A nonzero 2-cocycle $\Gamma\in Z^2(\g,\R)$ on the Lie algebra $\g$ of a compact Lie group $G$ defines a twisted version of the Lie-Poisson structure on the dual Lie algebra $\g^*$, leading to a Poisson algebra $C^{ \infty}(\g_{ (\Gamma)}^*)$. Similarly, a multiplier $c\in Z^2(G,U(1))$ on $G$ which is smooth near the identity defines a twist in the convolution product on $G$, encoded by the twisted group $C^*$-algebra C^*(G,c)$. Further to some superficial yet enlightening analogies between $C^{ \infty}(\g^*_{ (\Gamma)})$ and $C^*(G,c)$, it is shown that the latter is a strict quantization of the former, where Planck's constant $\hbar$ assumes values in $(\Z\backslash\{ 0\})^{ -1}$. This means that there exists a continuous field of $C^*$-algebras, indexed by $\hbar\in 0\cup (\Z\backslash\{ 0\})^{ -1}$, for which $\A^0=C_0(\g^*)$ and $\A_{ \hbar}=C^*(G,c)$ for $\hbar\neq 0$, along with a cross-section of the field satisfying Dirac's condition asymptotically relating the commutator in $\A_{ \hbar}$ to the Poisson bracket on $C^{ \infty}(\g^*_{ (\Gamma)})$. Note that the `quantization' of $\hbar$ does not occur for $\Gamma=0$.
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