Take an invertible \(n \times n\) matrix \(F\) over \(\mathbb{C}\) and consider the universal \(C^*\)-algebra \(A_u (F)\) generated by elements \(\{u_{ij} \mid i,j=1, \dots, n\}\) such that the matrices \(u\) and \(F \overline u F^{-1}\) (where \(\overline u_{ij}=u^*_{ij})\) are unitary in \(M_n (A_u(F))\). This \(C^*\)-algebra can be endowed with a comultiplication \(\delta\) satisfying \(\delta (u_{ij})=\sum_k u_{ik} \otimes u_{kj}\) for all \(i,j\). The pair \((A_u (F),u)\) is a compact matrix quantum group in the sense of S. L. Woronowicz [Commun. Math. Phys. 111, 613-665 (1987; Zbl 0627.58034)]. It is the universal (or free) quantum \(U(n)\) [see the reviewer and S. Wang, Int. J. Math. 7, 255-263 (1996)]. Similarly, the free quantum \(O(n)\) is defined as above for matrices \(F\) such that \(F \overline F\) is a scalar multiple of the identity and assuming the extra relation \(F\overline u F^{-1}=u\). The \(C^*\)-algebra is denoted by \(A_0 (F)\) in this case.
A representation of \((A_0 (F),u)\) is an invertible matrix \(r\) in some \(M_m (A_0 (F))\) such that \(\delta(r_{ij})=\sum_m r_{ik} \otimes r_{kj}\) for all \(i,j\). It is shown in this paper that, just as in the case of the (quantum) SU(2), the irreducible representations of \((A_0 (F),u)\) are self-adjoint, indexed by \(\mathbb{N}\) and satisfy \( r_0 =1\), \(r_1= u\) and \[ r_pr_q=r_{|p-q |} + r_{|p - q |+2} + \cdots + r_{p+ q} \] where \(r_pr_q\) denotes the tensor product of the representations \(r_p\) and \(r_q\) and where the sum is meant to be the direct sum of representations.
Also the converse is shown to be true.