Quasi-linear algebras and integrability (the Heisenberg picture)

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Abstract

We study Poisson and operator algebras with the "quasi-linear property" from the Heisenberg picture point of view. This means that there exists a set of one-parameter groups yielding an explicit expression of dynamical variables (operators) as functions of "time" t. We show that many algebras with nonlinear commutation relations such as the Askey-Wilson, q-Dolan-Grady and others satisfy this property. This provides one more (explicit Heisenberg evolution) interpretation of the corresponding integrable systems.

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Vinet, L., & Zhedanov, A. (2008). Quasi-linear algebras and integrability (the Heisenberg picture). Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 4. https://doi.org/10.3842/SIGMA.2008.015

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