The standard construction for cocycle twisted and braided tensor product W${}^{\ast }$-algebras

Given a locally compact quantum group $\mathbb{G}$ and a (generalized) dual unitary $2$-cocycle $\widehat{\mathrm{\Omega }}$, any W${}^{\ast }$-algebra $A$ with a $\mathbb{G}$-action can be twisted into a new W${}^{\ast }$-algebra $A_{\widehat{\mathrm{\Omega }}}$ with an action by the cocycle twist ${\mathbb{G}}_{\widehat{\mathrm{\Omega }}}$ of $\mathbb{G}$. We show how, in general, the standard space $L^2(A_{\widehat{\mathrm{\Omega }}})$, with its standard ${\mathbb{G}}_{\widehat{\mathrm{\Omega }}}$-representation, can be seen as a twist of $L^2(A)$ with its standard $\mathbb{G}$-representation. We then apply this general result in the special case of (generalized) Drinfeld doubles.