Let $Q\in GL\_n(\mathbb{C})$. The free unitary quantum group is the universal $C^\*$-algebra $A\_u(Q)$ with generators $u\_{ij},1\leq i,j\leq n$ and relations making $u=(u\_{ij})$ as well as $Q\bar{u}Q^{-1}$ unitary, where $\bar{u}=(u\_{ij}^\*)$. The comultiplication is defined by
\begin{align}
\Phi(u\_{ij})=\sum\_k u\_{ik}\otimes u\_{kj}.
\end{align}
$(A\_u(Q),\Phi)$ is a compact quantum group.


Let $Q\in GL\_n(\mathbb{C})$ such that $Q\bar{Q}=\pm 1$. The free orthogonal quantum group is the universal $C^\*$-algebra $A\_o(Q)$ with generators $u\_{ij},1\leq i,j\leq n$ such that $u=Q\bar{u}Q^{-1}$ is unitary. As above $(A\_o(Q),\Phi)$ is a compact quantum group.


It is well-known that every compact quantum group admits a unique Haar state. My question is that 


* What is the expliciit expression of the Haar state on $A\_u(Q)$ and $A\_o(Q)$?
* Are they faithful?


By the way, it is known for $SU\_q(2)$ and $SU\_q(2)\cong A\_o(Q)$ for
$Q=\begin{pmatrix}
0& |q|^{1/2} \newline
-\text{sign}(q)|q|^{-1/2}&0
\end{pmatrix}.$


I just find out that the Haar state is tracial if and only if the coinverse $\kappa$ on the dense Hopf $\*$-algebra satifies $\kappa^2=id$ (see "compact quantum groups" Thm 1.5 by Woronowicz).


Thanks!



The Haar state on compact quantum groups $A_u(Q)$ and  $A_o(Q)$
Have you seen [arXiv:math/0511253](http://arxiv.org/abs/math/0511253)?


INTEGRATION OVER COMPACT QUANTUM GROUPS


TEODOR BANICA AND BENOIT COLLINS 


Abstract. We find a combinatorial formula for the Haar functional of the orthogonal and unitary quantum groups. As an application, we consider diagonal coefficients of the fundamental representation, and we investigate their spectral measures.


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