In the matrix product states approach to n species diffusion processes the
stationary probability distribution is expressed as a matrix product state with
respect to a quadratic algebra determined by the dynamics of the process. The
quadratic algebra defines a noncommutative space with a SUq(n) quantum group
action as its symmetry. Boundary processes amount to the appearance of
parameter dependent linear terms in the algebraic relations and lead to a
reduction of the SUq(n) symmetry. We argue that the boundary operators of
the asymmetric simple exclusion process generate a tridiagonal algebra whose
irriducible representations are expressed in terms of the Askey-Wilson
polynomials. The Askey-Wilson algebra arises as a symmetry of the boundary
problem and allows to solve the model exactly.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium
on Non-Perturbative and Symmetry Methods in Field Theory (June 2006,
Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA