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

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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 ("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(slN), as t → 1. In the latter case, we produce representations of the double affine Hecke algebra of type A-1, for each n.
Original languageEnglish
Pages (from-to)2081-2105
Number of pages25
JournalInternational Mathematics Research Notices
Volume2009
Issue number11
DOIs
Publication statusPublished - 1 Feb 2009

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