Mladenov, Ivaïlo (ed.) et al., Proceedings of the 7th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 2--10, 2005. Sofia: Bulgarian Academy of Sciences (ISBN 954-8495-30-9/pbk). 57-78 (2006).

Summary: 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. For the entire collection see [

Zbl 1089.53004].

##### MSC:

81R50 | Quantum groups and related algebraic methods in quantum theory |