Common structures of quantum field theories and lattice systems through boundary symmetry

The sine-Gordon model and affine Toda field theories on the half-line, on the one hand, and the XXZ spin chain with nondiagonal boundary term and interacting many-body lattice systems with a flow, on the other, have a common characteristic. They possess nonlocal conserved boundary charges, generating the Askey-Wilson algebra, a coideal subalgebra of the bulk quantized affine symmetry. We argue that the boundary Askey-Wilson symmetry is the deep algebraic property allowing for integrability of the physical system in consideration