• Published : 2008.03.31


Suppose that D is a $C^*$-discrete quantum group and $D_0$ a discrete quantum group associated with D. If there exists a continuous action of D on an operator algebra L(H) so that L(H) becomes a D-module algebra, and if the inner product on the Hilbert space H is D-invariant, there is a unique $C^*$-representation $\theta$ of D associated with the action. The fixed-point subspace under the action of D is a Von Neumann algebra, and furthermore, it is the commutant of $\theta$(D) in L(H).


  1. E. Abe, Hopf Algebras, Cambridge Tracts in Mathematics, 74. Cambridge University Press, Cambridge-New York, 1980
  2. W. Arveson, An Invitation to C*-Algebras, Graduate Texts in Mathematics, no. 39. Springer-Verlag, New York-Heidelberg, 1976
  3. A. Van Daele, Multiplier Hopf algebras, Trans. Amer. Math. Soc. 342 (1994), no. 2, 917-932
  4. A. Van Daele, Discrete quantum groups, J. Algebra 180 (1996), no. 2, 431-444
  5. A. Van Daele, An algebraic framework for group duality, Adv. Math. 140 (1998), no. 2, 323-366
  6. A. Van Daele and Y. H. Zhang, Multiplier Hopf algebras of discrete type, J. Algebra 214 (1999), no. 2, 400-417
  7. E. G. Effros and Z. J. Ruan, Discrete quantum groups. I. The Haar measure, Internat. J. Math. 5 (1994), no. 5, 681-723
  8. W. Fulton and J. Harrie, Representation Theory: A First Course, Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991
  9. L. N. Jiang, M. Z. Guo, and M. Qian, The duality theory of a finite dimensional discrete quantum group, Proc. Amer. Math. Soc. 132 (2004), no. 12, 3537-3547
  10. L. N. Jiang and Z. D. Wang, The Schur-Weyl duality between quantum group of type A and Hecke algebra, Adv. Math. (China) 29 (2000), no. 5, 444-456
  11. M. Jimbo, A q-analogue of U(gl(N+1)), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), no. 3, 247-252
  12. G. J. Murphy, C*-Algebras and Operator Theory, Academic Press, Inc., Boston, MA, 1990
  13. P. Podles and S. L. Woronowicz, Quantum deformation of Lorentz group, Comm. Math. Phys. 130 (1990), no. 2, 381-431
  14. P. M. Soltan, Quantum Bohr compactification, Illinois J. Math. 49 (2005), no. 4, 1245-1270
  15. M. E. Sweedler, Hopf Algebras, Mathematics Lecture Note Series W. A. Benjamin, Inc., New York 1969
  16. K. Szlachanyi and P. Vecsernyes, Quantum symmetry and braid group statistics in G-spin models, Comm. Math. Phys. 156 (1993), no. 1, 127-168
  17. H. Weyl, The Classical Groups. Their Invariants and Representations., Princeton University Press, Princeton, N.J., 1939
  18. S. L. Woronowicz, Twisted SU(2) group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), no. 1, 117-181
  19. S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613-665

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