A complete classification of irreducible representations of the $C^*$-algebra $C(G_q)$, where $G_q$ is the $q$-deformation of a classical simply connected semisimple compact Lie group, was provided by Korogodski and Soĭbel’man in [1, 3]. This result was later extended by Neshveyev and Tuset [2] to classify representations of the $C^*$-algebra $C(G_q/H_q)$, where $H_q$ is the $q$-deformation of a closed Poison-Lie subgroup $H$ of $G$.
Question: If we remove the simply connected condition from $G$ (i.e., $G$ is a connected semisimple and compact classical Lie group, not simply connected), what is known about the representation theory of $C(G_q)$? Specifically, is there any classification for $C(G_q)$ for some not simply connected, but connected semisimple and compact classical matrix Lie group? More precisely, is the representation theory of $C\left(SO_q(n)\right)$ known or possible for $n=3$, if not for all $n$?
Thanks in advance for any input.
Korogodski, Leonid I.; Soĭbel’man, Yan S., Algebras of functions on quantum groups: Part I, Mathematical Surveys and Monographs. 56. Providence, RI: American Mathematical Society (AMS). ix, 150 p. (1998). ZBL0923.17017.
Neshveyev, Sergey; Tuset, Lars, Quantized algebras of functions on homogeneous spaces with Poisson stabilizers, Commun. Math. Phys. 312, No. 1, 223-250 (2012). ZBL1250.22017.
Soĭbel’man, Ya. S., The algebra of functions on a compact quantum group, and its representations, Leningr. Math. J. 2, No. 1, 161-178 (1991). ZBL0718.46012.
