Cocycle twists of algebras, representations and orders

Abstract: A way to deform an associative algebra $A$ is to twist the multiplication by a 2-cocycle on a group or a Hopf algebra acting on $A$. I am interested to know to what extent the representations (and ring-theoretic and homological properties) of the twist are determined by those of $A$. My case in point will be rational Cherednik algebras over complex reflection groups: twists of these well-studied objects give algebras, with similar PBW bases, over "mystic reflection groups", and for some of them we can give an explicit combinatorial description of standard modules(arXiv:2501.06673, with Jones-Healey). Twists descend to finite-dimensional quotients of Cherednik algebras at $t=0$, and over number fields, seem to produce their forms (in the sense that the twist trivializes over a field extension). This hints at an interplay between twists and arithmetic; if time permits, I will mention a possible connection to Hopf-Galois structures on fields.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( paper | video )

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Noncommutative geometry in NYC

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