  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 $\Gamma=0$.
