Geometry of representations of quantum spaces

TL;DR

This paper investigates the geometric structure of representations of quantum spaces, specifically the quantum plane at roots of unity, and explores the effectiveness of noncommutative blow-ups in resolving singularities.

Contribution

It demonstrates the limitations of noncommutative blow-ups for resolving singularities in the representation variety of quantum planes, contrasting with their success in Sklyanin algebras.

Findings

Noncommutative blow-up fails to resolve singularities in the representation variety of quantum planes.

The technique can improve the singularity of the center at the origin.

Contrast with Sklyanin algebras where the technique is more effective.

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.