Cocycle twists of 4-dimensional Sklyanin algebras

TL;DR

This paper investigates how cocycle twists by the Klein four-group alter the geometric properties of 4-dimensional Sklyanin algebras and related coordinate rings, revealing significant differences from their original forms.

Contribution

It demonstrates that cocycle twisting changes the geometric structure of Sklyanin algebras and their coordinate rings, providing explicit descriptions and classifications.

Findings

Twisted algebra $A^{G,0}$ has only 20 point modules.

Twisted algebra $A^{G,0}$ has infinitely many fat point modules of multiplicity 2.

The ring $B^{G,0}$ is classified as a sheaf of orders over an elliptic curve.

Abstract

We study cocycle twists of a 4-dimensional Sklyanin algebra $A$ and a factor ring $B$ which is a twisted homogeneous coordinate ring. Twisting such algebras by the Klein four-group $G$, we show that the twists $A^{G,\mu}$ and $B^{G,\mu}$ have very different geometric properties to their untwisted counterparts. For example, $A^{G,\mu}$ has only 20 point modules and infinitely many fat point modules of multiplicity 2. The ring $B^{G,\mu}$ falls under the purview of Artin and Stafford's classification of noncommutative curves, and we describe it using a sheaf of orders over an elliptic curve.