# What is the multiplicative unitary for SU_q(2) (or other quantum groups)?

Consider a (von Neumann algebraic) locally compact quantum group $(M, \Delta, \phi, \psi)$ where the von Neumann algebra $M$ is realized as operators on the Hilbert space $H$. There is a multiplicative unitary $W$ in $B(H\otimes H)$that generates the comultiplication $\Delta$ via $$\Delta(x)=W^*(1\otimes x)W.$$

For a group, the multiplicative unitary for $L^\infty(G)$ acts on $L^2(G\times G)$ via $$Wf(g, h) = f(g, g^{-1}h).$$ Relatedly, for $VN(G)$ the multiplicative unitary acts on $L^2(G\times G)$ via $$\hat{W}f(g, h) = f(hg, h).$$

Beyond these two examples, are there nice descriptions of any multiplicative unitaries? (I'd like to know about $SU_q(2)$, but any example would be nice.)

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