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)

 
 

 

Braided tensor C*-categories, Hecke symmetries
and actions on extended Cuntz algebras


Author: Anna Paolucci
Journal: Proc. Amer. Math. Soc. 127 (1999), 2249-2258
MSC (1991): Primary 46M05, 16W30, 81R50
DOI: https://doi.org/10.1090/S0002-9939-99-04693-6
Published electronically: April 16, 1999
MathSciNet review: 1476384
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we deal with braided tensor C*-categories. For every object $\rho $ of the category we associate a C*-algebra denoted by $O_\rho $. An analysis of the braiding is carried out by using the conjugate equations. If the braiding is a Hecke symmetry and the $q$-dimension is appropriately chosen, we characterize the C*-algebra as the one generated by the representation given by the Markov trace. This analysis leads to the existence of an action of $\mathcal{F}_{SU_q\left( d\right) }$ on $O_\rho $. Such actions (Theorem 1) correspond to *-monomorphisms of $\left( O_N\right) ^{SU_q\left( d\right) }$ on $O_\rho $ which generalize the ones obtained earlier by the author (1997).


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

  • 1. T. Banica, (1996), The quantization of $C\left( SU\left( N\right) \right) $ and some related amenability questions from the fusion semiring viewpoint, preprint, Marseille.
  • 2. J. Cuntz, (1977) Simple C*-algebras generated by isometries, Comm. Math. Phys. 57, p. 173-185. MR 57:7189
  • 3. S. Doplicher, J.E. Roberts, (1989), A new duality theory for compact groups, Invent. math., 98, 157-218. MR 90k:22005
  • 4. S. Doplicher, J.E. Roberts, (1987), Duals of compact Lie groups realized in the Cuntz algebras, J. Functional Analysis, 74, 90-120. MR 98a:22011
  • 5. V.F.R. Jones, (1987), Hecke algebra representations of braid groups and link polynomials, Annals of Mathematics, 126, p. 335-388. MR 89c:46092
  • 6. A. Joyal, R. Street, (1986) Braided monoidal categories, Macquarie Mathematics report n. 860081.
  • 7. C. Kassel, (1994), Quantum Groups, Springer Verlag. MR 96:17041
  • 8. R. Longo, J. E. Roberts, (1995), Theory of dimension, preprint, Rome.
  • 9. A. Paolucci, (1996), Remarks on braided C*-categories and endomorphisms of C*-algebras, J. Operator Theory, 36, p. 157-177. MR 97m:46113
  • 10. A. Paolucci, (1997), Coactions of Hopf Algebras on Cuntz algebras and their fixed point algebras, Proc. Amer. Math. Soc., 125, p. 1033-1042. MR 97f:46106
  • 11. S.L. Woronowicz, (1987), Compact matrix pseudogroups, Comm. Math. Phys. 111, 613-665. MR 88m:46079
  • 12. S.L. Woronowicz, (1987), Twisted SU(2) group. An example of a non commutative differential calculus, Publ. RIMS 23, 117-181.
  • 13. S.L. Woronowicz, (1988), Tannaka-Krein duality for compact matrix pseudogroups, twisted $SU\left( N\right) $ groups, Inventiones Math., 93, p. 35-76. MR 90e:22033
  • 14. S.L. Woronowicz, (1993), Compact Quantum Group, preprint, Warsaw.
  • 15. D.N. Yetter, (1990), Quantum groups and representations of monoidal categories, Math. Proceedings Camb. Phil. Soc. 108, p. 261-290. MR 91k:16028
  • 16. H. Wenzl, (1988), Hecke algebras of type $A_n$ and subfactors, Invent. Math. 92, p. 349-383. MR 90b:46118

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46M05, 16W30, 81R50

Retrieve articles in all journals with MSC (1991): 46M05, 16W30, 81R50


Additional Information

Anna Paolucci
Affiliation: School of Mathematics, University of Leeds, LS2 9JT, United Kingdom
Address at time of publication: Dipartimento di Matematica, Università di Torino, via Carlo Alberto, 10, 10124 Torino, Italy
Email: paolucci@dm.unito.it

DOI: https://doi.org/10.1090/S0002-9939-99-04693-6
Keywords: C*-algebras, Hilbert spaces, Hecke symmetries, braided C*-categories.
Received by editor(s): July 22, 1997
Received by editor(s) in revised form: August 11, 1997, August 18, 1997, and September 11, 1997
Published electronically: April 16, 1999
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society