Quantum D-modules, elliptic braid groups, and double affine Hecke algebras

TL;DR

This paper constructs representations of elliptic braid groups and double affine Hecke algebras using quantum D-modules over ribbon Hopf algebras, generalizing previous geometric and algebraic approaches.

Contribution

It introduces a new framework for building such representations from quantum D-modules, extending prior constructions by Lyubashenko, Majid, Calaque, Enriquez, and Etingof.

Findings

Constructed elliptic braid group representations from quantum D-modules.

Derived double affine Hecke algebra representations of type A_{n-1}.

Unified geometric and algebraic approaches to these representations.

Abstract

We build representations of the elliptic braid group from the data of a quantum D-module M over a ribbon Hopf algebra U. The construction is modelled on, and generalizes, similar constructions by Lyubashenko and Majid, and also certain geometric constructions of Calaque, Enriquez, and Etingof concerning trigonometric Cherednik algebras. In this context, the former construction is the special case where M is the basic representation, while the latter construction can be recovered as a quasi-classical limit of U=U_t(sl_N), as t limits 1. In the latter case, we produce representations of the double affine Hecke algebra of type A_{n-1}, for each n.