A polynomial class of $u(2)$ algebras

A $r$-parameter ${u}_{\{κ_1, κ_2, \cdots, κ_r\}}(2)$ algebra is introduced. Finite unitary representations are investigated. This polynomial algebra reduces via a contraction procedure to the generalized Weyl-Heisenberg algebra ${\cal A}_{\{κ_1, κ_2, \cdots, κ_r\}}$ (M. Daoud and M. Kibler, J. Phys. A: Math. Theor. {\bf 45} (2012) 244036). A pair of nonlinear (quadratic) bosons of type ${\cal A}_κ\equiv {\cal A}_{\{κ_1=κ, κ_2=0, \cdots, κ_r=0\}}$ are used to construct, à la Schwinger, a one parameter family of (cubic) $u_κ(2)$ algebra. The corresponding Hilbert space is constructed. The analytical Bargmann representation is also presented.

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