Skip to main content
SpringerLink
Log in

Skew derivations andU q (sl(2))

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

This note first describes the basic properties of the skew derivations on the polynomial ringk[X]. As a consequence of these properties it is proved that theq-analogue of the enveloping algebra of sl(2),U q(sl(2)), has a unique action on C[X], where “action” means that C[X] is a module algebra in the Hopf algebra sense. This depends on the fact that the generators of a subalgebra ofU q(sl(2)) described by Woronowicz must act via an automorphism, and the skew derivations associated to it.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. D. Bell and S. P. Smith,Some 3-dimensional skew polynomial rings, in preparation.

  2. G. Bergman,The Diamond Lemma for Ring Theory, Adv. Math.29 (1978), 178–218.

    Article  MATH  MathSciNet  Google Scholar 

  3. V. G. Drinfeld,Quantum Groups, Proc. Int. Congr. Math. Berkeley1 (1986), 798–820.

    Google Scholar 

  4. M. Jimbo,A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys.10 (1985), 63–69.

    Article  MATH  MathSciNet  Google Scholar 

  5. V. K. Kharchenko,Skew derivations of prime rings, Lecture, Stefan Banach Center, Warsaw, 1988.

    Google Scholar 

  6. G. Lusztig,Quantum deformations of certain simple modules over enveloping algebras, Adv. Math.70 (1988), 237–249.

    Article  MATH  MathSciNet  Google Scholar 

  7. T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi and K. Ueno,Representations of quantum groups and a q-analogue of orthogonal polynomials, C. R. Acad. Sci. Paris307 (1988), 559–564.

    MATH  MathSciNet  Google Scholar 

  8. O. Ore,Theory of non-commutative polynomials, Ann. of Math.34 (1933), 480–508.

    Article  MathSciNet  Google Scholar 

  9. M. Rosso,Comparaison des groupes SU(2)quantiques de Drinfeld et de Woronowicz, C. R. Acad. Sci. Paris304 (1987), 323–326.

    MATH  MathSciNet  Google Scholar 

  10. M. Rosso,Representations irreducibles de dimension finie du q-analogue de l’algebre enveloppante d’une algebre de Lie semisimple, C. R. Acad. Sci. Paris305 (1987), 587–590.

    MATH  MathSciNet  Google Scholar 

  11. M. Rosso,Finite dimensional representations of the quantum analogue of the enveloping algebra of a complex simple Lie algebra, Commun. Math. Phys.117 (1988), 581–593.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Sweedler,Hopf Algebras, Benjamin, New York, 1969.

    Google Scholar 

  13. E. J. Taft,The order of the antipode of finite dimensional Hopf algebras, Proc. Natl. Acad. Sci. U.S.A.68 (1971), 2631–2633.

    Article  MATH  MathSciNet  Google Scholar 

  14. S. L. Woronowicz,Twisted SU(2)-group. An example of a non-commutative differential calculus, Publ. R.I.M.S., Kyoto Univ.23 (1987), 117–181.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Southern California, 90089, Los Angeles, CA, USA

    S. Montgomery

  2. Department of Mathematics, University of Washington, 98195, Seattle, WA, USA

    S. Paul Smith

Authors
  1. S. Montgomery
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Paul Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Both authors were supported by the NSF, S. Montgomery by grant DMS 87-00641, and S. P. Smith by DMS 87-02447.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Montgomery, S., Paul Smith, S. Skew derivations andU q (sl(2)). Israel J. Math. 72, 158–166 (1990). https://doi.org/10.1007/BF02764618

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02764618

Keywords

Search

Navigation