Cocycle Twists of Algebras

TL;DR

This paper studies cocycle twists of algebras, especially Sklyanin algebras, showing that many algebraic and geometric properties are preserved or transformed, with applications to noncommutative geometry and module theory.

Contribution

It introduces a framework for cocycle twists of graded algebras, demonstrating property preservation and analyzing geometric implications for Sklyanin algebras under group actions.

Findings

Many properties like noetherianity and regularity are preserved under twists.

Twisted Sklyanin algebras have fewer point modules, affecting their geometric interpretation.

The structure of twisted coordinate rings aligns with noncommutative curve classifications.

Abstract

Let $A$ be a $k$-algebra where $k$ is an algebraically closed field and $G$ be a finite abelian group for which the characteristic of $k$ does not divide $|G|$. If $G$ acts on $A$ by $k$-algebra automorphisms then the action induces a $G$-grading on $A$ which, in conjunction with a normalised 2-cocycle of the group, can be used to twist the multiplication of the algebra. Such twists can be formulated as Zhang twists as well as in the language of Hopf algebras. We investigate such cocycle twists with an emphasis on the situation where $A$ also possesses a connected graded structure and the action of $G$ respects this grading. We show that many properties are preserved under such twists; for example the strongly noetherian property, finite global dimension and Artin-Shelter regularity. The above concepts are then applied to the 4-dimensional Sklyanin algebras, $A:=A(\alpha,\beta,\gamma)$. We define an action of the Klein four-group on $A$ such that the action restricts to a special geometric factor ring $B$ (a twisted homogeneous coordinate ring). The cocycle twists of these algebras, denoted by $A^{G,\mu}$ and $B^{G,\mu}$ respectively, have very different geometric properties to their untwisted counterparts. While $A$ has point modules parameterised by an elliptic curve $E$ and four extra points, $A^{G,\mu}$ has only 20 point modules (when an automorphism associated to $E$ has infinite order). The point modules over $A$ can be used to construct fat point modules of multiplicity 2 over $A^{G,\mu}$, and there are isomorphisms among such objects corresponding to orbits of a natural action of $G$ on $E$. Furthermore, the ring $B^{G,\mu}$ can be described in terms of Artin and Stafford's classification of noncommutative curves.