Completeness of the \(3j\)-symbols for the group \(\mathrm{SL}(2, \mathbb{C} )\)
type: Item
instance of: scholarly article
title: Completeness of the \(3j\)-symbols for the group \(\mathrm{SL}(2, \mathbb{C} )\)
zbMATH Open document ID: 1482.22013
author: N. M. Belousov
author: S. É. Derkachov
published in: Journal of Mathematical Sciences (New York)
publication date: 2021-09-03T00:00:00Z
review text: The problem of decomposing the tensor product of two unitary principal series representations into irreducible representations was solved in a classical work of \textit{M. A. Naĭmark} [Transl., Ser. 2, Am. Math. Soc. 36, 101--136 (1964; Zbl 0132.11202); translation from Tr. Mosk. Mat. Obshch. 8, 121--153 (1959)]. The goal of the paper under review is to simplify some proofs given there: the authors offer a new proof of formula (2), see below. We use the following notations. For complex numbers, the overline means complex conjugation, for complex \(s\) and \(z\) we write \[ \widehat s=-1-\overline s, \ \ \ [z]^s=z^s \, \overline z ^{ \, \widehat s} \] The group \(\mathrm{SL}(2, \mathbb{C})\) consists of complex matrices \[ g=\left( \begin{array} {ll} a & b\\ c & d \end{array} \right), \ \ \ ad-bc=1. \] A unitary principal series representation \(T^{(s)}\) of the group \(\mathrm{SL}(2, \mathbb{C})\) acts on the functions \(\Phi(z)\) on the complex plane by \[ (T^{(s)}(g)\Phi)(z) = [d-bz]^{2s} \Phi\left(\frac{-c+az}{d-bz}\right), \eqno(1) \] here \[ s=-\frac{1}{2} +\frac{n}{4} +\frac{i\sigma}{2} \, , \, \, n \in \mathbb Z, \, \, \sigma \in \mathbb R, \ \ \text{so that} \ \ \widehat s=-\frac{1}{2} -\frac{n}{4} +\frac{i\sigma}{2} \, . \] The tensor product of such representations is reducible and decomposes into direct integral of them: \[ T^{(s_1)}\otimes T^{(s_2)} = \frac{1}{2}\sum \int_{\mathbb R} \, \rho(s) T^{(s)} d\sigma, \] where the summation is taken over \(n\) such that \(n_1+n_2+n\) is even, \[ \rho(s) = - \frac{(2s + 1)(2\widehat s + 1)}{4\pi^4} = \frac{1}{16\pi^4} \left( n^2+4\sigma^2 \right) \geqslant 0. \] The components \(\Phi_s(z)\) of this decomposition transformed by \(T^{(s)}\) are given by \[ \Phi_s(z)=\int W\left( \begin{array} {lll} s_1, & s_2, & s\\ z_1, & z_2, & z \end{array} \right) \Phi(z_1, z_2) \, dz_1 \, dz_2 \] where \[ W\left( \begin{array} {lll} s_1, & s_2, & s\\ z_1, & z_2, & z \end{array} \right)=[z_2-z_1]^{-2-s_1-s_2-s} [z-z_1]^{-s_1+s…
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 7390404
zbMATH Keywords: representations of the group \(\mathrm{SL}(2, \mathbb{C})\)
zbMATH Keywords: unitary principal series representations
zbMATH Keywords: orthogonality relations, the decomposition of the tensor product of representations
reviewed by: V. F. Molchanov
MaRDI profile type: Publication
full work available at URL: s10958-021-05493-1
OpenAlex ID: W3197109496
cites work: Q5520376
cites work: Q5802185
cites work: Q3943153
cites work: Q5724171
cites work: Infinite conformal symmetry in two-dimensional quantum field theory
cites work: On Racah operators
cites work: Racah operators for principal series of representations of the group $ \mathrm {SL}(2,\mathbb C)$
cites work: The \(6j\)-symbols for the \(SL(2,\mathbb{C})\) group
DOI: 10.1007/S10958-021-05493-1
description: scientific article; zbMATH DE number 7390404