Ukrainian Mathematical Journal

, Volume 60, Issue 4, pp 648–662 | Cite as

A locally compact quantum group of triangular matrices

  • P. Fima
  • L. Vainerman


We construct a one-parameter deformation of the group of 2 × 2 upper triangular matrices with determinant 1 using the twisting construction. An interesting feature of this new example of a locally compact quantum group is that the Haar measure is deformed in a nontrivial way. We also give a complete description of the dual C*-algebra and the dual comultiplication.


Quantum Group Compact Group Haar Measure Abelian Subgroup Triangular Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Enock and L. Vainerman, “Deformation of a Kac algebra by an abelian subgroup,” Commun. Math. Phys., 178, No. 3, 571–596 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    L. Vainerman, “2-Cocycles and twisting of Kac algebras,” Commun. Math. Phys., 191, No. 3, 697–721 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    M. Rieffel, “Deformation quantization for actions of ℝd,” Mem. Amer. Math. Soc., 506 (1993).Google Scholar
  4. 4.
    M. Landstad, “Quantization arising from abelian subgroups,” Int. J. Math., 5, 897–936 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    P. Kasprzak, Deformation of C*-Algebras by an Action of Abelian Groups with Dual 2-cocycle and Quantum Groups, Preprint, arXiv:math.OA/0606333.Google Scholar
  6. 6.
    P. Fima and L. Vainerman, Twisting of Locally Compact Quantum Groups. Deformation of the Haar Measure, Preprint (2000).Google Scholar
  7. 7.
    S. Stratila, Modular Theory in Operator Algebras, Abacus Press, Tunbridge Wells (1981).zbMATHGoogle Scholar
  8. 8.
    G. K. Pedersen, C*-Algebras and Their Automorphism Groups, Academic Press, New York (1979).zbMATHGoogle Scholar
  9. 9.
    J. Kustermans and S. Vaes, “Locally compact quantum groups,” Ann. Sci. Ecole Norm. Super. Ser. 33, 4, No. 6, 547–934 (2000).Google Scholar
  10. 10.
    J. Kustermans and S. Vaes, “Locally compact quantum groups in the von Neumann algebraic setting,” Math. Scand., 92, No. 1, 68–92 (2003).zbMATHMathSciNetGoogle Scholar
  11. 11.
    S. Vaes, “A Radon-Nikodym theorem for von Neumann algebras,” J. Oper. Theory, 46, No. 3, 477–489 (2001).zbMATHMathSciNetGoogle Scholar
  12. 12.
    M. Enock and J.-M. Schwartz, Kac Algebras and Duality of Locally Compact Groups, Springer, Berlin (1992).zbMATHGoogle Scholar
  13. 13.
    J. Bichon, J. A. de Rijdt, and S. Vaes, “Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups,” Commun. Math. Phys., 22, 703–728 (2006).CrossRefGoogle Scholar
  14. 14.
    M. Takesaki and N. Tatsuuma, “Duality and subgroups,” Ann. Math., 93, 344–364 (1971).CrossRefMathSciNetGoogle Scholar
  15. 15.
    S. L. Woronowicz, “Quantum E(2) group and its Pontryagin dual,” Lett. Math. Phys., 23, 251–263 (1991).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    S. L. Woronowicz, “C*-algebras generated by unbounded elements,” Rev. Math. Phys., 7, No. 3, 481–521 (1995).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  • P. Fima
    • 1
  • L. Vainerman
    • 2
  1. 1.University of Franche-ComtéBesançonFrance
  2. 2.University of CaenCaenFrance

Personalised recommendations