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

SIGMA 4 (2008), 056, 16 pages      arXiv:0807.4391      https://doi.org/10.3842/SIGMA.2008.056
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Tridiagonal Symmetries of Models of Nonequilibrium Physics

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

Received March 03, 2008, in final form July 14, 2008; Published online July 28, 2008

Abstract
We study the boundary symmetries of models of nonequilibrium physics where the steady state behaviour strongly depends on the boundary rates. Within the matrix product state approach to many-body systems the physics is described in terms of matrices defining a noncommutative space with a quantum group symmetry. Boundary processes lead to a reduction of the bulk symmetry. We argue that the boundary operators of an interacting system with simple exclusion generate a tridiagonal algebra whose irreducible representations are expressed in terms of the Askey-Wilson polynomials. We show that the boundary algebras of the symmetric and the totally asymmetric processes are the proper limits of the partially asymmetric ones. In all three type of processes the tridiagonal algebra arises as a symmetry of the boundary problem and allows for the exact solvability of the model.

Key words: driven many-body systems; nonequilibrium; tridiagonal algebra; Askey-Wilson polynomials.

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