2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/126973Baxter's Q-operator for the quantum transfer matrix of the XXZ spin-chain is constructed employing the representation theory of quantum groups. The spectrum of this Q-operator is discussed and novel functional relations which describe the finite temperature regime of the XXZ spin-chain are derived. For non-vanishing magnetic field the previously known Bethe ansatz equations can be replaced by a system of quadratic equations which is an important advantage for numerical studies. For vanishing magnetic field and rational coupling values it is argued that the quantum transfer matrix exhibits a loop algebra symmetry closely related to the one of the classical six-vertex transfer matrix at roots of unity.20 pages, v2: some minor style improvementsMathematical PhysicsStatistical MechanicsExactly Solvable and Integrable Systemstext