TY - JOUR

T1 - Quantum D-modules, elliptic braid groups, and double affine hecke algebras

AU - Jordan, D.

PY - 2009/2/1

Y1 - 2009/2/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?partnerID=yv4JPVwI&eid=2-s2.0-67650803047&md5=95087181a051e0a376ebba16a1d80bb3

U2 - 10.1093/imrp/rnp012

DO - 10.1093/imrp/rnp012

M3 - Article

AN - SCOPUS:67650803047

VL - 2009

SP - 2081

EP - 2105

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 11

ER -