- journal article metadata
European Mathematical Society Publishing House
2017-06-01 23:45:01
Quantum Topology
Quantum Topol.
QT
1663-487X
1664-073X
General
10.4171/QT
http://www.ems-ph.org/doi/10.4171/QT
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
8
2017
2
Fourier transform for quantum $D$-modules via the punctured torus mapping class group
Adrien
Brochier
Universität Hamburg, HAMBURG, GERMANY
David
Jordan
University of Edinburgh, EDINBURGH, UNITED KINGDOM
Quantum $D$-modules, elliptic braid group, mapping class groups
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 $q\to 1$.
Associative rings and algebras
Group theory and generalizations
361
379
10.4171/QT/92
http://www.ems-ph.org/doi/10.4171/QT/92