Universal integrability objects

TL;DR

This paper reviews the quantum group approach to quantum integrable systems, focusing on the quantum group $U_q( ext{L}( ext{sl}_2))$, clarifies functional relations, and explores the connection between representations and transfer operators.

Contribution

It provides a complete set of functional relations and clarifies the connection between representations and universal transfer and Q-operators in quantum integrable systems.

Findings

Corrected previous inaccuracies in functional relations

Established detailed connection between representations and transfer operators

Enhanced understanding of quantum group approach in integrable systems

Abstract

We discuss the main points of the quantum group approach in the theory of quantum integrable systems and illustrate them for the case of the quantum group $U_q(\mathcal L(\mathfrak{sl}_2))$. We give a complete set of the functional relations correcting inexactitudes of the previous considerations. A special attention is given to the connection of the representations used to construct the universal transfer operators and $Q$-operators.