## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Operator Algebras and Mathematical Physics, M. Izumi, Y. Kawahigashi, M. Kotani, H. Matui and N. Ozawa, eds. (Tokyo: Mathematical Society of Japan, 2019), 153 - 178

### Braided multiplicative unitaries as regular objects

Ralf Meyer and Sutanu Roy

#### 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**

Received: 31 December 2016

Revised: 20 April 2017

First available in Project Euclid:
21 August 2019

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/aspm/08010153

**Mathematical Reviews number (MathSciNet)**

MR3966588

**Zentralblatt MATH identifier**

07116427

**Subjects**

Primary: 46L89: Other "noncommutative" mathematics based on C-algebra theory [See also 58B32, 58B34, 58J22]

Secondary: 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37] 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]

**Keywords**

quantum group braided quantum group multiplicative unitary braided multiplicative unitary tensor category quantum group representation quantum group morphism Tannaka–Krein Theorem

#### 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