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Completeness of the 3j-Symbols for the Group SL(2, ℂ)

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The basic formulas for the unitary principal series representations of the group SL(2, ℂ) are given, and the decomposition of a tensor product of two representations into irreducible representations is considered. A simple proof of the completeness of the 3j-symbols is given.

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References

  1. I. M. Gelfand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol. 5, Academic Press (1966).

  2. I. M. Gelfand and M. A. Naimark, “Unitary representations of the classical groups,” Trudy Mat. Inst. Steklov, 36, 3–288 (1950).

    Google Scholar 

  3. M. A. Naimark and A. I. Stern, Theory of Group Representations, Springer-Verlag, New York (1982).

    Book  Google Scholar 

  4. I. M. Gelfand and G. E. Shilov, Generalized Functions, Volume 1: Properties and Operations, AMS Chelsea Publishing (1964).

  5. M. A. Naimark, “Decomposition of a tensor product of irreducible representations of the proper Lorentz group into irreducible representations,” AMS Transl., Ser. 2, Vol. 36, 101–229 (1964).

  6. A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory,” Nucl. Phys. B, 241, No. 2, 333–380 (1984).

    Article  MathSciNet  Google Scholar 

  7. R. S. Ismagilov, “On Racah operators,” Funct. Anal. Appl., 40, No. 3, 222–224 (2006).

    Article  MathSciNet  Google Scholar 

  8. R. S. Ismagilov, “Racah operators for principal series of representations of the group SL(2, ℂ),” Sb. Math., 198, No. 3, 369–381 (2007).

    Article  MathSciNet  Google Scholar 

  9. S. E. Derkachov and V. P. Spiridonov, “On the 6j-symbols for SL(2,C) group,” arXiv:1711.07073 (2017).

  10. S. E. Derkachov and L. D. Faddeev, “3j-symbol for the modular double SLq(2, ℝ) revisited,” arXiv:1302.5400 (2013).

  11. V. F. Molchanov and Yu. A. Neretin, “A pair of commuting hypergeometric operators on the complex plane and bispectrality,” arXiv:1812.06766 (2018).

  12. Yu. A. Neretin, “An analog of the Dougall formula and of the de Branges–Wilson integral,” arXiv:1812.07341 (2018).

  13. Yu. A. Neretin, “Barnes-Ismagilov integrals and hypergeometric functions of the complex field,” arXiv:1910.10686 (2019).

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Authors and Affiliations

  1. St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia

    N. M. Belousov & S. E. Derkachov

Authors
  1. N. M. Belousov
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  2. S. E. Derkachov
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Correspondence to N. M. Belousov.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 487, 2019, pp. 40–52.

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Belousov, N.M., Derkachov, S.E. Completeness of the 3j-Symbols for the Group SL(2, ℂ). J Math Sci 257, 450–458 (2021). https://doi.org/10.1007/s10958-021-05493-1

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  • DOI: https://doi.org/10.1007/s10958-021-05493-1

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