Resolutions of Type $\mathbb{A}$ Quantum Surface Singularities

Let $B={\mathbb{k}}_q[u,v{]}^{C_{n+1}}$ be a Type ${\mathbb{A}}_n$ quantum Kleinian singularity, which is an example of a noncommutative surface singularity. This singularity is known to have a noncommutative quasi-crepant resolution $\mathrm{\Lambda }$, which is an “algebraic” resolution of $B$. We construct a category $\mathcal{X}$ which serves as a “geometric” resolution of $B$ by adapting techniques from quiver GIT and show that $\mathcal{X}$ and $\text{mod-}\mathrm{\Lambda }$ are derived equivalent. Furthermore, we show that the intersection arrangement of lines in the exceptional locus of $\mathcal{X}$ corresponds to a Type ${\mathbb{A}}_n$ Dynkin diagram. This generalises the geometric McKay correspondence for classical Kleinian singularities.