Project Euclid

We use the theory of regular objects in tensor categories to clarify the passage between braided multiplicative unitaries and multiplicative unitaries with projection. The braided multiplicative unitary and its semidirect product multiplicative unitary have the same Hilbert space representations. We also show that the multiplicative unitaries associated to two regular objects for the same tensor category are equivalent and hence generate isomorphic $\mathrm{C}^*$-quantum groups. In particular, a $\mathrm{C}^*$-quantum group is determined uniquely by its tensor category of representations on Hilbert space, and any functor between representation categories that does not change the underlying Hilbert spaces comes from a morphism of $\mathrm{C}^*$-quantum groups.