Skip to main content
SpringerLink
Log in

Bicross-Product Structure of Affine Quantum Groups

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We show that the affine quantum group \({\text{U}}_q (\widehat{{\text{sl}}}_{\text{2}} )\) is isomorphic to a bicross-product central extension \(\mathbb{C}\mathbb{Z}_\chi \blacktriangleright \triangleleft {\text{U}}_q (L{\text{sl}}_2 )\) of the quantum loop group \({\text{U}}_q (L{\text{sl}}_2 )\) by a quantum cocycle \({\text{U}}_q (\hat g)\) in R-matrix form.

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. Jimbo, M. and Miwa, T.: Algebraic Analysis of Solvable Lattice Models, Amer. Math. Soc., Providence, 1995.

  2. Frenkel, I. B. and Reshetikhin, N. Yu.: Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146(1992), 1–60.

    Google Scholar 

  3. Pressley, A. and Segal, G.: Loop Groups, OUP, Oxford.

  4. Majid, S.: More examples of bicross-product and double cross product Hopf algebras, Israel J. Math. 72(1990), 133–148.

    Google Scholar 

  5. Majid, S. and Soibelman, Ya. S.: Bicross-product structure of the quantum Weyl group, J. Algebra 163(1994), 68–87.

    Google Scholar 

  6. Reshetikhin, N. Yu. and Semenov-Tian-Shansky, M. A.: Central extensions of quantum current groups, Lett. Math. Phys. 19(1990), 133–142.

    Google Scholar 

  7. Sweedler, M. E.: Hopf Algebras, Benjamin, New York, 1969.

    Google Scholar 

  8. Majid, S.: Foundations of Quantum Group Theory, Cambridge Univeristy Press, 1995.

  9. Doi, Y.: Equivalent crossed products for a Hopf algebra, Comm. Algebra 17(1989), 3053–3085.

    Google Scholar 

  10. Blattner, R. J. and Montgomery, S.: Crossed-products and Galois extensions of Hopf algebras, Pacific J. Math. 137(1989), 37–54.

    Google Scholar 

  11. Brzeziński, T. and Majid, S.: Quantum group gauge theory on quantum spaces, Comm. Math. Phys. 157(1993), 591–638. Erratum 167(1995), 235.

    Google Scholar 

  12. Majid, S.: Non-commutative-geometric groups by a bicross-product construction, PhD thesis, Harvard Mathematical Physics, 1988.

  13. Drinfeld, V. G.: Quantum groups, in: A. Gleason (ed.), Proc. ICM, Amer. Math. Soc., Providence, 1987, pp. 798–820.

  14. Faddeev, L. D., Reshetikhin, N. Yu. and Takhtajan, L. A.:Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1(1990), 193–225.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 9EW, U.K.

    S. MAJID

  2. Research Institute of Mathematical Sciences, Kyoto, 606, Japan

    S. MAJID

Authors
  1. S. MAJID
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

MAJID, S. Bicross-Product Structure of Affine Quantum Groups. Letters in Mathematical Physics 39, 243–252 (1997). https://doi.org/10.1023/A:1007318108923

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1007318108923

Search

Navigation