Skip to main content
Log in

Inductive Limit Violates Quasi-cocommutativity

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

We show that the inductive limit of a certain inductive system of quasi-cocommutative C*-bialgebras is not quasi-cocommutative. This implies that the category of quasi-cocommutative C*-bialgebras is not closed with respect to the inductive limit.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Agore, A.L.: Limits of coalgebras, bialgebras and Hopf algebras. Proc. Am. Math. Soc. 139(3), 855–863 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baaj, S., Skandalis, G.: Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres. Ann. Sci. Ec. Norm. Super. 4e série 26, 425–488 (1993)

    MathSciNet  Google Scholar 

  3. Baxter, R.J.: Partition function of the eight-vertex lattice model. Ann. Phys. 70, 193–228 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blackadar, B.: Operator Algebras. Theory of C*-algebras and von Neumann Algebras. Springer-Verlag, Berlin, Heidelberg, New York (2006)

    MATH  Google Scholar 

  5. Brzeziński, T., Wisbauer, R.: Corings and Comodules. Cambridge University Press (2003)

  6. Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press (1994)

  7. Drinfel’d, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematicians, pp. 798–820. Berkeley, California (1987)

    Google Scholar 

  8. Enock, M., Schwartz, J.M.: Kac Algebras and Duality of Locally Compact Groups. Springer-Verlag (1992)

  9. Glimm, J.: On a certain class of operator algebras. Trans. Am. Math. Soc. 95, 318–340 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jimbo, M.: A q-difference analogue of U(\(\mathfrak{g}\)) and Yang-Baxter equation. Lett. Math. Phys. 10, 63–69 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kassel, C.: Quantum Groups. Springer-Verlag (1995)

  12. Kawamura, K.: A tensor product of representations of Cuntz algebras. Lett. Math. Phys. 82, 91–104 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kawamura, K.: C*-bialgebra defined by the direct sum of Cuntz algebras. J. Algebra 319, 3935–3959 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kawamura, K.: C*-bialgebra defined as the direct sum of Cuntz-Krieger algebras. Commun. Algebra 37, 4065–4078 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kawamura, K.: Triangular C*-bialgebra defined as the direct sum of matrix algebras. Linear Algebra Appl. 436, 2638–2652 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kawamura, K.: Non-existence of universal R-matrix for some C*-bialgebras. math.OA/0912.3578v1

  17. Kawamura, K.: C*-bialgebra defined as the direct sum of UHF algebras. math.OA/1109.2962v1

  18. Kustermans, J., Vaes, S.: The operator algebra approach to quantum groups. Proc. Natl. Acad. Sci. USA 97(2), 547–552 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Masuda, T., Nakagami, Y., Woronowicz, S.L.: A C *-algebraic framework for quantum groups. Int. J. Math. 14, 903–1001 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Phillips, N.C.: Inverse limits of C *-algebras. J. Oper. Theory 19, 159–195 (1988)

    MATH  Google Scholar 

  21. Podleś, P., Woronowicz, S.L.: Quantum deformation of Lorentz group. Commun. Math. Phys. 130, 381–431 (1990)

    Article  MATH  Google Scholar 

  22. Powers, R.T.: Representations of uniformly hyperfinite algebras and their associated von Neumann rings. Ann. Math. 86, 138–171 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wang, S.: Free products of compact quantum groups. Commun. Math. Phys. 167, 671–692 (1995)

    Article  MATH  Google Scholar 

  24. Woronowicz, S.L.: C *-algebras generated by unbounded elements. Rev. Math. Phys. 7, 481–521 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yang, C.N.: Some exact results for the many-body problem in one dimension with repulsive delta-function integration. Phys. Rev. Lett. 19(23), 1312–1315 (1967)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Katsunori Kawamura.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kawamura, K. Inductive Limit Violates Quasi-cocommutativity. Appl Categor Struct 21, 837–849 (2013). https://doi.org/10.1007/s10485-012-9288-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-012-9288-1

Keywords

Mathematics Subject Classifications (2010)

Navigation