Quasi-Linear Algebras and Integrability (the Heisenberg Picture)

2008/02/06 by Luc Vinet, Alexei Zhedanov · 3 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons #math-ph #math.MP #math.QA

paper · pdf · doi:10.3842/sigma.2008.015

published as SIGMA 4 (2008), 015, 22 pages · This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

arxiv created 2008/02/06 · openalex publication_date 2008/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• “Hidden symmetry” of Askey-Wilson polynomials
• Two relations that generalize the q-Serre relations and the Dolan-Grady relations
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other
• Generalized Bochner theorem: characterization of the Askey-Wilson polynomials

Cited by

• On the shape of a tridiagonal pair
• Exactly Solvable 'Discrete' Quantum Mechanics; Shape Invariance, Heisenberg Solutions, Annihilation-Creation Operators and Coherent States
• Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity

Related

• Nonlinear Reformulation of Heisenberg's Dynamics
• On the Liouville-Arnold integrable flows related with quantum algebras and their Poissonian representations
• Adler-Oevel-Ragnisco type operators and Poisson vertex algebras
• Classical dynamics, arrow of time, and genesis of the Heisenberg commutation relations
• Solving the Poisson Yang-Baxter equation via deformation-to-quasiclassical-limits