Geodesic
Completeness of the $3j$-symbols for $SL(2,\mathbb C)$ group
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 26, Tome 487 (2019), pp. 40-52 Cet article a éte moissonné depuis la source Math-Net.Ru

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The main formulas for the unitary series of representations of the group $SL(2,\mathbb C)$ are given, and the decomposition of a tensor product of two representations into irreducible is considered. A simple proof of completeness of the $3j$-symbols is given. 
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N. M. Belousov; S. È. Derkachev. Completeness of the $3j$-symbols for $SL(2,\mathbb C)$ group. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 26, Tome 487 (2019), pp. 40-52. http://geodesic.mathdoc.fr/item/ZNSL_2019_487_a2/

[1] I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin, Generalized functions, v. 5, Academic Press, 1966 | MR | Zbl

[2] I. M. Gelfand, M. A. Naimark, “Unitary representations of the classical groups”, Trudy Mat. Inst. Steklov, 36, 1950, 3–288 | MR

[3] M. A. Naimark, Teoriya predstavlenii grupp, Nauka, M., 1976

[4] I. M. Gelfand, G. E. Shilov, Obobschennye funktsii. Vyp 1. Obobschennye funktsii i deistviya nad nimi, Nauka, M., 1959

[5] M. A. Naimark, “Decomposition of a tensor product of irreducible representations of the proper Lorentz group into irreducible representations”, Tr. Mosk. Mat. Obs., 8, 1959, 121–153 | MR

[6] A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory”, Nucl. Phys. B, 241:2 (1984), 333–380 | DOI | MR | Zbl

[7] R. S. Ismagilov, “On Racah operators”, Funktsional. Anal. Prilozhen., 40:3 (2006), 69–72 | DOI | MR | Zbl

[8] R. S. Ismagilov, “Racah operators for principal series of representations of the group $SL(2,\mathbb C)$”, Mat. Sbornik, 198:3 (2007), 77–90 | MR | Zbl

[9] S. E. Derkachov, V. P. Spiridonov, On the 6j-symbols for SL(2,C) group, arXiv: 1711.07073 | MR

[10] S. E. Derkachov, L. D. Faddeev, 3j-symbol for the modular double $SL_q(2,\mathbb{R})$ revisited, arXiv: 1302.5400

[11] Vladimir F. Molchanov, Yu. A. Neretin, A pair of commuting hypergeometric operators on the complex plane and bispectrality, arXiv: 1812.06766

[12] Yu. A. Neretin, An analog of the Dougall formula and of the de Branges–Wilson integral, arXiv: 1812.07341

[13] Yu. A. Neretin, Barnes-Ismagilov integrals and hypergeometric functions of the complex field, arXiv: 1910.10686