Unipotent quantum coordinate ring and prefundamental representations for types $A_n^{(1)}$ and $D_n^{(1)}$

Abstract

We give a new realization of the prefundamental representation L ȓ,a introduced by Hernandez and Jimbo, when the quantum loop algebra Uqpgq is of types A p1q n and D p1q n , and the r-th fundamental weight ̟r for types An and Dn is minuscule. We define an action of the Borel subalgebra Uqpbq of Uqpgq on the unipotent quantum coordinate ring associated to the translation by the negative of ̟r, and show that it is isomorphic to L ȓ,a . We then give a combinatorial realization of L r,a in terms of the Lusztig data of the dual PBW vectors.