The quantum metric structure of quantum SU(2)
Jens Kaad, David Kyed
TL;DR
The paper constructs a two-parameter family of Dirac operators $D_{t,q}$ on quantum $SU(2)$, decomposing them into horizontal and vertical parts with bounded twisted commutators, and proves that the associated Lip-norms make $(C(SU_q(2)),L_{t,q}^{ ext{max}})$ a compact quantum metric space. It develops a comprehensive toolkit, including a quantum Berezin transform and Schur multiplier techniques, to lift the Podleś sphere’s metric structure to spectral bands and then to the full quantum group, establishing quantum Gromov–Hausdorff continuity in the deformation parameters $(t,q)$ and convergence to the classical $(C(SU(2)),L_{ ext{Lip}})$ as $(t,q) o(1,1)$. The work integrates twisted noncommutative geometry with quantum group symmetry, providing a robust framework for approximating quantum metric spaces via fuzzy models and offering a precise bridge between quantum and classical metric geometries. These results have potential implications for noncommutative geometry on $q$-deformed spaces and for the study of metric aspects of quantum groups. The combination of spectral-bands analysis, Berezin-type approximations, and operator-algebraic tools yields a versatile approach to metric aspects in noncommutative quantum groups.
Abstract
We introduce a two parameter family of Dirac operators on quantum SU(2) and analyse their properties from the point of view of non-commutative metric geometry. It is shown that these Dirac operators give rise to compact quantum metric structures, and that the corresponding two parameter family of compact quantum metric spaces varies continuously in Rieffel's quantum Gromov-Hausdorff distance. This continuity result includes the classical case where we recover the round 3-sphere up to a global scaling factor on the metric. Our main technical tool is a quantum SU(2) analogue of the Berezin transform, together with its associated fuzzy approximations, the analysis of which also leads to a systematic way of approximating Lipschitz operators by means of polynomial expressions in the generators.