Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 068, 12 pages      arXiv:0705.2671      https://doi.org/10.3842/SIGMA.2007.068
Contribution to the Proceedings of the O'Raifeartaigh Symposium

Hidden Symmetries of Stochastic Models

Boyka Aneva
Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria

Received November 23, 2006, in final form May 04, 2007; Published online May 18, 2007

Abstract
In the matrix product 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. The quadratic algebra defines a noncommutative space with a SUq(n) quantum group action as its symmetry. Boundary processes amount to the appearance of parameter dependent linear terms in the algebraic relations and lead to a reduction of the SUq(n) symmetry. We argue that the boundary operators of the asymmetric simple exclusion process generate a tridiagonal algebra whose irriducible representations are expressed in terms of the Askey-Wilson polynomials. The Askey-Wilson algebra arises as a symmetry of the boundary problem and allows to solve the model exactly.

Key words: stohastic models; tridiagonal algebra; Askey-Wilson polynomials.

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