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 AG,μAG,μ and BG,μBG,μ have very different geometric properties to their untwisted counterparts. For example, AG,μAG,μ has only 20 point modules and infinitely many fat point modules of multiplicity 2. The ring BG,μBG,μ 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.