The geometry of representations of 3-dimensional Sklyanin algebras

Abstract

The representation scheme ${\mathtt{rep}}_{n}A$ of the 3-dimensional Sklyanin algebra $A$ associated to a plane elliptic curve and n-torsion point contains singularities over the augmentation ideal $\mathfrak{m}$. We investigate the semi-stable representations of the noncommutative blow-up algebra $B=A\oplus \mathfrak{m}t\oplus {\mathfrak{m}}^2{t}^2\oplus ...$ to obtain a partial resolution of the central singularity $\mathtt{proj}Z(B)\to \mathtt{spec}Z(A)$ such that the remaining singularities in the exceptional fiber determine an elliptic curve and are all of type $\mathbb{C}\times {\mathbb{C}}^2/{\mathbb{Z}}_n$.

Cited by in corpus (2)

• Poisson geometry of PI 3-dimensional Sklyanin algebras
• Explicit representations of 3-dimensional Sklyanin algebras associated to a point of order 2