Abstract
A Poisson bracket structure having the commutation relations of the quantum group SL q (2) is quantized by means of the Moyal star-product on C ∞(ℝ2), showing that quantum groups are not exactly quantizations, but require a quantization (with another parameter) in the background. The resulting associative algebra is a strongly invariant nonlinear star-product realization of the q-algebra U q (sl(2)). The principle of strong invariance (the requirement that the star-commutator is star-expressed, up to a phase, by the same function as its classical limit) implies essentially the uniqueness of the commutation relations of U q (sl(2)).
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