Quantum symmetries of the deformation quantization of $SU(3)$

TL;DR

This paper establishes a criterion for extending classical group actions to deformation quantizations, computes an explicit example with $SU(3)_ heta$, and explores coactions on noncommutative spheres, revealing their properties and associated modules.

Contribution

It introduces a criterion for extending group actions to deformation quantizations and applies it to $SU(3)$, providing explicit examples and analyzing coactions on noncommutative spheres.

Findings

Explicit example of $SU(3)_ heta$ as a deformation of $SU(3)$

Extension criterion for group actions to quantum groups

Construction of coactions on noncommutative spheres and modules

Abstract

We prove a criterion of when a coaction of a compact Lie group on an algebra of continuous functions on a compact manifold extends to a coaction of deformation quantizations of the Lie group and the algebra. We compute an explicit example of a compact quantum group $SU(3)_\theta$, which arises as a deformation quantization of the Lie group $SU(3)$ by an action of its maximal torus. Using the criterion, we determine exactly when the action of $SU(3)$ on $S^5$ extends to a coaction of $SU(3)_\theta$ on the noncommutative 5-sphere $S^5_{\theta'}$. Furthermore, this coaction is shown to be cotransitive. A coaction of $SU(3)_\theta$ on the product of two noncommutative 5-sphere $(S^5\times S^5)_{\theta'}$ with nontrivial algebra of coinvariant elements is given and associated projective modules are constructed.