Quantum metrics from length functions on quantum groups
Are Austad, David Kyed
TL;DR
This work develops a framework for obtaining compact quantum metric spaces from length functions on compact quantum groups. It shows that for coamenable quantum groups of Kac type the quantum metric data is captured by the central function algebra $C_z({\mathbb{G}})$, enabling new genuine quantum-group examples via a central-approximation mechanism. A Dirac-type operator built from a length function together with a conditional expectation $E:C({\mathbb{G}})\to C_z({\mathbb{G}})$ lifts Lip-norm information from the fusion algebra to the full quantum group, and Kaad’s finite-diameter criterion yields compact quantum metric spaces. The paper also provides explicit examples, including $SU_q(2)$ (at $q=\pm1$), $O_2^+$, $SO(3)$, and $S_4^+$, showing that length-function–induced metrics yield Lip-norms on genuine quantum groups and connect to finite-dimensional truncations and truncation frameworks. Overall, it broadens the catalog of quantum groups equipped with natural quantum metrics arising from length functions and clarifies the role of central data in these metric properties.
Abstract
We study the quantum metric structure arising from length functions on quantum groups and show that for coamenable quantum groups of Kac type, the quantum metric information is captured by the algebra of central functions. Using this, we provide the first examples of length functions on (genuine) quantum groups which give rise to compact quantum metric spaces.