Суперсиметрии, деформирани симетрии и взаимодействие Много-системите в тялото

The dissertation thesis is a study of the symmetries of physical systems and their application for constructing of exactly solvable models. The research follows two main lines: realization and representations of space-time noncommutative algebraic structures and application of the obtained realizations for deriving exact analytic results of models of statistical ensembles with stochastic dynamics. The stochastic processes are the ground of the nonequilibrium systems, which are everywhere in nature and their behaviour is not yet well understood. Deformed symmetries are the proper method for their study, because the deformation parameter has a direct physical meaning. For multiparticle diffusion processes it is the left/right diffusion rate ratio. In the dissertation thesis probability processes of open diffusion lattice systems for modeling processes of biologic transport and traffic flow, interface growth etc., are considered and studied. The recently developed new algebras of Askey-Wilson and tridiagonal algebras are applied and they are shown to define the boundary symmetry of the open nonequilibrium processes wirh simple exclusion. The explicit form of the operatorvalued reflection matrix is constructed in terms of the Askey-Wilson algebra generators and is proved to be a solution of the boundary Yang-Baxter equation, the latter being the basic ingredient of the quantum inverse scattering method to integrable systems and puts the algebraic Bethe Ansatz into perspective. In the dissertation thesis the tridiagonal approach is derived, based on the boundary symmetry of the multiparticle processes with simple exclusion, as a generalization of the known matrixproduct- state algebraic approach for exact solution to the stationary problem. Within the tridiagonal approach the diagonalization of the transition matrix is found in a complete basis of eigenstates, including the unique ground state which is the stationary state of the asymmetric process with exclusion. The tridiagonal approach allows for unified exact description of all three versions of the open process with simple exclusion (namely, the symmetric one, the partially asymmetric and the totally asymmetric ones), which is the fundamental process of nonequilibrium physics, as on its turn is the Ising model lof the equilibrium statistical physics.