Twists of rational Cherednik algebras

TL;DR

This paper demonstrates that braided Cherednik algebras are cocycle twists of rational Cherednik algebras for certain complex reflection groups, providing new insights into their structure and representations.

Contribution

It establishes a connection between braided and rational Cherednik algebras via cocycle twists, offering a novel construction of mystic reflection groups.

Findings

Braided Cherednik algebras are cocycle twists of rational Cherednik algebras for even m.

A braided Cherednik algebra has a finite-dimensional representation iff its rational counterpart does.

New construction of mystic reflection groups with Artin-Schelter regular rings.

Abstract

We show that braided Cherednik algebras introduced by the first two authors are cocycle twists of rational Cherednik algebras of the imprimitive complex reflection groups $G(m,p,n)$, when $m$ is even. This gives a new construction of mystic reflection groups which have Artin-Schelter regular rings of quantum polynomial invariants. As an application of this result, we show that a braided Cherednik algebra has a finite-dimensional representation if and only if its rational counterpart has one.