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The phase space given by the cotangent bundle of a Lie group appears in thecontext of several models for physical systems. A representation for thequantum system in terms of non-commutative functions on the (dual) Lie algebra,and a generalized notion of (non-commutative) Fourier transform, different fromstandard harmonic analysis, has been recently developed, and found severalapplications, especially in the quantum gravity literature. We show that thisalgebra representation can be defined on the sole basis of a quantization mapof the classical Poisson algebra, and identify the conditions for itsexistence. In particular, the corresponding non-commutative star-productcarried by this representation is obtained directly from the quantization mapvia deformation quantization. We then clarify under which conditions a unitaryintertwiner between such algebra representation and the usual grouprepresentation can be constructed giving rise to the non-commutative planewaves and consequently, the non-commutative Fourier transform. The compactgroups U(1) and SU(2) are considered for different choices of quantizationmaps, such as the symmetric and the Duflo map, and we exhibit the correspondingstar-products, algebra representations and non-commutative plane waves.

Quantization maps, algebra representation and non-commutative Fourier transform for Lie groups