Skip to main content

Quantum groups, filtered rings and Gelfand-Kirillov dimension

Part of the Lecture Notes in Mathematics book series (LNM,volume 1448)

Abstract

It is shown here that, for a number of quantum groups, there exists a finite dimensional ‘filtration’ for which the associated graded algebra has a simple form. It follows from this that Gelfand-Kirillov dimension behaves particularly well for these algebras.

Keywords

  • Hopf Algebra
  • Quantum Group
  • Short Exact Sequence
  • Cartan Matrix
  • Coordinate Ring

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   29.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   39.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Humphreys, J.E. Introduction to Lie algebras and representation theory. Graduate Texts No. 9. Springer Verlag, New York-Berlin, 1972.

    MATH  Google Scholar 

  2. Kac, V. Infinite dimensional Lie algebras, 2nd ed. Cambridge University Press, Cambridge-New York, 1985.

    MATH  Google Scholar 

  3. Lorenz, M. Gelfand-Kirillov dimension and Poincaré series. Cuadernos de Algebra No. 7. Universidad de Granada, 1988, pp. 68.

    Google Scholar 

  4. Lusztig, G. Canonical bases arising from quantized enveloping algebras. Preprint, M.I.T., 1989.

    Google Scholar 

  5. McConnell, J.C. and Pettit, J.J. Crossed products and multiplicative analogues of Weyl algebras. J. London Math. Soc. 38 (1988) 47–55.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. McConnell, J.C. and Robson, J.C. Noncommutative Noetherian rings. J. Wiley and Sons, Chichester-New York, 1987.

    MATH  Google Scholar 

  7. McConnell, J.C. and Stafford, J.T. Gelfand-Kirillov dimension and associated graded modules. J. Algebra 125 (1989) 197–214.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Manin, Yu. I. Quantum group and non-commutative geometry. Les. Publ. du Centre de Récherches Math. Université de Montreal, 1988.

    Google Scholar 

  9. Smith, S.P. Quantum groups: An introduction and survey for ring theorists. Preprint, University of Washington, Seattle, 1989.

    Google Scholar 

  10. Takeuchi, M. The q-bracket product and the P.B.W. theorem for quantum enveloping algebras of classical types (A n ), (B n ), (C n ) and (D n ). Preprint, University of Tsukuba, 1989.

    Google Scholar 

  11. Yamane, H. A P.B.W. theorem for quantized universal enveloping algebras of type A n . Publ. R.I.M.S. Kyoto 25 (1989) 503–520.

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1990 Springer-Verlag

About this paper

Cite this paper

McConnell, J.C. (1990). Quantum groups, filtered rings and Gelfand-Kirillov dimension. In: Jain, S.K., López-Permouth, S.R. (eds) Non-Commutative Ring Theory. Lecture Notes in Mathematics, vol 1448. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091258

Download citation

  • DOI: https://doi.org/10.1007/BFb0091258

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53164-7

  • Online ISBN: 978-3-540-46745-8

  • eBook Packages: Springer Book Archive