Quantum Groups and Stohastic Models

Abstract

The aim of this paper is to show that stochastic models provide a very good playground to enhance the utility of quantum groups. Quantum groups arise naturally and the deformation parameter has a direct physical meaning for diffusion systems where it is just the ratio of left/right probability rate. In the matrix product state approach to diffusion processes the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra which defines a noncommutative space with a quantum group action as its symmetry. Boundary processes amount for the appearance of parameter-dependent linear terms in the algebra which leads to a reduction of the bulk symmetry.