q-Deformed Onsager symmetry in boundary integrable models related to twisted U$_{q^{1/2}}(\hat{sl_2})$ symmetry

We consider an unified model, called ancestor model, associated with twisted trigonometric $R$ matrix which model leads to several descendant integrable lattice models related to the U$_{q^{1/2}}(\hat{sl_2})$ symmetry. Boundary operators compatible with integrability are introduced to this model. Reflection and dual reflection equations to ensure integrability of the system are shown to be same as the untwisted case. It follows that underlying symmetry of the ancestor model with integrable boundaries is identified with the q-deformed analogue of Onsager's symmetry. The transfer matrix and its related mutually commuting quantities are expressed in terms of an abelian subalgebra in the $q$-Onsager algebra. It is illustrated that the generalized McCoy-Wu model with general open boundaries enjoys this symmetry.

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