Quasi-Linear Algebras and Integrability (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.

arXiv commentThis 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/

References in corpus (6)

• Deformed Dolan-Grady relations in quantum integrable models
• Unified Theory of Annihilation-Creation Operators for Solvable (`Discrete') Quantum Mechanics
• Exact solution in the Heisenberg picture and annihilation-creation operators
• Some algebra related to $P$-and $Q$-polynomial association schemes
• Two relations that generalize the $q$-Serre relations and the Dolan-Grady relations
• Generalized Bochner theorem: characterization of the Askey-Wilson polynomials

Cited by in corpus (10)

• The Universal Askey-Wilson Algebra
• Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states
• The Askey-Wilson algebra and its avatars
• A classification of sharp tridiagonal pairs
• The dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$, $q$-oscillators and Askey-Wilson algebras
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• On the shape of a tridiagonal pair
• Exactly Solvable Quantum Mechanics
• Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity
• Quantised Painlevé monodromy manifolds, Sklyanin and Calabi-Yau algebras