## Advanced Studies in Pure Mathematics

### Braided multiplicative unitaries as regular objects

#### 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 $\mathrm{C}^*$-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.

#### Article information

Dates
Revised: 20 April 2017
First available in Project Euclid: 21 August 2019

https://projecteuclid.org/ euclid.aspm/1566404314

Digital Object Identifier
doi:10.2969/aspm/08010153

Mathematical Reviews number (MathSciNet)
MR3966588

Zentralblatt MATH identifier
07116427

#### Citation

Meyer, Ralf; Roy, Sutanu. Braided multiplicative unitaries as regular objects. Operator Algebras and Mathematical Physics, 153--178, Mathematical Society of Japan, Tokyo, Japan, 2019. doi:10.2969/aspm/08010153. https://projecteuclid.org/euclid.aspm/1566404314