Conjectures on hidden onsager algebra symmetries in interacting quantum lattice models

We conjecture the existence of hidden Onsager algebra symmetries in two interacting quantum integrable lattice models, i.e. spin-1=2 XXZ model and spin-1 Zamolodchikov- Fateev model at arbitrary root of unity values of the anisotropy. The conjectures relate the Onsager generators to the conserved charges obtained from semi-cyclic transfer matrices. The conjectures are motivated by two examples which are spin-1=2 XX model and spin-1 U(1)-invariant clock model. A novel construction of the semi-cyclic transfer matrices of spin-1 Zamolodchikov-Fateev model at arbitrary root of unity values of the anisotropy is carried out via the transfer matrix fusion procedure.