Indecomposable representations and boson realizations of the nonlinear deformed angular momentum algebra of Witten's first type (Q1936787)

The article investigates irreducible representations of the nonlinear angular momentum algebra \(\mathcal{R}_{q,p}^{c_1,c_2,c_3}\) introduced in [\textit{E. Witten}, ``Gauge theories, vertex models, and quantum groups'', Nucl. Phys., B 330, No. 2--3, 285--346 (1990; \url{doi:10.1016/0550-3213(90)90115-T})], a multi-parametric deformation of \(\mathrm{SU}(2)\) in which \([J_+,J_-]\) is a polynomial of degree \(3\) in \(J_3\). Boson realizations are constructed for this algebra, namely the generalization of the Holstein--Primakoff (single boson), Dyson (inverse single inverse boson) and the Jordan--Schwinger (double boson) realizations of \(\mathrm{SU}(2)\); some of these are then used to construct the irreducible representations of \(\mathcal{R}_{q,p}^{c_1,c_2,c_3}\) in the angular momentum basis.