Skip to main content

The two-dimensional Euclidean quantum algebra at roots of unity


The two-dimensional Euclidean quantum algebra is considered at roots of unity. The algebraic properties and the Poisson-Hopf structure are first investigated. We then determine the irreducible representations and study the reduction of the tensor product of two of them.

This is a preview of subscription content, access via your institution.


  1. Vaksman, L. L. and Korogodski, L. I.,Soviet Math. Dokl. 39, 173 (1989).

    Google Scholar 

  2. Celeghini, E., Giachetti, R., Sorace, E., and Tarlini M.,J. Math. Phys. 31, 2548 (1990) and Contractions of quantum groups, inLecture Notes in Mathematics 1510, Springer-Verlag, Berlin, 1992, p. 221.

    Google Scholar 

  3. Woronowicz, S. L.,Lett. Math. Phys. 23, 251 (1991);Comm. Math. Phys. 144, 417 (1992);Comm. Math. Phys. 149, 637 (1992).

    Google Scholar 

  4. Ballesteros, A., Celeghini, E., Giachetti, R., Sorace, E., and Tarlini, M.,J. Phys. A: Math. Gen. 26, 7495 (1993).

    Google Scholar 

  5. Bonechi, F., Celeghini, E., Giachetti, R., Pereña, C. M., Sorace, E., and Tarlini, M.,J. Phys. A: Math. Gen., in press, 1994.

  6. De Concini, C. and Kac, V. G., Representations of quantum groups at roots of 1, inProgr. Math. 92, Birkhäuser, Boston, Mass. 1990, p. 471.

    Google Scholar 

  7. De Concini, C., Kac, V. G., and Procesi, C.,Comm. Math. Phys. 157, 405 (1993).

    Google Scholar 

  8. Lusztig, G.,Geom. Dedicata 35, 89 (1990).

    Google Scholar 

  9. Reshetikhin, N. Y. and Turaev, V. G.,Inv. Math. 103, 547 (1991).

    Google Scholar 

  10. Kirby, R. and Melvin, P.,Inv. Math. 105, 473 (1991).

    Google Scholar 

  11. Roche, P. and Arnaudon, D.,Lett. Math. Phys. 17, 295 (1989).

    Google Scholar 

  12. Chari, V. and Pressley, A.,Lett. Math. Phys. 26, 133 (1992).

    Google Scholar 

  13. Arnaudon, D.,Comm. Math. Phys. 159, 175 (1994).

    Google Scholar 

  14. McConnell, J. C. and Robson, J. C.,Noncommutative Noetherian Rings, Wiley, New York, 1987.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ciccoli, N., Giachetti, R. The two-dimensional Euclidean quantum algebra at roots of unity. Lett Math Phys 34, 37–48 (1995).

Download citation

  • Received:

  • Issue Date:

  • DOI:

Mathematical Subject Classifications (1991)

  • 16S32
  • 81R50