On the structure and representations of quantum graph algebras at roots of unity

We study the specializations ${\mathcal{L}}_{g,n}^{\epsilon }$ at roots of unity $\epsilon$ of odd order of the graph algebras, associated to a simply-connected complex semi-simple algebraic group $G$ and a compact oriented surface ${\mathrm{\Sigma }}_{g,n}^{\circ }$ with genus $g$, $n$ punctures, and one boundary component. We prove that the central localizations of ${\mathcal{L}}_{g,n}^{\epsilon }$ and of its subalgebra ${\mathcal{L}}_{g,n}^{u_{\epsilon }}$ of invariant elements under the coadjoint action of a small quantum group, are central simple algebras of PI degrees that we compute. Also, we describe their centers, and show they are integrally closed rings.