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 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 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 

Pages (fromto)  361379 
Number of pages  18 
Journal  Quantum Topology 
Volume  8 
Issue number  2 
DOIs  
Publication status  Published  2017 
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David Jordan
 School of Mathematics  Personal Chair of Categorical Symmetry
Person: Academic: Research Active