Generalized quadratic commutator algebras of PBW-type

Abstract

In recent years, various nonlinear algebraic structures have been obtained in the context of quantum systems as symmetry algebras, Painlevé transcendent models and missing label problems. In this paper we treat all of these algebras as instances of the class of quadratic (and higher degree) commutator bracket algebras of PBW type. We provide a general approach for simplifying the constraints arising from the diamond lemma, and apply this in particular to give a comprehensive analysis of the quadratic case. We present new examples of quadratic algebras, which admit a cubic Casimir invariant. The connection with other approaches such as Gröbner bases is developed, and we suggest how our explicit and computational techniques can be relevant in other contexts.

References in corpus (30)

• Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems
• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics II:Painleve transcendent potentials
• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics I:Rational function potentials
• Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials
• Bispectrality of multivariable Racah-Wilson polynomials
• Diffusion algebras
• A higher rank Racah algebra and the ${\mathbb{Z}}_2^n$ Laplace-Dunkl operator
• Algebraic Heun operator and band-time limiting
• Quadratic Algebra Approach to an Exactly Solvable Position-Dependent Mass Schrödinger Equation in Two Dimensions
• Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
• New Insights into Superintegrability from Unitary Matrix Models
• Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D
• Spectrum generating algebras for position-dependent mass oscillator Schrodinger equations
• The generalized Racah algebra as a commutant
• On realizations of polynomial algebras with three generators via deformed oscillator algebras
• Embedding of the Racah Algebra R($\mathbf{n}$) and Superintegrability
• Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials
• The Heun-Askey-Wilson algebra and the Heun operator of Askey-Wilson type
• The Racah algebra: An overview and recent results
• Integrable and superintegrable systems associated with multi-sums of products
• Racah algebras, the centralizer $Z_n({\mathfrak{sl}}_2)$ and its Hilbert-Poincaré series
• Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures
• Heun algebras of Lie type
• Polynomial algebras from $su(3)$ and the generic model on the two sphere
• A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$
• The general Racah algebra as the symmetry algebra of generic systems on pseudo--spheres
• A class of quadratic deformations of Lie superalgebras
• Degenerate Sklyanin algebras, Askey-Wilson polynomials and Heun operators
• A fourth-order superintegrable system with a rational potential related to Painleve VI
• Hidden supersymmetry and quadratic deformations of the space-time conformal superalgebra

Cited by in corpus (1)

• Infinite dimensional representations of cubic and quintic algebras and special functions