Heisenberg realization for U_q(sln) on the flag manifold

TL;DR

This paper presents a Heisenberg realization of the quantum algebra U_q(sl_n) using q-difference operators on the flag manifold, derived via a Borel-Weil like approach, facilitating free field constructions.

Contribution

It introduces a novel Heisenberg realization of U_q(sl_n) on the flag manifold through a q-difference operator framework, expanding the tools for quantum algebra representations.

Findings

Realization expressed via q-difference operators

Constructed from U_q(sl_n) action on q-symmetric algebra

Applicable to free field realization of U_q(ŝl_n)

Abstract

We give the Heisenberg realization for the quantum algebra $U_q(sl_n)$, which is written by the $q$-difference operator on the flag manifold. We construct it from the action of $U_q(sl_n)$ on the $q$-symmetric algebra $A_q(Mat_n)$ by the Borel-Weil like approach. Our realization is applicable to the construction of the free field realization for the $U_q(\widehat{sl_n})$ [AOS].