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.)



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


E. Christopher Lance, An explicit description of the fundamental unitary for SUq(2), Communications in Mathematical Physics, Volume 164, Issue 1, pp 1-15, 1994, <http://link.springer.com/article/10.1007/BF02108804>


or also


Janusz Wysoczański, Twisted product structure and representation theory of the quantum group Uq(2), Reports on Mathematical Physics, Volume 54, Issue 3, Pages 327–347, 2004, <http://www.sciencedirect.com/science/article/pii/S0034487704800235>


