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SIGMA 4 (2008), 015, 22 pages arXiv:0802.0744
https://doi.org/10.3842/SIGMA.2008.015
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics
Quasi-Linear Algebras and Integrability (the Heisenberg Picture)
Luc Vinet a and Alexei Zhedanov b
a) Université de Montréal PO Box 6128, Station
Centre-ville, Montréal QC H3C 3J7, Canada
b) Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine
Received November 16, 2007, in final form January 19, 2008; Published online February 06, 2008
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.
Key words:
Lie algebras; Poisson algebras; nonlinear algebras; Askey-Wilson algebra; Dolan-Grady relations.
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