**Quantum symmetries and exactly solvable models of nonequilibrium physics**

**Abstract:**

We consider driven diffusive systems on one dimensional lattice
as models of nonequilibrium physics. These systems provide a very good
playground to enhance utility of quantum group symmetries. The
deformation parameter has a direct physical meaning -
it is the ratio of left/right diffusion rates. We apply an approach
inspired by the inverse scttering method where the stationary probability
distribution is expressed in terms of noncommutative matrices that
form a comodule of SU_q(n) and this is the model bulk
symmetry. Boundary processes amount to a reduction of the bulk symmetry.
We argue that for open systems the
boundary operators generate a tridiagonal algebra whose irriducible
representations are given by the
Askey-Wilson polynomials and allow to solve the model exactly.