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

TL;DR

This paper presents a new realization of prefundamental representations for quantum loop algebras of types A and D, using unipotent quantum coordinate rings and Lusztig data, advancing understanding of their structure.

Contribution

It introduces a novel realization of prefundamental representations via unipotent quantum coordinate rings and Lusztig data for types A and D quantum loop algebras.

Findings

Isomorphism between the Borel subalgebra action and prefundamental representations

Explicit combinatorial realization using Lusztig data

Extension of known structures to types A and D

Abstract

We give a new realization of the prefundamental representations $L^\pm_{r,a}$ introduced by Hernandez and Jimbo, when the quantum loop algebra $U_q(\mathfrak{g})$ is of types $A_n^{(1)}$ and $D_n^{(1)}$, and the $r$-th fundamental weight $\varpi_r$ for types $A_n$ and $D_n$ is minuscule. We define an action of the Borel subalgebra $U_q(\mathfrak{b})$ of $U_q(\mathfrak{g})$ on the unipotent quantum coordinate ring associated to the translation by $-\varpi_r$, and show that it is isomorphic to $L^\pm_{r,a}$. We then give a combinatorial realization of $L^+_{r,a}$ in terms of the Lusztig data of the dual PBW vectors.