  Stationary probability distributions for stochastic processes on linear
chains with closed or open ends are obtained using the matrix product Ansatz.
The matrices are representations of some quadratic algebras. The algebras and
the types of representations considered depend on the boundary conditions. In
the language of quantum chains we obtain the ground state of N-state quantum
chains with free boundary conditions or with non-diagonal boundary terms at one
or both ends. In contrast to problems involving the Bethe Ansatz, we do not
have a general framework for arbitrary N which when specialized, gives the
known results for N=2; in fact, the N=2 and N>2 cases appear to be very
different.
