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

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 $B^{\underline{\mathrm{op}}}⋊A⋉B^{\ast }$. As an application we find new generators for $c_q[S{L}_2]$ 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 $u_q(s{l}_2)$. Our methods apply in principle for general $c_q[G]$ as we demonstrate for $c_q[S{L}_3]$ at certain odd roots of unity.

30 pages latex; shortened at request of a referee, made clarifications and minor corrections