Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras

Abstract

Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial symmetry algebra is proposed. The case of the special linear Lie algebra $\mathfrak{sl}(n)$ is discussed in detail, where an explicit basis for the commutant with respect to the Cartan subalgebra is obtained, and the order of the polynomial algebra is computed. It is further shown that, with an appropriate realization of $\mathfrak{sl}(n)$, this provides an explicit connection with the generic superintegrable model on the $(n-1)$-dimensional sphere ${\mathbb{S}}^{n-1}$ and the related Racah algebra $R(n)$. In particular, we show explicitly how the models on the $2$-sphere and $3$-sphere and the associated symmetry algebras can be obtained from the quadratic and cubic polynomial algebras generated by the commutants defined in the enveloping algebra of $\mathfrak{sl}(3)$ and $\mathfrak{sl}(4)$, respectively. The construction is performed in the classical (or Poisson-Lie) context, where the Berezin bracket replaces the commutator.

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Cited by in corpus (9)

• Polynomial algebras from Lie algebra reduction chains $\mathfrak{g}\supset {\mathfrak{g}}^{\prime }$
• Polynomial algebras from commutants: Classical and Quantum aspects of ${\mathcal{A}}_3$
• Discrete-time systems in quasi-standard form and the ${\mathfrak{h}}_6$ coalgebra symmetry
• Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum
• Algebraic structures and Hamiltonians from the equivalence classes of 2D conformal algebras
• On the construction of polynomial Poisson algebras: a novel grading approach
• Polynomial algebra from the Lie algebra reduction chain $\mathfrak{su}(4)\supset \mathfrak{su}(2)\times \mathfrak{su}(2)$: The supermultiplet model
• Subalgebra chains and nuclear physics: Commutant approach and construction of polynomial algebras
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions