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 $\Sigma_{g,n}^{\circ}$ 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.