Quasi-Linear Algebras and Integrability (the Heisenberg Picture)

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.