Geometry of representations of quantum spaces

Abstract

The quantum plane $A=\mathbb{C}_\rho[x,y,z]$ with $\rho$ a root of unity has singularities in its representation variety $\mathbf{trep}_nA$ and its center $Z(A)$. Using the technique of a noncommutative blow-up, we prove that this technique fails in contrast to the 3-dimensional Sklyanin algebras if we want to resolve the singularities in the representation variety. However, we will see that the singularity of the center in the origin can be made better using this technique.