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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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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

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