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Representations and Classification of the compact quantum groups $U_q(2)$ for complex deformation parameters

In this article, we obtain a complete list of inequivalent irreducible representations of the compact quantum group $U_q(2)$ for non-zero complex deformation parameters $q$, which are not roots of unity. The matrix coefficients of these representations are described in terms of the little $q$-Jacobi polynomials. The Haar state is shown to be faithful and an orthonormal basis of $L^2(U_q(2))$ is obtained. Thus, we have an explicit description of the Peter-Weyl decomposition of $U_q(2)$. As an application, we discuss the Fourier transform and establish the Plancherel formula. We also describe the decomposition of the tensor product of two irreducible representations into irreducible components. Finally, we classify the compact quantum groups $U_q(2)$.