Comodules of $U_q(sl_2)$ and modules of $SL_q(2)$ via quiver

Xiao-Wu Chen; Pu Zhang

paper

Comodules of $U_{q} (s l_{2})$ and modules of $S L_{q} (2)$ via quiver

arXiv:math/0409159

Abstract

The aim of this paper is to construct comodules of $U_{q} (s l_{2})$ and modules of $S L_{q} (2)$ via quiver, where $q$ is not a root of unity. By embedding the quantized algebra $U_{q} (s l_{2})$ into the path coalgebra $k D^{c}$ , where $D$ is the Gabriel quiver of $U_{q} (s l_{2})$ as a coalgebra, we obtain a basis of $U_{q} (s l_{2})$ in terms of combinations of paths in the quiver $D$ ; this special basis enable us to describe the category of $U_{q} (s l_{2})$ -comodules by certain representations of $D$ ; and this description further permits us to construct a class of modules of the quantum special linear group $S L_{q} (2)$ , from certain representations of $D$ , via the duality between $U_{q} (s l_{2})$ and $S L_{q} (2)$ .

19 pages

Comodules of Uq​(sl2​) and modules of SLq​(2) via quiver

Abstract

Comodules of $U_{q} (s l_{2})$ and modules of $S L_{q} (2)$ via quiver