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Symmetries of quantum group coupling coefficients

  • Part IV Representations of Groups and Algebras
  • Conference paper
  • First Online:
Differential Geometry, Group Representations, and Quantization

Part of the book series: Lecture Notes in Physics ((LNP,volume 379))

Abstract

Quantum groups are new algebraic symmetry structures which have recently found application in many diverse fields of physics. We discuss the simplest of such structures: SUq(2) - the q-analog of the familiar quantal angular momentum group SU(2) - and determine in complete detail the symmetries, under exchange of q-angular momenta, of the (q-3j) and (q-6j) coefficients.

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References

  1. E. Sklyanin, L. Takhtajan, L. Faddeev: Theor. Math. Phys. 40 194 (1979) (in Russian)

    Article  Google Scholar 

  2. P. Kulish, N. Reshetikhin: Zap. nauch. seminarov LOMI 101 101 (1981) (in Russian)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  5. M. Jimbo: Lett. Math. Phys. 11 247 (1986)

    Article  Google Scholar 

  6. A. Shapere, F. Wilczek: Geometric Phases in Physics, Advanced Series in Mathematical Physics, Vol. 5 (World Scientific, Singapore, 1989)

    Google Scholar 

  7. R. Askey, J.A. Wilson: Siam J. Math. Anal. 10 1008 (1985)

    Article  Google Scholar 

  8. V.G. Drinfeld: Sov. Math. Dokl. 32 254 (1985)

    Google Scholar 

  9. M. Jimbo: Lett. Math. Phys. 10 63 (1985)

    Article  Google Scholar 

  10. L.C. Biedenharn, J.D. Louck: Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and Its Applications, 8, (Addison Wesley, 1981)

    Google Scholar 

  11. A.J. Macfarlane: J. Phys. A. Math. Gen. 22 4581 (1989)

    Article  Google Scholar 

  12. L.C. Biedenharn: J. Phys. A. Math. Gen. 22 L873 (1989)

    Article  Google Scholar 

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

    Article  Google Scholar 

  14. L.C. Biedenharn, M. Tarlini: Lett. Math. Phys. 20 271 (1990)

    Article  Google Scholar 

  15. I.M. Gel'fand, A.V. Zelevinsky: Société Math. de France, Astérisque, hors series, 117 (1985)

    Google Scholar 

  16. G. Lusztig: Adv. in Math. 70 237 (1988)

    Article  Google Scholar 

  17. M. Rosso: Commun. Math. Phys. 117 581 (1988)

    Article  Google Scholar 

  18. V. Pasquier, H. Saleur: Nucl. Phys. B330 523 (1990)

    Article  Google Scholar 

  19. A.N. Kirillov, N.Yu. Reshetikhin: “Representations of the Algebra Uq(sl(2)), qOrthogonal Polynomials and Invariants of Links”, USSR Academy of Sciences (preprint) 1988

    Google Scholar 

  20. L. Vaksman: Sov. Math. Dokl. 39 467 (1989)

    Google Scholar 

  21. M. Jimbo: Commun. Math. Phys. 102 537 (1986)

    Article  Google Scholar 

  22. H. Ruegg: J. Math. Phys. 31 1085 (1990)

    Article  Google Scholar 

  23. W. N. Bailey: Generalized Hypergeometric Series (Cambridge Univ. Press, Cambridge, 1935, reprinted Hafner, New York, 1972)

    Google Scholar 

  24. L.J. Slater: Generalized Hypergeometric Functions (Cambridge University Press, Cambridge, 1966)

    Google Scholar 

  25. R. Askey, J.A. Wilson: Memoirs Amer. Math. Soc. 319 (1985)

    Google Scholar 

  26. S.C. Milne: Advances in Math. 72 59 (1988)

    Article  Google Scholar 

  27. G. Gasper, M. Rahman: Basic Hypergeometric Series, Encyclopedia. of Mathematics and its Applications, 35 (Cambridge University Press, 1990)

    Google Scholar 

  28. J. Thomae: J. für Math. 87 26–73 (1879)

    Google Scholar 

  29. L.C. Biedenharn, J.D. Louck: Racah-Wigner Algebra in Quantum Theory, Encyclopedia of Mathematics and Its Applications, 9 (Addison Wesley, 1981)

    Google Scholar 

  30. M. Huszár: Acta Phys. Acad. Scient. Hung. 32 181–185 (1972)

    Google Scholar 

  31. M. Nomura: J. Math. Phys. 30 2397 (1989); J.Phys. Soc. Jap. 58 2694 (1989); ibid. 59 439 (1990)

    Article  Google Scholar 

  32. D.B. Sears: Proc. London Math. Soc. (2) 53 158–180 (1951)

    Google Scholar 

  33. S.D. Majumdar: Prog. of Theor. Phys. 20 798–803 (1958)

    Google Scholar 

  34. M.E. Rose: Multipole Fields, p. 92 (Wiley, New York, 1955)

    Google Scholar 

  35. A. Erdélyi: Math. Rev. 14 642 (1957)

    Google Scholar 

  36. H.A. Jahn, K.M. Howell: Proc. Cambridge Phil. Soc. 55 338 (1959)

    Google Scholar 

  37. K.S. Rao, T.S. Santhanam, K. Venkatesh: J. Math. Phys. 16 1528 (1975)

    Article  Google Scholar 

  38. K.S. Rao, K. Venkatesh: “Representation of the Racah coefficient as a generalized hypergeometricfunction,” in Proceedings Fifth International Colloquium on Group Theoretical Methods in Physics, ed. by R.T. Sharp and B. Kolman, pp. 649–656 (Academic Press, New York, 1977)

    Google Scholar 

  39. K. Venkatesh: J. Math. Phys. 19 1973 (1978); ibid., 2060

    Article  Google Scholar 

  40. L.C. Biedenharn: J. Math. and Phys. (MIT) XXXI 287 (1953)

    Google Scholar 

  41. H.D. Doebner, J. Tolar: in Symposium on Symmetries in Science, Carbondale, Illinois, 1979, ed. by B. Gruber and R.S. Millman, p. 475 (Plenum, New York; 1980)

    Google Scholar 

  42. L. Biedenharn, E. Lieb, B. Simon, F. Wilczek: Physics Today, p. 90 (Aug. 1990)

    Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Duke University, 27706, Durham, NC, USA

    L. C. Biedenharn & M. A. Lohe

Authors
  1. L. C. Biedenharn
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  2. M. A. Lohe
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Editor information

Jö-Dieter Hennig Wolfgang Lücke Jiří Tolar

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© 1991 Springer-Verlag

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Biedenharn, L.C., Lohe, M.A. (1991). Symmetries of quantum group coupling coefficients. In: Hennig, JD., Lücke, W., Tolar, J. (eds) Differential Geometry, Group Representations, and Quantization. Lecture Notes in Physics, vol 379. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-53941-7_12

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  • DOI: https://doi.org/10.1007/3-540-53941-7_12

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  • Print ISBN: 978-3-540-53941-4

  • Online ISBN: 978-3-540-46473-0

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