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)

 
 

 

Aspects of compact quantum group theory


Authors: G. J. Murphy and L. Tuset
Journal: Proc. Amer. Math. Soc. 132 (2004), 3055-3067
MSC (2000): Primary 46L89, 58B32
DOI: https://doi.org/10.1090/S0002-9939-04-07400-3
Published electronically: June 2, 2004
MathSciNet review: 2063127
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that if a compact quantum semigroup satisfies certain weak cancellation laws, then it admits a Haar measure, and using this we show that it is a compact quantum group. Thus, we obtain a new characterization of a compact quantum group. We also give a necessary and sufficient algebraic condition for the Haar measure of a compact quantum group to be faithful, in the case that its coordinate $C^*$-algebra is exact. A representation is given for the linear dual of the Hopf $*$-algebra of a compact quantum group, and a functional calculus for unbounded linear functionals is derived.


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

  • 1. E. Abe, Hopf Algebras, Cambridge University Press, Cambridge, 1980. MR 83a:16010
  • 2. K. Abodayeh and G. J. Murphy, Compact topological semigroups, Proc. Royal Irish Acad. 97A (1997), 131-137. MR 99g:22002
  • 3. E. Bedos, G. J. Murphy and L. Tuset, Co-amenability of compact quantum groups, J. Geom. Phys. 40 (2001), 130-153. MR 2002m:46100
  • 4. E. Bedos, G. J. Murphy and L. Tuset, Amenability and coamenability of algebraic quantum groups, International J. Math. and Math. Sci. 31 (2002), 577-601. MR 2003j:46107
  • 5. V. G. Drinfeld, Quantum groups, Proceedings of the International Congress of Mathematicians, Berkeley (1986), 793-820. MR 89f:17017
  • 6. M. Enock and J.-M. Schwartz, Kac Algebras and Duality of Locally Compact Groups, Springer, Berlin-Heidelberg, 1992. MR 94e:46001
  • 7. C. Kassel, Quantum Groups, Springer, Berlin-Heidelberg, 1995. MR 96e:17041
  • 8. E. C. Lance, Hilbert $C^*$-Modules--A Toolkit for Operator Algebraists, Cambridge University Press, Cambridge, 1995. MR 96k:46100
  • 9. A. Maes and A. Van Daele, Notes on compact quantum groups, Nieuw Arch. Voor Wisk. 16 (1998), 73-112. MR 99g:46105
  • 10. G. J. Murphy, $C^*$-Algebras and Operator Theory, Academic Press, San Diego, 1990. MR 91m:46084
  • 11. A. Van Daele, The Haar measure on a compact quantum group, Proc. Amer. Math. Soc. 123 (1995), 3125-3128. MR 95m:46097
  • 12. S. Wassermann, Exact $C^*$-Algebras and Related Topics, University Press, National University Seoul, 1994. MR 95b:46081
  • 13. S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-665. MR 88m:46079
  • 14. S. L. Woronowicz, Twisted $SU(2)$ groups--an example of a noncommutative differential calculus, Publ. RIMS Kyoto Univ. 23 (1987), 117-181. MR 88h:46130
  • 15. S. L. Woronowicz, Compact quantum groups, in Symétries Quantiques, North Holland, Amsterdam, 1998, pp. 845-884. MR 99m:46164

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46L89, 58B32

Retrieve articles in all journals with MSC (2000): 46L89, 58B32


Additional Information

G. J. Murphy
Affiliation: Department of Mathematics, National University of Ireland, Cork, Ireland

L. Tuset
Affiliation: Faculty of Engineering, University College, Oslo, Norway

DOI: https://doi.org/10.1090/S0002-9939-04-07400-3
Keywords: Quantum group, Haar measure
Received by editor(s): December 10, 2001
Received by editor(s) in revised form: June 3, 2003
Published electronically: June 2, 2004
Communicated by: David R. Larson
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society