# Double-Bosonization and dual bases of quantum groups $\mathbb C_q [\mathop{SL} (2)]$ and $\mathbb C_q [\mathop{SL} (3)]$

2017/12/22 Fri 16:30 - 18:00

RIMS402号室

Shahn Majid

Queen Mary University of London

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 $B$ in the category of comodules of a coquasitriangular Hopf algebra $A$ has an associated coquasitriangular Hopf algebra $coD_A(B)$. As an application we find new generators for $\mathbb C_q [\mathop{SL} (2)]$ reduced at $q$ a primitive odd root of unity with the remarkable property that their monomials are essentially a dual basis to the standard PBW basis of the reduced Drinfeld-Jimbo quantum enveloping algebra $u_q(\mathfrak{sl} (2))$. Our methods apply in principle for general $\mathbb C_q [G]$ as we demonstrate for $\mathbb C_q [\mathop{SL} (3)]$ at certain odd roots of unity.