Skip to main content

An overview of quantum groups

  • II. Quantum Groups and Algebras, Non-Commutative Geometry
  • Conference paper
  • First Online:
Group Theoretical Methods in Physics

Part of the book series: Lecture Notes in Physics ((LNP,volume 382))

Abstract

Quantum groups are remarkable new mathematical structures of considerable current interest in both physics and mathematics. The origin, structure, representations and properties of quantum groups are reviewed with emphasis on q-group tensor operator algebra.

Supported, in part, by the National Science Foundation, PHY-9008007.

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

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Sklyanin, L. Takhatajan, and L. Faddeev, Teor. Math. Phys., 40 (1979) 194.

    Google Scholar 

  2. P. Kulish, N. Reshetikhin, Zap. nauch. seminarov LOMI, 101 (1981) 101, ibid. J. Soviet Math. 23 (1983) 2435.

    Google Scholar 

  3. V. G. Drinfeld, Quantum Groups, Proc. Int. Congr. of Math., MSRI Berkeley, CA, (1986) 798; Sov. Math. Dokl. 36 (1988) 212.

    Google Scholar 

  4. M. Jimbo, Lett. Math. Phys. 10 (1985) 63.

    Article  Google Scholar 

  5. G. Lusztig, Adv. in Math. 70 (1988) 237.

    Article  Google Scholar 

  6. M. Rosso, Commun. Math. Phys. 117 (1988) 581.

    Article  Google Scholar 

  7. L.C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and Its Applications, 8, Addison Wesley, 1981; reprinted Cambridge University Press (1989).

    Google Scholar 

  8. M. Jimbo, Commun. Math. Phys. 102 (1986) 537.

    Article  Google Scholar 

  9. G. Lusztig, Cont. Math. 82 (1989) 59.

    Google Scholar 

  10. P. Roche and D. Arnaudon, Lett. Math. Phys. 17 (1989) 295.

    Article  Google Scholar 

  11. L. Alvarez-Gaumé, C. Gomez, and G. Sierra, Nucl. Phys. B319 (1989) 155.

    Article  Google Scholar 

  12. V. Pasquier and H. Saleur, Nucl. Phys. B330 (1990) 523.

    Google Scholar 

  13. L. C. Biedenharn, J. Phys. A. Math. Gen. 22 (1989) L873.

    Article  Google Scholar 

  14. A. J. Macfarlane, J. Phys. A. Math. Gen. 22 (1989) 4581.

    Article  Google Scholar 

  15. C.-P. Sun and H.-C Fu, J. Phys. A. Math. Gen. 22 (1989) L983.

    Article  Google Scholar 

  16. P. Kulish and E. Damashinsky, J. Phys. A23 (1990) L415.

    Google Scholar 

  17. M. Chaichian and P. Kulish, Phys. Lett. 234B (1990) 72.

    Google Scholar 

  18. M. Chaichian, P. Kulish, and J. Lukierski, Phys. Lett. 237B (1990) 401.

    Google Scholar 

  19. L. C. Biedenharn and M. A. Lohe, invited paper at the Rochester Conference (honoring Prof. S. Okubo) 4–5 May 1990 (to appear in the Proceedings, World Scientific, Singapore).

    Google Scholar 

  20. T. Curtright and C. Zachos, Phys. Lett. 243B (1990) 237.

    Google Scholar 

  21. T. Curtright, Miami prepring TH/1/89, to appear in Physics and Geometry, L-L. Chau and W. Nahm, eds., Plenum, 1990.

    Google Scholar 

  22. D. Fairlie, J. Phys. A23 (1990) L183.

    Google Scholar 

  23. A. Polychronakos, Florida preprint, UFIFT-90-14, 1990, to appear in Proceedings of the Argonne Workshop on. Quantum Groups, T. Curtright, D. Fairlie, and C. Zachos (eds.), World Scientific, 1990.

