The noncommutative space of stochastic diffusion systems

In the matrix product ground 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. We show that the quadratic algebra defines a noncommutative space with a GL(q) (n) quantum group action as its symmetry. Boundary processes account for the appearance of parameter-dependent linear terms in the algebraic relations. We argue that for systems with boundary conditions the diffusion algebras are also obtained either by a shift of basis in the n-dimensional quantum plane or by an appropriate change of basis in a lower dimensional one which leads to a reduction of the GL(q) (n) symmetry.