Braided multiplicative unitaries as regular objects

TL;DR

This paper explores the relationship between braided multiplicative unitaries and multiplicative unitaries with projection using regular objects in tensor categories, showing their equivalence and implications for C*-quantum groups.

Contribution

It establishes a connection between braided and projection-based multiplicative unitaries via regular objects, and proves the uniqueness of C*-quantum groups determined by their representation tensor categories.

Findings

Braided multiplicative unitaries and their semidirect products have the same Hilbert space representations.

Multiplicative unitaries from regular objects are equivalent and generate isomorphic C*-quantum groups.

C*-quantum groups are uniquely determined by their tensor categories of representations.

Abstract

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 C*-quantum groups. In particular, a 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 C*-quantum groups.