# Braided algebra and dual bases of quantum groups

### APA

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

### MLA

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

### BibTex

@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)

## Abstract

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.