A polynomial class of $u(2)$ algebras

A $r$-parameter $u_{\{{\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_r\}}(2)$ algebra is introduced. Finite unitary representations are investigated. This polynomial algebra reduces via a contraction procedure to the generalized Weyl-Heisenberg algebra ${\mathcal{A}}_{\{{\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_r\}}$ (M. Daoud and M. Kibler, J. Phys. A: Math. Theor. {\bf 45} (2012) 244036). A pair of nonlinear (quadratic) bosons of type ${\cal A}{\kappa}\equiv {\cal A}{{\kappa_1=\kappa, \kappa_2=0, \cdots, \kappa_r=0}}$ are used to construct, `a la Schwinger, a one parameter family of (cubic) $u_{\kappa }(2)$ algebra. The corresponding Hilbert space is constructed. The analytical Bargmann representation is also presented.