## Abstract

We construct a certain cross product of two copies of the braided dual H~ of a quasitriangular Hopf algebra H, which we call the elliptic double EH, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attached to H. We show that the elliptic double is the universal source of such representations. We recover the representations of the punctured torus braid group obtained in arXiv:0805.2766, and hence construct a homomorphism to the Heisenberg double DH, which is an isomorphism if H is factorizable.

The universal property of EH endows it with an action by algebra automorphisms of the mapping class group SL2(Z)˜ of the punctured torus. One such automorphism we call the quantum Fourier transform; we show that when H=Uq(g), the quantum Fourier transform degenerates to the classical Fourier transform on D(g) as q→1.

The universal property of EH endows it with an action by algebra automorphisms of the mapping class group SL2(Z)˜ of the punctured torus. One such automorphism we call the quantum Fourier transform; we show that when H=Uq(g), the quantum Fourier transform degenerates to the classical Fourier transform on D(g) as q→1.

Original language | English |
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Pages (from-to) | 361-379 |

Number of pages | 18 |

Journal | Quantum Topology |

Volume | 8 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2017 |