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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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