# Highest $\ell$-Weight Representations and Functional Relations

## TL;DR

This paper explores highest $ ext{ell}$-weight representations of quantum loop algebras, comparing different representations and deriving new expressions for $L$-operators and their functional relations, contributing to the understanding of integrability structures.

## Contribution

It introduces new expressions for $L$-operators and clarifies relationships between representations, advancing the study of functional relations in quantum loop algebras.

## Key findings

- Derived explicit $L$-operator expressions.
- Established relationships between different representations.
- Analyzed functional relations in quantum integrability.

## Abstract

We discuss highest $\ell$-weight representations of quantum loop algebras and the corresponding functional relations between integrability objects. In particular, we compare the prefundamental and $q$-oscillator representations of the positive Borel subalgebras of the quantum group $\mathrm{U}_q(\mathcal L(\mathfrak{sl}_{l+1}))$ for arbitrary values of $l$. Our article has partially the nature of a short review, but it also contains new results. These are the expressions for the $L$-operators, and the exact relationship between different representations, as a byproduct resulting in certain conclusions about functional relations.