Fourier transform for quantum $D$modules via the punctured torus mapping class group
Abstract
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 wellknown 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 $D_H$, 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 $\widetilde{SL_2(\mathbb{Z})}$ 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 $q\to 1$.
 Publication:

arXiv eprints
 Pub Date:
 March 2014
 arXiv:
 arXiv:1403.1841
 Bibcode:
 2014arXiv1403.1841B
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Geometric Topology;
 16T05;
 16T25;
 20F36
 EPrint:
 12 pages, 1 figure. Final version, to appear in Quantum Topology