Quantum groups and functional relations for lower rank

TL;DR

This paper constructs universal integrability objects for quantum integrable systems related to U_q(sl_2), proving functional relations independent of specific representations, and applies these to spin chain models.

Contribution

It provides a detailed construction and proof of universal functional relations for quantum groups of lower rank, extending previous work to general gradation and twisting cases.

Findings

Universal integrability objects are constructed for U_q(sl_2).

Functional relations are proven independently of quantum space representations.

Application to discrete spin chain models is described.

Abstract

A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group $\mathrm U_q(\mathcal L(\mathfrak{sl}_2))$ is given. The full proof of the functional relations in the form independent of the representation of the quantum group on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described. This is a degression of the corresponding consideration for the case of the quantum group $\mathrm U_q(\mathcal L(\mathfrak{sl}_3))$ with an extensions to the higher spin case.