Braided quantum $\mathrm{SU}(2)$ group - a case study

TL;DR

This paper advances the understanding of braided quantum SU(2) groups by analyzing Haar measures, antipodes, and braided tensor products, providing new insights into their algebraic structures and equivalences.

Contribution

It constructs the scaling group, antipode, and describes the braided Hopf algebra for the braided SU(2) quantum group, clarifying its algebraic properties and relations.

Findings

Established existence of Haar measure for braided SU(2)

Constructed the scaling group and antipode with polar decomposition

Proved the equivalence of different approaches to bosonization and braided tensor products

Abstract

We continue the study of the braided compact quantum group $\mathrm{SU}_q(2)$ for complex $q$ satisfying $0<|q|<1$ introduced by Kasprzak, Meyer, Roy and Woronowicz (J. Noncommut. Geom. 10(4):1611-1625, 2016). We address such aspects as existence of the Haar measure, construct the scaling group, the antipode and its polar decomposition and describe the related braided Hopf algebra. We also study when the braided flip extends to a completely bounded map and establish equivalence between the two approaches to bosonization and braided tensor product taken in the literature (Kasprzak, Meyer, Roy, Woronowicz J. Noncommut. Geom. 10(4):1611-1625, 2016 vs. Meyer, Roy Woronowicz Internat. J. Math. 25(2):1450019, 37, 2014, Roy Int. Math. Res. Not. (14):11791--11828, 2023 and De Commer, Krajczok arXiv:2412.17444, to appear in J. Operator Th.).