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Whenever locally compact group acts on von Neumann algebras $M,N$, it gives rise to a canonical "diagonal" action on their tensor product $M\overline{\otimes}N$. This is no longer true, if we consider actions of locally compact quantum groups. Nonetheless, not all is lost. If the quantum group acting on von Neumann algebras $M,N$ is quasi-triangular (i.e. it is equipped with an $\operatorname{R}$-matrix), then one can form a twisted version of tensor product, called the braided tensor product $M\overline{\boxtimes}N$. This is a new von Neumann algebra which contains $M,N$ as subalgebras and which carries a canonical action of $\mathbb{G}$. As a special case, $\mathbb{G}$ can be taken to be the Drinfeld double of some (quantum) group $\mathbb{H}$, then action of $\mathbb{G}=D(\mathbb{H})$ on $M,N$ amounts to a pair of actions of $\mathbb{H}$ and its dual quantum group $\widehat{\mathbb{H}}$, satisfying Yetter-Drinfeld condition. I will discuss the construction of $M\overline{\boxtimes}N$, some examples and properties. Later, I will discuss results concerning the standard form of $M\overline{\boxtimes} N$. This talk is based on a joint work with Kenny De Commer.