MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.)

share|cite|improve this question
up vote 4 down vote accepted


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,

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,

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.