# Co-double bosonisation and dual bases of c_q[SL_2] and c_q[SL_3]

## TL;DR

This paper introduces a dual version of double-bosonisation theory and applies it to find new generators and dual bases for quantum coordinate algebras c_q[SL_2] and c_q[SL_3] at roots of unity.

## Contribution

It develops a dual double-bosonisation theorem and constructs new generators for quantum groups at roots of unity, revealing dual bases related to PBW bases.

## Key findings

- New generators for c_q[SL_2] at odd roots of unity.
- Dual bases approximately dual to PBW bases.
- Method extends to c_q[SL_3] at certain roots.

## Abstract

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 $B^{\underline{\rm op}}\rtimes A \ltimes B^*$. As an application we find new generators for $c_q[SL_{2}]$ 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 $u_q(sl_{2})$. Our methods apply in principle for general $c_{q}[G]$ as we demonstrate for $c_q[SL_3]$ at certain odd roots of unity.