Quasi-Linear Algebras and Integrability (the Heisenberg Picture)

TL;DR

This paper explores quasi-linear algebraic structures in the Heisenberg picture, revealing explicit dynamical expressions for operators in integrable systems with nonlinear relations like Askey-Wilson and q-Dolan-Grady.

Contribution

It introduces the quasi-linear property in Poisson and operator algebras, offering a new explicit Heisenberg evolution perspective for integrable systems with nonlinear commutation relations.

Findings

Many nonlinear algebras satisfy the quasi-linear property.

Explicit expressions for dynamical variables as functions of time.

Provides a new interpretation of integrability in the Heisenberg picture.

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.