Polynomial algebras from $su(3)$ and the generic model on the two sphere

Abstract

Construction of superintegrable systems based on Lie algebras have been introduced over the years. However, these approaches depend on explicit realisations, for instance as a differential operators, of the underlying Lie algebra. This is also the case for the construction of their related symmetry algebra which take usually the form of a finitely generated quadratic algebra. These algebras often display structure constants which depend on the central elements and in particular on the Hamiltonian. In this paper, we develop a new approach reexamining the case of the generic superintegrable systems on the 2-sphere for which a symmetry algebra is known to be the Racah algebra $R(3)$. Such a model is related to the 59 $2D$ superintegrable systems on conformally flat spaces and their 12 equivalence classes. We demonstrate that using further polynomials of degree 2,3 and 4 in the enveloping algebra of $su(3)$ one can generate an algebra based only on abstract commutation relations of $su(3)$ Lie algebra without explicit constraints on the representations or realisations. This construction relies on the maximal Abelian subalgebra, also called MASA, which are the Cartan generators and their commutant. We obtain a new 6-dimensional cubic algebra where the structure constant are integer numbers which reduce from a quartic algebra for which the structure constant depend on the Cartan generator and the Casimir invariant. We also present other form of the symmetry algebra using the quadratic and cubic Casimir invariants of $su(3)$. It reduces as the known quadratic Racah algebra $R(3)$ only when using an explicit realization. This algebraic structure describe the symmetry of the generic superintegrable systems on the 2 sphere. We also present a contraction to another 6-dimensional cubic algebra which would corresponding to the symmetry algebra of a Smorodinsky-Winternitz model.

11 pages

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Cited by in corpus (10)

• Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras
• Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrödinger algebras
• Demkov-Fradkin tensor for curved harmonic oscillators
• Polynomial algebras from commutants: Classical and Quantum aspects of ${\mathcal{A}}_3$
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
• On the construction of polynomial Poisson algebras: a novel grading approach
• Algebraic structures and Hamiltonians from the equivalence classes of 2D conformal algebras
• Generalized quadratic commutator algebras of PBW-type
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions
• Non-Hermitian superintegrable systems