$U_q(sl(n))$ Difference Operator Realization

TL;DR

This paper develops a systematic method for constructing differential and q-difference operator realizations of irreducible representations of $sl(n)$ and its quantum algebra $U_q(sl(n))$, providing explicit results for low-rank cases.

Contribution

It introduces a unified scheme for differential and q-difference operator realizations of irreducible representations of $sl(n)$ and $U_q(sl(n))$, extending previous methods.

Findings

Explicit q-difference operator realizations for $U_q(sl(2))$, $U_q(sl(3))$, and $U_q(sl(n))$

A systematic scheme applicable to all irreducible representations

Extension of classical realization methods to quantum groups

Abstract

A unified and systematic scheme for constraction of differential opreator realization of any irreducible representation of $sl(n)$ is developed. The $q$-analogue of this unified scheme is used to constract $q$-difference operator realization of any irreducible representation of $U_q(sl(n))$. Explicit results for $U_q(sl(2))$, $U_q(sl(3))$ and $U_q(sl(n))$ are given.