Representations and classification of the compact quantum groups Uq(2) for complex deformation parameters

Abstract

In this paper, we obtain a complete list of inequivalent irreducible representations of the compact quantum group [Formula: see text] for nonzero complex deformation parameters [Formula: see text], which are not roots of unity. The matrix coefficients of these representations are described in terms of the little [Formula: see text]-Jacobi polynomials. The Haar state is shown to be faithful and an orthonormal basis of [Formula: see text] is obtained. Thus, we have an explicit description of the Peter–Weyl decomposition of [Formula: see text]. 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 group [Formula: see text].