-235
%V 61
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-036-3/
%R 10.4153/CMB-2017-036-3
%F 10_4153_CMB_2017_036_3

[Ban97a] [Ban97a] Banica, T., Legroupe quantique compact libre U(n). Comm. Math. Phys. 190(1997), no. 1, 143–172. http://dx.doi.Org/10.1007/s002200050237 Google Scholar

[BS09] [BS09] Banica, T. and Speicher, R., Liberation orthogonal Lie groups. Adv. Math. 222(2009), no. 4, 1461–1501. http://dx.doi.Org/10.1016/j.aim.2009.06.009 Google Scholar

[BV10] [BV10] Banica, T. and Vergnioux, R., Invariants ofthe half-liberated orthogonal group. Ann. Inst. Fourier 60(2010), no. 6, 2137–2164. Google Scholar | DOI

[BMT01] [BMT01] Bédos, E., Murphy, G. J., and Tuset, L., Co-amenability of compact quantum groups. J. Geom. Phys. 40(2001), no. 2, 130–153. Google Scholar | DOI

[vdB98] [vdB98] van den Bergh, M., A relation between Hochschild homology and cohomology for Gorenstei rings. Proc. Amer. Math. Soc. 126(1998), no. 5,1345–1348. Google Scholar | DOI

[BDD11] [BDD11] Bhowmick, J., D'Andrea, F., and Dabrowski, L., Quantum isometries ofthe finite noncommutative geometry of the Standard model. Comm. Math. Phys. 307(2011), 101–131. http://dx.doi.Org/10.1007/s00220-011-1301-2 Google Scholar

[Bicl6] [Bicl6] Bichon, J., Cohomological dimensions of universal cosoverign Hopf algebras. Publicacions Matemàtiques, to appear. arxiv:1611.02069 Google Scholar

[BNY15] [BNY15] Bichon, J., Neshveyev, S., and Yamashita, M., Graded twisting of categories and quantum groups by group actions. Ann. Inst. Fourier 66(2016), no. 6, 2299–2338. Google Scholar | DOI

[BNY16] [BNY16] Bichon, J., Neshveyev, S., and Yamashita, M., Graded twisting of comodule algebras and module categories. arxiv:1 604.02078 Google Scholar

[Bral2] [Bral2] Brannan, M., Approximation properties for free orthogonal andfree unitary quantum groups. J. Reine Angew. Math. 672(2012), 223–251. Google Scholar

[Chil4] [Chil4] Chirvasitu, A., Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras. Algebra Number Theory 8(2014), no. 5,1179–1199. http://dx.doi.Org/10.214O/ant.2O14.8.1179 Google Scholar

[CHT09] [CHT09] Collins, B., Härtel, Johannes, and Thom, A., Homology offree quantum groups. C. R. Math. Acad. Sei. Paris 347(2009), no. 5-6, 271–276. http://dx.doi.Org/10.1016/j.crma.2009.01.021 Google Scholar

[FimlO] [FimlO] Fima, P. Kazhdan's property T for discrete quantum groups. Internat. J. Math. 221(2010), no. 1, 47–65. http://dx.doi.Org/10.1142/S0129167X1000591X Google Scholar

[KVOO] [KVOO] Kustermans, J. and Vaes, S., Locally compact quantum groups. Ann. Sei. École Norm. Sup. (4) 33(2000), no. 6, 837–934. http://dx.doi.Org/10.1016/S0012-9593(00)01055-7 Google Scholar

[KyeO8a] [KyeO8a] Kyed, D., L2-Betti numbers of coamenable quantum groups. Münster J. Math. 1(2008), 143–179. Google Scholar

[KyeO8b] [KyeO8b] Kyed, D., L2-homology for compact quantum groups. Math. Scand. 103(2008), no. 1,111–129. Google Scholar | DOI

[Kyell] [Kyell] Kyed, D., On the zeroth L2-homology of a quantum group. Münster J. Math. 4(2011), 119–127. Google Scholar

[Kyel2] [Kyel2] Kyed, D., An L2-Kunneth formula for tracial algebras. J. Operator Theory 67(2012), 317–327. Google Scholar

[KR16] [KR16] Kyed, D. and Raum, S., On the l2-Betti numbers of universal quantum groups. Math. Ann., to appear. arxiv:1 610.05474 Google Scholar

[LücO2] [LücO2] Lück, W., L2-invariants: theory and applications to geometry and K-theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, 44, Folge, A Series of Modern Surveys in Mathematics, Springer-Verlag, Berlin, 2002. Google Scholar

[MN06] [MN06] Meyer, R. and Nest, R., The Baum-Connes conjeeture via localisation of categories. Topology 45(2006), no. 2, 209–259. http://dx.doi.Org/10.1016/j.top.2005.07.001 Google Scholar

[ReiOl] [ReiOl] Reich, H., On the K-and L-theory ofthe algebra of Operators affiliated to a finite von Neumann algebra. Jf-Theory 24, no. 4, 303–326. http://dx.doi.Org/10.1023/A:1014078228859 Google Scholar

[SauO2] [SauO2] Sauer, R., L2-invariants of groups and discrete measured groupoids. Ph.D. Dissertation, University of Münster, 2002. Google Scholar

[Tho08] [Tho08] Thom, A., L2 *-cohomology for von Neumann algebras. Geom. Funct. Anal. 18(2008), no. 1, 251–270. Google Scholar | DOI

[VW96] [VW96] Van Daele, A. and Wang, S., Universal quantum groups. Internat. J. Math. 7(1996), no. 2, 255–263. http://dx.doi.Org/10.1142/S0129167X96000153 Google Scholar

[VerO7] [VerO7] Vergnioux, R., The property of rapid decayfor discrete quantum groups. J. Operator Theory 57(2007), 303–324. Google Scholar

[Verl2] [Verl2] Vergnioux, R., Paths in quantum Cayley trees and L2-cohomology. Adv. Math. 229(2012), 2686–2711. http://dx.doi.Org/10.1016/j.aim.2O12.01.011 Google Scholar

[Voill] [Voill] Voigt, C., The Baum-Connes conjeeture for free orthogonal quantum groups. Adv. Math. 227(2011), no. 5, 1873–1913. http://dx.doi.Org/10.1016/j.aim.2O11.04.008 Google Scholar

[Wan95] [Wan95] Wang, S., Free produets ofcompact quantum groups. Comm. Math. Phys. 167(1995), no. 3, 671–692. http://dx.doi.Org/10.1007/BF02101540 Google Scholar

[Wan98] [Wan98] Wang, S., Quantum symmetry groups of finite Spaces. Comm. Math. Phys. 195(1998), 195–211. Google Scholar | DOI

[Wei94] [Wei94] Weibel, C. A., An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994. http://dx.doi.Org/10.1017/CBO9781139644136 Google Scholar

[Wor87] [Wor87] Woronowicz, S. L., Twisted SU(2) group. An example ofa noncommutative differential calculus. Publ. Res. Inst. Math. Sei. 23(1987), 117–181. Google Scholar | DOI

[Wor98] [Wor98] Woronowicz, S. L., Compact quantum groups. In: Symetries quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, pp. 845–884. Google Scholar

Cité par Sources :