Coupling coefficients for tensor product representations of quantum \(\mathrm{SU}(2)\) (Q2928092)

The quantum \(\mathrm{SU}(2)\) group is described by a Woronowicz \(C^*\)-algebra (a \(C^*\)-algebra with a coproduct) or by a dense Hopf \(*\)-subalgebra \(\mathcal{A}(\mathrm{SU}_q(2))\), which is a deformation of the algebra of (continuous or representative) functions on the classical \(\mathrm{SU}(2)\) group.NEWLINENEWLINEGiven a Hopf algebra, one can look at the fusion rules for its (co-)representation theory: one labels the irreducible (co-)representations and describes how a tensor product of (two or more) irreducible (co-)representations breaks up into a direct sum of irreducible (co-)representations.NEWLINENEWLINEPassing from the monoidal category of (co-)representations to the fusion rules one loses some information: for example the fusion rules for corepresentations of \(\mathcal{A}(\mathrm{SU}_q(2))\) are the same as the ones for \(q=1\). One may recover information about the monoidal category by looking not only at fusion rules, but also at coupling coefficients between eigenvectors of suitable selfadjoint operators.NEWLINENEWLINEIn this paper, following some previous works of T. H. Koornwinder, the author studies the fusion rules for irreducible representations (rather than corepresentations) of \(\mathcal{A}(\mathrm{SU}_q(2))\). He looks at infinite-dimensional irreducible representations (which exist if \(q\in\mathbb{R}^+\smallsetminus\{1\}\)) and studies the coupling coefficients for eigenvectors of a \(2\)-parameter family of selfadjoint operators \(\rho_{\tau,\sigma}\) -- also introduced by \textit{T. H. Koornwinder} in [SIAM J. Math. Anal. 24, No. 3, 795--813 (1993; Zbl 0799.33015)] -- and their relation with \(q\)-special functions. In particular, the author obtains several identities for some \(q\)-analogue of Bessel functions.