Cocycle twists of 4-dimensional Sklyanin algebras

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.