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

TL;DR

This paper investigates the structure of quantum graph algebras at roots of unity, revealing their centers, simple algebra properties, and PI degrees, with implications for algebraic and geometric representation theory.

Contribution

It provides a detailed analysis of the centers and algebraic properties of specialized quantum graph algebras at roots of unity, including their PI degrees and integrally closed centers.

Findings

Centers are explicitly described and are integrally closed rings.

Central localizations are central simple algebras with computable PI degrees.

The structure of invariant subalgebras under the small quantum group action is characterized.

Abstract

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.