Skip to main content
SpringerLink
Log in

Explicit coproduct formula for quantum groups of infinite series

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

Abstract

We find an explicit form for the coproduct formula for PBW generators of quantum groups of infinite series U q (sp2n ) and U q (so2n ). Similar formulas for U q (sl n+1) and U q (so2n+1) are already known.

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. N. Andruskiewitsch and H.-J. Schneider, Pointed Hopf algebras, in New Directions in Hopf Algebras, Mathematical Siences Research Iinstitute Publications, Vol. 43, Cambridge University Press, Cambridge, 2002, pp. 1–68.

    Google Scholar 

  2. I. Angiono, Nichols algebras with standard braiding, Algebra & Number Theory 3 (2009), 35–106.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Beattie, S. Dǎscǎlescu and Ş. Raianu, Lifting of Nichols algebras of type B 2, Israel Journal of Mathematics 132 (2002), 1–28.

    Article  MATH  MathSciNet  Google Scholar 

  4. D. Flores de Chela and J. A. Green, On quantum symmetric algebra, Algebras and Representation Theory 4 (2001), 55–76.

    Article  MATH  MathSciNet  Google Scholar 

  5. I. Heckenberger and H.-J. Schneider, Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid, Israel Journal of Mathematics 197 (2013), 139–187.

    Article  MATH  MathSciNet  Google Scholar 

  6. N. Jacobson, Lie Algebras, Interscience Publishers, New York-London, 1962.

    MATH  Google Scholar 

  7. V. Kac, Infinite Dimensional Lie Algebras, Cambridge University Press, 1990.

  8. V. K. Kharchenko, An algebra of skew primitive elements, Algebra i Logica 37 (1998),181–224; English translation: Algebra and Logic 37 (1998), 101–127, arXiv: math. QA/0006077.

    Article  MATH  MathSciNet  Google Scholar 

  9. V. K. Kharchenko, A combinatorial approach to the quantifications of Lie algebras, Pacific Journal of Mathematics 203 (2002), 191–233.

    Article  MATH  MathSciNet  Google Scholar 

  10. V. K. Kharchenko, Quantum Lie algebras and related problems, in Proceedings of the Third International Algebra Conference (Tainan, 2002), Kluwer Academic Publishers, Dordrecht, 2003, pp. 57–114.

    Google Scholar 

  11. V. K. Kharchenko, PBW bases of coideal subalgebras and a freeness theorem, Transactions of the American Mathematical Society 360 (2008), 5121–5143.

    Article  MATH  MathSciNet  Google Scholar 

  12. V. K. Kharchenko, Right coideal subalgebras of U + q (so 2n+1), Journal of the European Mathematical Society 13 (2011), 1677–1735.

    MATH  MathSciNet  Google Scholar 

  13. V. K. Kharchenko and A. A. Alvarez, On the combinatorial rank of Hopf algebras, Contemporary Mathematics 376 (2005), 299–308.

    Article  Google Scholar 

  14. V. K. Kharchenko and A. V. Lara Sagahon, Right coideal subalgebras in U q(sl n+1), Journal of Algebra 319 (2008), 2571–2625.

    Article  MATH  MathSciNet  Google Scholar 

  15. S. Z. Levendorskii and Ya. S. Soibelman, Some applications of the quantum Weyl groups, Journal of Geometry and Physics 7 (1990), 241–254.

    Article  MathSciNet  Google Scholar 

  16. M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, 2002.

  17. G. Lusztig, Introduction to Quantum Groups, Birkhäuser, Boston, MA, 1993.

    MATH  Google Scholar 

  18. M. Rosso, An analogue of the Poincaré-Birkhoff-Witt theorem and the universal Rmatrix of U q(sl(N +1)), Communications in Mathematical Physics 124 (1989), 307–318.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Rosso, Quantum groups and quantum shuffles, Inventiones Mathematicae 113 (1998), 399–416.

    Article  MathSciNet  Google Scholar 

  20. P. Schauenberg, A characterization of the Borel-like subalgebras of quantum enveloping algebras, Communications in Algebra 24 (1996), 2811–2823.

    Article  MathSciNet  Google Scholar 

  21. J.-P. Serre, Algêbres de Lie semi-simples complexes, Benjamin, New York-Amsterdam, 1966.

    MATH  Google Scholar 

  22. A. I. Shirshov, On free Lie rings, Matematicheskiĭ Sbornik 45,87(2) (1958), 113–122.

    Google Scholar 

  23. A. I. Shirshov, Some algorithmic problems for Lie algebras, Sibirskiĭ Matematicheskiĭ Zhurnal 3 (1962), 292–296.

    MATH  Google Scholar 

  24. M. Takeuchi, Survey of braided Hopf algebras, in New Trends in Hopf Algebra Theory (La Falda, 1999), Contemporary Mathematics, Vol. 267, American Mathematical Society, Providence, RI, 2000, pp. 301–324.

    Chapter  Google Scholar 

  25. H. Yamane, A Poincaré-Birkhoff-Witt theorem for quantized universal enveloping algebras of type A N , Publications of the Research Institute for Mathematical Siences 25 (1989), 503–520.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Facultad de Estudios Superiores Cuautitlán, Universidad Nacional Autónoma de México, Primero de Mayo s/n Campo 1, CIT, Cuautitlan Izcalli, Edstado de México, 54768, Mexico

    V. K. Kharchenko

Authors
  1. V. K. Kharchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. K. Kharchenko.

Additional information

The author was supported by PAPIIT IN 112913, UNAM, and PIAPIVC06, FESC UNAM, México.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kharchenko, V.K. Explicit coproduct formula for quantum groups of infinite series. Isr. J. Math. 208, 13–43 (2015). https://doi.org/10.1007/s11856-015-1192-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-015-1192-6

Keywords

Search

Navigation