q-difference intertwining operators for $U_q(sl(n))$: general setting and the case $n=3$

TL;DR

This paper constructs and analyzes specific representations and q-difference intertwining operators for the quantum algebra U_q(sl(n)), providing explicit forms for n=3 and general results for arbitrary n.

Contribution

It introduces a new class of representations of U_q(sl(n)) labeled by complex parameters and explicitly constructs q-difference intertwining operators, extending understanding of quantum group symmetries.

Findings

Explicit representations for n=3 are provided.

Conditions for reducibility of the representations are established.

General results for arbitrary n are discussed.

Abstract

We construct representations $\hat\pi_{\br}$ of the quantum algebra $U_q(sl(n))$ labelled by $n-1$ complex numbers $r_i$ and acting in the space of formal power series of $n(n-1)/2$ non-commuting variables. These variables generate a flag manifold of the matrix quantum group $SL_q(n)$ which is dual to $U_q(sl(n))$. The conditions for reducibility of $\hat\pi_{\br}$ and the procedure for the construction of the $q$ - difference intertwining operators are given. The representations and $q$ - difference intertwining operators are given in the most explicit form for $n=3$. In the Note Added some general results for arbitrary $n$ are given.