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Twists of rational Cherednik algebras

Yuri Bazlov, Arkady Berenstein, Edward Jones-Healey, Alexander McGaw

TL;DR

The paper shows that for even $m$, braided Cherednik algebras $\underline{H}_{\underline c}(\mu(G(m,p,n)))$ are cocycle twists of rational Cherednik algebras $H_c(G(m,p,n))$ via a finite abelian group cocycle $\mathcal{F}$. It constructs an explicit isomorphism $\underline H_{\underline c}(\mu(G(m,p,n))) \cong (H_c(G(m,p,n)),\star)$, with a detailed mapping of generators and a verification of all defining relations, aided by a PBW-type argument. This twist transfers finite-dimensional representations from the rational to the braided setting, and, in particular, provides a method to obtain nontrivial finite-dimensional representations of braided algebras from rational ones; it also shows that twisting can nontrivially permute one-dimensional characters for $G(2,1,n)$. The framework clarifies the role of mystic reflection groups $\mu(G(m,p,n))$ and opens a pathway to study the representation theory of braided Cherednik algebras through their rational counterparts.

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.

Twists of rational Cherednik algebras

TL;DR

The paper shows that for even , braided Cherednik algebras are cocycle twists of rational Cherednik algebras via a finite abelian group cocycle . It constructs an explicit isomorphism , with a detailed mapping of generators and a verification of all defining relations, aided by a PBW-type argument. This twist transfers finite-dimensional representations from the rational to the braided setting, and, in particular, provides a method to obtain nontrivial finite-dimensional representations of braided algebras from rational ones; it also shows that twisting can nontrivially permute one-dimensional characters for . The framework clarifies the role of mystic reflection groups and opens a pathway to study the representation theory of braided Cherednik algebras through their rational counterparts.

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 , when 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.
Paper Structure (27 sections, 17 theorems, 56 equations)

This paper contains 27 sections, 17 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Reflection groups
  3. Complex reflection groups.
  4. Mystic reflection groups.
  5. A presentation of mystic reflection groups.
  6. Rational and braided Cherednik algebras
  7. Rational Cherednik algebras.
  8. Braided Cherednik algebras.
  9. The cocycle twist
  10. Quasitriangular structures and $2$-cocycles.
  11. Twists.
  12. The cocycle $\mathcal{F}$.
  13. $\mathcal{F}$-twisted product of $T$-eigenvectors.
  14. The Main Result
  15. Statement of the main theorem.
  16. ...and 12 more sections

Key Result

Theorem 2.1

For $V$ and $G$ as above, the invariant ring $S(V)^G$ is a polynomial algebra if and only if $G$ is a complex reflection group.

Theorems & Definitions (39)

  • Theorem 2.1: Chevalley-Shephard-Todd
  • Definition 2.2
  • Theorem 2.3: a presentation of $\mu(G(m,p,n))$
  • proof
  • Definition 3.1
  • Theorem 3.2: The PBW theorem for rational Cherednik algebras, etingof2002symplectic
  • Definition 3.3
  • Definition 3.4
  • Definition 4.1
  • Definition 4.2
  • ...and 29 more