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

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 $S_{{\theta }^{\prime }}^5$. Furthermore, this coaction is shown to be cotransitive. A coaction of $SU(3{)}_{\theta }$ on the product of two noncommutative 5-sphere $(S^5\times S^5{)}_{{\theta }^{\prime }}$ with nontrivial algebra of coinvariant elements is given and associated projective modules are constructed.

14 pages