Boundary Algebra and Exact Solvability of the Asymmetric Exclusion Process

We consider a lattice driven diffusive system with Uq(su(2)) invariance in the bulk. Within the matrix product states approach the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra. Boundary processes amount to the appearance of parameter dependent linear terms in the algebraic relations and lead to a reduction of the bulk symmetry. We find the boundary quantum group of the process to be a tridiagonal algebra, the linear covariance algebra for the bulk Uq(su(2)) symmetry, which allows for the exact solvability.