  We study the boundary symmetries of models of nonequilibrium physics where
the steady state behaviour strongly depends on the boundary rates. Within the
matrix product state approach to many-body systems the physics is described in
terms of matrices defining a noncommutative space with a quantum group
symmetry. Boundary processes lead to a reduction of the bulk symmetry. We argue
that the boundary operators of an interacting system with simple exclusion
generate a tridiagonal algebra whose irreducible representations are expressed
in terms of the Askey-Wilson polynomials. We show that the boundary algebras of
the symmetric and the totally asymmetric processes are the proper limits of the
partially asymmetric ones. In all three type of processes the tridiagonal
algebra arises as a symmetry of the boundary problem and allows for the exact
solvability of the model.
