Symmetry, integrable chain models and stochastic processes

A general way to construct chain models with certain Lie algebraic or quantum Lie algebraic symmetries is presented. These symmetric models give rise to series of integrable systems. As an example the chain models with A, symmetry and the related Temperley-Lieb algebraic structures and representations are discussed. It is shown that corresponding to these A, symmetric integrable chain models there are exactly solvable stationary discrete-time (resp. continuous-time) Markov chains with transition matrices (resp. intensity matrices) having spectra which coincide with the ones of the corresponding integrable models.