Quantum D-modules, Elliptic Braid Groups, and Double Affine Hecke Algebras

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 (“Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity,” Communications in Mathematical Physics 172 (1995): 467–516; “Braided Groups and Quantum Fourier Transform,” Journal of Algebra 166 (1994): 506–28), and also certain geometric constructions of Calaque, Enriquez, and Etingof (“Universal KZB equations I: The elliptic case,” (2007): preprint arXiv:math/0702670) 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 $t\rightarrow1$. In the latter case, we produce representations of the double affine Hecke algebra of type $A_{n-1}$, for each n.