Braided algebra and dual bases of quantum groups


Majid, S. (2017). Braided algebra and dual bases of quantum groups. Perimeter Institute. https://pirsa.org/17070060


Majid, Shahn. Braided algebra and dual bases of quantum groups. Perimeter Institute, Jul. 19, 2017, https://pirsa.org/17070060


          @misc{ pirsa_17070060,
            doi = {10.48660/17070060},
            url = {https://pirsa.org/17070060},
            author = {Majid, Shahn},
            keywords = {Mathematical physics},
            language = {en},
            title = {Braided algebra and dual bases of quantum groups},
            publisher = {Perimeter Institute},
            year = {2017},
            month = {jul},
            note = {PIRSA:17070060 see, \url{https://pirsa.org}}

Shahn Majid Queen Mary - University of London (QMUL)


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.