Twisting of the Quantum double and the Weyl algebra

Understand

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 ${\mathcal K}(L^2(G))$.