Braided tensor C*-categories, Hecke symmetries and actions on extended Cuntz algebras

In this paper we deal with braided tensor C*-categories. For every object $\rho$ of the category we associate a C*-algebra denoted by $O_\rho$. An analysis of the braiding is carried out by using the conjugate equations. If the braiding is a Hecke symmetry and the $q$-dimension is appropriately chosen, we characterize the C*-algebra as the one generated by the representation given by the Markov trace. This analysis leads to the existence of an action of $\mathcal{F}_{SU_q\left( d\right) }$ on $O_\rho$. Such actions (Theorem 1) correspond to *-monomorphisms of $\left( O_N\right) ^{SU_q\left( d\right) }$ on $O_\rho$ which generalize the ones obtained earlier by the author (1997).