The Wigner-Weyl isomorphism for quantum mechanics on a compact simple Liegroup G is developed in detail. Several features are shown to arise which have nocounterparts in the familiar Cartesian case. Notable among these is the notion of asemiquantized phase space, a structure on which the Weyl symbols of operatorsturn out to be naturally defined and, figuratively speaking, located midway betweenthe classical phase space T*G and the Hilbert space of square integrable functionson G. General expressions for the star product for Weyl symbols are presented andexplicitly worked out for the angle-angular momentum case.

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