  Quantum double construction, originally due to Drinfeld and has been since
generalized even to the operator algebra framework, is naturally associated
with a certain (quasitriangular) $R$-matrix ${\mathcal R}$. It turns out that
${\mathcal R}$ determines a twisting of the comultiplication on the quantum
double. It then suggests a twisting of the algebra structure on the dual of the
quantum double. For $D(G)$, the $C^*$-algebraic quantum double of an ordinary
group $G$, the "twisted $\hat{D(G)}$" turns out to be the Weyl algebra
$C_0(G)\times_{\tau}G$, which is in turn isomorphic to ${\mathcal K}(L^2(G))$.
This is the $C^*$-algebraic counterpart to an earlier (finite-dimensional)
result by Lu. It is not so easy technically to extend this program to the
general locally compact quantum group case, but we propose here some possible
approaches, using the notion of the (generalized) Fourier transform.

Operator Algebras;Quantum Algebra