Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 015, 22 pages      arXiv:0802.0744      http://dx.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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