Cocycle Twists of Algebras
Andrew Davies

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 be a -algebra where is an algebraically closed field and be a finite abelian group for which the characteristic of does not divide . If acts on by -algebra automorphisms then the action induces a -grading on 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 also possesses a connected graded structure and the action of 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, .…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Algebraic Geometry and Number Theory
