Quantum groups and functional relations for arbitrary rank

TL;DR

This paper explores quantum integrable systems linked to quantum loop algebras, deriving transfer operators, functional relations, and Bethe equations for arbitrary rank, advancing the mathematical understanding of these models.

Contribution

It introduces new determinant forms of transfer operators and derives master $TQ$- and $TT$-relations for arbitrary rank quantum loop algebras.

Findings

Factorized form of transfer operators for infinite dimensional representations

Determinant form of transfer operators for finite dimensional representations

Derivation of master $TQ$- and $TT$-relations and nested Bethe equations

Abstract

The quantum integrable systems associated with the quantum loop algebras $\mathrm U_q(\mathcal L(\mathfrak{sl}_{\, l + 1}))$ are considered. The factorized form of the transfer operators related to the infinite dimensional evaluation representations is found and the determinant form of the transfer operators related to the finite dimensional evaluation representations is obtained. The master $TQ$- and $TT$-relations are derived. The operatorial $T$- and $Q$-systems are found. The nested Bethe equations are obtained.