Skip to main content
SpringerLink
Log in

Canonical Basis for Quantum \({\mathfrak{osp}(1|2)}\)

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

Abstract

We introduce a modified quantum enveloping algebra as well as a (modified) covering quantum algebra for the ortho-symplectic Lie superalgebra \({\mathfrak{osp}(1|2)}\). Then we formulate and compute the corresponding canonical bases, and relate them to the counterpart for \({\mathfrak{sl}(2)}\). This provides a first example of canonical basis for quantum superalgebras.

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. Arnaudon D., Bauer M.: Scasimir operator, scentre and representations of U q (osp(1|2)). Lett. Math. Phys. 40, 307–320 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beilinson A., Lusztig G., MacPherson R.: A geometric setting for quantum groups. Duke Math. J. 61, 655–677 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blumen S.: On the U q (osp(1|2n)) and U −q (so(2n + 1)) uncoloured quantum Link invariants. J. Knot Theory Ramifications. 19, 335–353 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Clark, S., Hill, D., Wang, W. (to appear)

  5. Hill, D., Wang, W.: Categorification of quantum Kac-Moody superalgebras. Preprint, arXiv:1202.2769 (2012)

    Google Scholar 

  6. Jantzen, J.C.: Lectures on quantum groups. In: Graduate Studies in Mathematics, vol. 6. AMS, Providence (1996)

  7. Jeong K.: Crystal bases for Kac-Moody superalgebras. J. Algebra 237, 562–590 (2002)

    Article  Google Scholar 

  8. Kulish P., Reshetikhin N.: Universal R-matrix of the quantum superalgebra osp(2|1). Lett. Math. Phys. 18, 143–149 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Kashiwara M.: On crystal bases of the Q-analogue of universal enveloping algebras. Duke Math. J. 63, 456–516 (1991)

    Article  MathSciNet  Google Scholar 

  10. Lauda A.: A categorification of quantum \({\mathfrak{sl}(2)}\). Adv. Math. 225, 3327–3424 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lusztig G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3, 447–498 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lusztig G.: Introduction to quantum groups. Birkhäuser, Basel (1994)

    Google Scholar 

  13. Zou Y.M.: Integrable representations of U q (osp(1, 2n)). J. Pure Appl. Alg. 130, 99–112 (1998)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904, USA

    Sean Clark & Weiqiang Wang

Authors
  1. Sean Clark
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weiqiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weiqiang Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Clark, S., Wang, W. Canonical Basis for Quantum \({\mathfrak{osp}(1|2)}\) . Lett Math Phys 103, 207–231 (2013). https://doi.org/10.1007/s11005-012-0592-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-012-0592-3

Mathematics Subject Classification (1991)

Keywords

Search

Navigation