The free unitary compact quantum group

The free analogues of $U(n)$ in Woronowicz’s compact quantum group theory are the quantum groups $\{A_u(F)\mathrm{\mid }F\in GL(n,\mathbb{C})\}$ introduced by Van Daele and Wang. We classify here their irreducible representations. Their fusion rules turn to be related to the combinatorics of Voiculescu’s circular variable. If $F\bar{F}\in \mathbb{R}{I}_n$ we find an embedding $A_u(F{)}_{red}\subset C(\mathbb{T}){\ast }_{red}{A}_o(F)$, where $A_o(F)$ is the deformation of $SU(2)$ that we previously studied. We use the representation theory and Powers’ method for showing that the reduced algebras $A_u(F{)}_{red}$ are simple, with at most one trace.