This article is cited in 11 papers

Quasi-Linear Algebras and Integrability (the Heisenberg Picture)

Luc Vinet^a, Alexei Zhedanov<sup>b

a</sup> Université de Montréal PO Box 6128, Station Centre-ville, Montréal QC H3C 3J7, Canada
^b Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine

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.

Keywords: Lie algebras; Poisson algebras; nonlinear algebras; Askey–Wilson algebra; Dolan–Grady relations.

MSC: 17B63; 17B37; 47L90

Received: November 16, 2007; in final form January 19, 2008; Published online February 6, 2008

Language: English

DOI: 10.3842/SIGMA.2008.015

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ArXiv: 0802.0744