# Quantum Topology

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**Volume 8, Issue 2, 2017, pp. 361–379**

**DOI: 10.4171/QT/92**

Published online: 2017-05-29

Fourier transform for quantum $D$-modules via the punctured torus mapping class group

Adrien Brochier^{[1]}and David Jordan

^{[2]}(1) University of Edinburgh, UK

(2) University of Edinburgh, UK

We construct a certain cross product of two copies of the braided dual $\tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic double $E_H$, 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 [15], and hence construct a homomorphism to the Heisenberg double $D_H$, which is an isomorphism if $H$ is factorizable.

The universal property of $E_H$ endows it with an action by algebra automorphisms of the mapping class group ${\widetilde{\operatorname{SL}_2(\mathbb Z)}}$ of the punctured torus. One such automorphism we call the quantum Fourier transform; we show that when $H=U_q(\mathfrak{g})$, the quantum Fourier transform degenerates to the classical Fourier transform on $D(\mathfrak{g})$ as $q\to 1$.

*Keywords: *Quantum $D$-modules, elliptic braid group, mapping class groups

Brochier Adrien, Jordan David: Fourier transform for quantum $D$-modules via the punctured torus mapping class group. *Quantum Topol.* 8 (2017), 361-379. doi: 10.4171/QT/92