    Google Scholar 

  24. C. Zachos, ANL-HEP-PR-90-61 to appear in Symmetries in Science V, B. Gruber (ed.), Plenum (N.Y.).

    Google Scholar 

  25. D. Fairlie, J. Nuyts, and C. Zachos, Phys. Lett. 202B, (1988) 320.

    Google Scholar 

  26. H. Itoyama and A. Sevrin, Stony Brook preprint, ITP-SB-90-12, February 1990.

    Google Scholar 

  27. E. Sklyanin, Funct. Anal. Appl. 16 (1982) 263.

    Article  Google Scholar 

  28. S. Woronowicz, Comm. Math. Phys. 111 (1987) 613; 130 (1990) 381.

    Article  Google Scholar 

  29. E. Witten, Nucl. Phys. B330 (1990) 285.

    Article  Google Scholar 

  30. D. Aebersold and L. C. Biedenharn, Phys. Rev. A15, (1977) 441.

    Google Scholar 

  31. L. C. Biedenharn and M. Tarlini, Lett. Math. Phys. 20, (1990) 271.

    Article  Google Scholar 

  32. A. N. Kirillov and N. Reshetikhin, Harvard Preprint 90/B261 (April 1990).

    Google Scholar 

  33. I. M. Gel'fand and A. V. Zelevinsky, Societé Math. de France, Astérique, hors serie, (1985) 117.

    Google Scholar 

  34. D. Flath and L.C. Biedenharn, Can. J. Math. 37 (1985) 710; L.C. Biedenharn and D. Flath, Commun. Math. Phys. 93 (1984) 143.

    Google Scholar 

  35. L. Faddeev, N. Reshetikhin, and L. Takhtajan, Alg. Anal. 1 (1988) 129; also in Braid Group, Knot Theory and Statistical Mechanics, C. Yang and M. Ge (eds.), World Scientific, 1989.

    Google Scholar 

  36. P. Kulish and N. Reshetikhin, Lett. Math. Phys. 18 (1989) 143.

    Google Scholar 

  37. E. Sklyanin, Uspekhi, Mat. Nauk. 40 (1985) 214.

    Google Scholar 

  38. M. Jimbo, Int. J. Mod. Phys. A4 (1989) 3759.

    Google Scholar 

  39. M.-L. Ge, Y.-S. Wu, and K. Xue, Stony Brook preprint, ITP-SB-90-02.

    Google Scholar 

  40. N. Burroughs, Comm. Math. Phys. 127 (1990) 109.

    Google Scholar 

  41. H. J. de Vega, Int. J. Mod. Phys., 4 (1989) 2371.

    Article  Google Scholar 

  42. M. Batchelor, L. Mezincescu, R. Nepomechie, and V. Rittenberg, J. Phys. A23 (1990) L141.

    Google Scholar 

  43. G. Moore and N. Reshetikhin, Nucl. Phys. 8328 (1989) 557.

    Article  Google Scholar 

  44. C. Gómez and G. Sierra, Phys. Lett. 2408 (1990) 149.

    Google Scholar 

  45. J.-L. Gervais, Comm. Math. Phys. 130 (1990) 257; Phys. Lett. 243B (1990) 85.

    Google Scholar 

  46. E. Guadagnini et al., Phys. Lett. 235B (1990) 275.

    Google Scholar 

  47. S. Majid, Int. J. Mod. Phys. A5 (1990) 1.

    Google Scholar 

  48. L. Kauffman, Int. J. Mod. Phys. A5 (1990) 93.

    Google Scholar 

  49. L.C. Biedenharn and M.A. Lohe, “Quantum Groups and Basic Hypergeometric Functions”, to appear in the Proceedings of the Argonne Workshop on Quantum Groups, T. Curtright, D. Fairlie, and C. Zachos (eds.), World Scientific, 1990.

    Google Scholar 

  50. L. Romans, in Strings '89, R. Arnowitt et al. (eds.), World Scientific, (1990).

    Google Scholar 

  51. Y. Manin, Comm. Math. Phys. 123 (1989) 163.

    Google Scholar 

  52. O. W. Greenberg, to appear in the Proceedings of the Argonne Workshop on Quantum Groups, T. Curtright, D. Fairlie, and C. Zachos (eds.), World Scientific, 1990.

    Google Scholar 

  53. P.P. Raychev, R. P. Roussev and Yu F. Smirnov, J. Phys. G: Nucl. Part. Phys. 16 (1990) L137.

    Article  Google Scholar 

  54. M. Nomura, J. Math. Phys. 30 (1989) 2397.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Victor V. Dodonov Vladimir I. Man'ko

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag

About this paper

Cite this paper

Biedenharn, L.C. (1991). An overview of quantum groups. In: Dodonov, V.V., Man'ko, V.I. (eds) Group Theoretical Methods in Physics. Lecture Notes in Physics, vol 382. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54040-7_102

Download citation

  • DOI: https://doi.org/10.1007/3-540-54040-7_102

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54040-3

  • Online ISBN: 978-3-540-47363-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics