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Braided algebra and dual bases of quantum groups

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The talk is based on my recent work with Ryan Aziz. We find a dual version of a previous double-bosonisation theorem whereby each finite-dimensional braided-Hopf algebra in the category of corepresentations of a coquasitriangular Hopf algebra gives a new larger coquasitriangular Hopf algebra, for example taking c_q[SL_2] to c_q[SL_3] for these quantum groups reduced at certain odd roots of unity. As an application we find new generators for c_q[SL2] with the remarkable property that their monomials are essentially a dual basis to the standard PBW basis of the reduced quantum enveloping algebra u_q(sl2). This allows one to calculate  Fourier transform and other results for such quantum groups. Our method also works for even roots of unity where we obtain new finite-dimensional quantum groups, including an 8-dimensional one at q=-1. Our method
can be used to construct  many other new finite-dimensional  quasitriangular Hopf algebras and their duals that could be fed into applications in quantum gravity and quantum computing.