The representation category of the Woronowicz compact quantum group S_muU(d) as a braided tensor C*-category

An abstract characterization of the representation category of the Woronowicz twisted SU(d) group is given, generalizing analogous results known in the classical case. We generalize the theorem about the restriction of the ‘statistics parameter’ values to the values lambda_d := q^d (q?1)/(q^d?1) , with d ? Z. Here we consider both the case where q is real and the case where q is a root of unity. We show that each of these allowed values determines uniquely the kernel of the Hecke algebra representations given by the symmetry: If q > 0 and d is negative, then the kernel is the same as the kernel of the Jimbo–Woronowicz representation on H ? H with H a d–dimensional Hilbert space. Symmetries where this situation occurs will be then called of dimension d. In the root of unity case we find out that the given Hecke algebra representation with parameter \lambda_d has the same kernel as the Wenzl’s representation \pi(d,m) (see Theorem 3.3 for a precise statement). Later on we specialize to the case where q > 0. In section 5 we introduce the basic intertwiner, namely the quantum determinant, S of S?U(d), with ? = ?q, and we compute the conjugate representation of the defining representation (Theorem 5.5). For future reference, we perform the computations in a slight more generality, considering also quantum determinants of proper subspaces of H. In doing so we discover that in the case where q is not 1 there exist more left inverses of H which are faithful on the image of the Jimbo–Woronowicz representation than in the classical case (Cor. 5.2). We realize also that these smaller quantum determinants do not satisfy the afore mentioned braid relation. In section 6 we prove the main result which characterizes the representation category of S?U(d) among tensor C?–categories by means of its Hecke symmetry and the conjugate representation of the fundamental representation