Cocycle twists

A cocycle twist deforms the multiplication of a graded algebra \(A\) using a \(2\)-cocycle \(\mu\) on a finite group \(G\) acting on \(A\) (equivalently, a comodule algebra structure). The twisted algebra \(A^{G,\mu}\) has the same Hilbert series and is again Artin–Schelter regular, but its finer geometry — the point scheme and the centre — can change completely.

For the four-dimensional Sklyanin algebra the relevant group is the Klein four-group \(G = (\mathbb{Z}/2)^2\). Twisting trades the elliptic point scheme for exactly 20 point modules (together with fat points), a phenomenon studied by Davies and placed in the framework of exotic elliptic algebras by Chirvasitu–Smith.

Several families on this site arise this way: the Sklyanin twist, the \(\mathrm{S}_\infty\) twist, and the Vancliff twist are all cocycle twists of their untwisted counterparts. Comparing a family with its twist shows that neither the point scheme nor the centre is a twist invariant.