Indecomposable representations and boson realizations of the nonlinear deformed angular momentum algebra of Witten’s first type

In this paper indecomposable representations and boson realizations of the nonlinear angular momentum algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R}_{q,p}^{c_1,c_2,c_3}$$\end{document} of Witten’s first type are investigated in a purely algebraic manner. Explicit form of the master representation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R}_{q,p}^{c_1,c_2,c_3}$$\end{document} on the space of its universal enveloping algebra is given. Then, from this master representation, other indecomposable representations are obtained in explicit form. Various kinds of single-boson, single inverse boson, and double-boson realizations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R}_{q,p}^{c_1,c_2,c_3}$$\end{document} are respectively obtained by generalizing the Holstein–Primakoff realization, the Dyson realization, and the Jordan–Schwinger realization of the Lie algebras SU(2) and SU(1,1). For each kind, the unitary realization, the nonunitary realization, and their connection by the corresponding similarity transformation are respectively discussed. Using a kind of double-boson realizations, the irreducible representation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R}_{q,p}^{c_1,c_2,c_3}$$\end{document} in the angular momentum basis is given.