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Deformations of thermo-algebras and of Fock spaces on a ring, and open quantum systems

Using a hyperbolic ring based formalism, it is shown that special unitary groups correspond to thermo-algebras for closed quantum systems, while for open systems, the full unitary algebra is the suitable underlying structure. Specifically the groups U(2) and U(1, 1) are realized as the dynamical symmetry groups for open quantum systems through continuous tensorial products; these unitary groups are obtained by mean of abelian extensions through U(1)-factors, from the standard thermo-algebras SU(2), and SU(1, 1).